Ratios and percentages questions, answered

Scale Calculator

What does a scale of 1:N mean?

A scale of 1:N means one unit on the drawing or model represents N of the same units in real life. At a scale of 1:100, one centimetre on the plan represents 100 centimetres, or one metre, in the real object.

How do I convert a real length to a scale length?

Divide the real length by the scale factor N. For a scale of 1:100, a real length of 5 metres becomes 5 divided by 100, which is 0.05 metres, or 5 centimetres on the drawing.

How do I convert a scale length back to a real length?

Multiply the scale length by the scale factor N. For a scale of 1:100, a drawing length of 0.05 metres represents 0.05 times 100, which is 5 metres in real life.

Does the unit matter in a scale ratio?

The ratio itself is unitless, so 1:100 holds true whatever unit you choose, as long as both lengths use the same unit. This calculator keeps the real length and the scale length in the unit you select so the result is easy to read.

A/B Test Calculator

How does an A/B test calculator work?

It runs a two-proportion z-test. The two conversion rates are pooled into one overall rate, the standard error of the difference is calculated from that pooled rate and both sample sizes, and the difference between the rates is divided by the standard error to give a z-score, which converts to a two-tailed p-value. With 500 conversions from 10,000 visitors against 560 from 10,000, the rates are 5.00% and 5.60%, z is about 1.89 and the p-value is about 0.058.

What p-value counts as statistically significant?

Most teams use 0.05: a p-value below 0.05 means less than a 5% chance of seeing a difference this large if the variants really performed the same, so the result is called significant at the 95% confidence level. Stricter tests use 0.01. A p-value of 0.058 just misses the 95% bar, which usually means keep the test running rather than declare a winner.

How many visitors do I need for an A/B test?

It depends on your baseline conversion rate and the smallest uplift you want to detect. Detecting a 12% relative lift on a 5% baseline at 95% confidence and 80% power needs roughly 22,000 visitors per variant, and smaller lifts need far more. Run a sample size calculation before you start so you know when the test can fairly be judged.

Absence Percentage Calculator

How is absence percentage calculated?

Absence percentage (also called absence rate) is calculated as: (days absent / total working days available) x 100. For example, if an employee was absent for 10 days out of 200 working days in a year, their absence rate is (10 / 200) x 100 = 5%. The total working days figure should exclude public holidays and pre-approved leave such as annual leave, counting only scheduled working days.

What is the Bradford Factor and how is it calculated?

The Bradford Factor is a formula used by HR teams to identify frequent short-term unplanned absences, which tend to be more disruptive than a single long absence. It is calculated as: S x S x D, where S is the number of separate absence episodes and D is the total days absent. For example, if someone had 3 separate absences totalling 10 days, their Bradford Factor is 3 x 3 x 10 = 90. A score over 200 is often used as a trigger point for an informal discussion with a manager.

What is a good absence rate?

The average absence rate in New Zealand is approximately 3 to 4 percent per year, equating to around 6 to 8 days absent per employee. Rates below 2.5% are generally considered low and healthy. Rates above 5% may signal wellbeing, workload, or management issues worth investigating. The appropriate benchmark varies by industry: physical roles in healthcare, construction, and manufacturing typically see higher absence rates than office-based roles.

Absolute Error Calculator

What is absolute error?

The absolute value of the difference between a measured value and the true value, in the same units as the measurement.

How is it different from relative error?

Absolute error is the raw gap. Relative error divides that gap by the true value, usually as a percentage, to show how significant it is.

Can absolute error be negative?

No. It uses the absolute value, so it is always zero or positive.

Absolute Uncertainty Calculator

How do you calculate absolute uncertainty?

For a set of repeated readings, take the best estimate as the mean, then take the absolute uncertainty as half the range: the largest reading minus the smallest, divided by two. For readings of 24.1, 24.3, 23.9, 24.0 and 24.2 cm, the mean is 24.1 cm and the range is 0.4 cm, so the absolute uncertainty is 0.2 cm and the result is written 24.1 plus or minus 0.2 cm.

What is the difference between absolute and relative uncertainty?

Absolute uncertainty is quoted in the same units as the measurement, such as plus or minus 0.2 cm. Relative uncertainty expresses that as a fraction or percentage of the best estimate, so 0.2 cm on a 24.1 cm reading is 0.83%. Absolute uncertainty tells you the size of the doubt, while relative uncertainty tells you how significant that doubt is compared with the quantity measured.

How do you write a measurement with its uncertainty?

Write it as best estimate, a plus-or-minus sign, then the absolute uncertainty, with both in the same unit: 24.1 plus or minus 0.2 cm. Convention rounds the uncertainty to one significant figure and rounds the best estimate to the same decimal place, so a mean of 24.13 with an uncertainty of 0.2 is written 24.1 plus or minus 0.2 cm.

Absolute Value Calculator

What is absolute value?

The absolute value of a number is its distance from zero on the number line, always expressed as a non-negative number. Written with vertical bars as |x|, it strips the sign: |7| equals 7 and |-7| also equals 7. The absolute value of zero is zero.

How do you calculate the absolute value of an expression like |3 minus 10|?

You evaluate the expression inside the bars first: 3 minus 10 equals -7. Then you take the absolute value of -7, which is 7. The absolute value bars act like brackets, so you always resolve what is inside them before applying the modulus.

What is the difference between absolute value and absolute error?

Absolute value is the mathematical operation |x| that returns the non-negative magnitude of any number. Absolute error is a specific application of absolute value used in measurement: it is the absolute value of the difference between a measured value and the true value, |measured minus true|. The absolute value calculator on this page handles both, since the difference mode computes |a minus b|.

Adjugate Calculator

What is the adjugate?

The transpose of the cofactor matrix. For a 2 by 2 it swaps the diagonal entries and negates the off-diagonal ones.

How does it give the inverse?

The inverse is the adjugate divided by the determinant, valid when the determinant is non-zero.

Is adjoint the same as adjugate?

In this classical sense yes, though adjoint also has a different meaning for conjugate transposes in other contexts.

Albumin Globulin Ratio Calculator

What is a normal albumin globulin ratio?

The normal A/G ratio is generally between 1.0 and 2.5, with most laboratories citing a typical reference range of 1.1 to 2.1. A ratio above 1.0 means there is more albumin than globulin in the blood, which is normal. A ratio below 1.0 (reversed A/G ratio) can indicate elevated globulin production, reduced albumin synthesis, or both. A very high ratio above 2.5 may suggest low globulin levels. Always interpret your result alongside your laboratory's own reference range and in the context of your symptoms and other test results.

How is the A/G ratio calculated?

The A/G ratio is calculated by dividing serum albumin by serum globulin. Globulin is not usually measured directly; it is derived by subtracting albumin from total protein (Globulin = Total Protein - Albumin). The formula is therefore: A/G Ratio = Albumin / (Total Protein - Albumin). For example, if albumin is 42 g/L and total protein is 72 g/L, then globulin is 30 g/L and the A/G ratio is 42 / 30 = 1.40.

What does a low A/G ratio mean?

A low A/G ratio (below 1.0) usually means that globulin is elevated relative to albumin, or that albumin is reduced. Common causes of a low A/G ratio include liver disease (which reduces albumin production), kidney disease (which causes albumin to be lost in urine), inflammatory conditions, autoimmune diseases (such as lupus or rheumatoid arthritis), and certain blood cancers such as multiple myeloma. A low ratio is not a diagnosis on its own and needs further investigation. Discuss any abnormal result with your doctor.

Alligation Calculator

What is the alligation method?

Alligation is a shortcut for finding the ratio to mix two ingredients of different strengths so the blend reaches a target strength between them. The parts of the strong ingredient equal the target minus the weak strength, and the parts of the weak ingredient equal the strong strength minus the target.

How do I turn the ratio into amounts?

Add the parts to get the total parts, then multiply your total amount by each ingredient's share of the parts. For a ratio of 7 parts strong to 8 parts weak and a total of 100, you use 100 x 7 / 15, which is 46.67 of the strong, and 100 x 8 / 15, which is 53.33 of the weak.

Does the target have to be between the two strengths?

Yes. Mixing two ingredients can only produce a strength between the weaker and the stronger one, so the target must sit between them. If your target is outside that range, you cannot reach it by blending those two ingredients alone.

Amplitude Period and Phase Shift Calculator

How do I find the period?

Period = 2 pi divided by the absolute value of B, the coefficient of x inside the sine.

What is the phase shift?

Phase shift = minus C over B. A positive value shifts the curve right, a negative value shifts it left.

What does D do?

D is the vertical shift, the height of the midline the wave oscillates around.

Annulus Calculator

What is an annulus?

An annulus is the region between two concentric circles (circles that share the same centre). It looks like a flat ring or washer. The annulus is defined by its outer radius R and inner radius r, where R must be greater than r. It appears in engineering for cross-sections of hollow pipes and tubes, in optics for lens apertures, and in statistics for confidence bands around a circle.

What is the area formula for an annulus?

The area of an annulus equals the area of the outer circle minus the area of the inner circle: pi times R squared minus pi times r squared, which simplifies to pi times (R squared minus r squared), or equivalently pi times (R + r) times (R - r). This last form is useful when you know the sum and difference of the radii rather than the individual values.

What is the width of an annulus?

The width (or thickness) of an annulus is simply the difference between the outer and inner radii: width = R minus r. It tells you how wide the ring is at any point, since the gap between the two circles is constant all the way around.

Arccosine Calculator

What does arccosine give me?

The angle whose cosine is the number you enter. It is the inverse of cosine.

What is the valid input range?

The input must be between -1 and 1, because cosine never goes outside that range.

In what range is the angle returned?

Arccosine returns an angle between 0 and 180 degrees.

Arcsine Calculator

What does arcsine give me?

The angle whose sine is the number you enter. It is the inverse of sine.

What is the valid input range?

The input must be between -1 and 1, because sine never goes outside that range. Values outside it have no real angle.

In what range is the angle returned?

Arcsine returns an angle between -90 and 90 degrees.

Arctangent Calculator

What does arctangent give me?

The angle whose tangent is the number you enter. It is the inverse of tangent.

What is the valid input range?

Any real number is valid, because tangent takes every value.

In what range is the angle returned?

Arctangent returns an angle between -90 and 90 degrees.

Ascending Order Calculator

What does ascending order mean?

Ascending order means arranging numbers from smallest to largest, so each value is greater than or equal to the one before it. For example, 3, 4, 8, 15 is in ascending order. The opposite is descending order, which runs from largest to smallest.

Does it handle negative numbers and decimals?

Yes. You can enter negative numbers, decimals and whole numbers together, separated by commas, spaces or new lines. The calculator sorts by numeric value, so minus 5 comes before minus 2, which comes before 0 and then the positive numbers.

How are duplicate values handled?

Duplicates are kept and placed next to each other. If you enter 4, 2, 4, 1 the ascending order is 1, 2, 4, 4, so the count still reflects every value you typed rather than only the unique ones.

Associative Property Calculator

What is the associative property?

The associative property says that when you add or multiply three or more numbers, the way you group them with brackets does not change the result. For addition, (a+b)+c equals a+(b+c). For multiplication, (a x b) x c equals a x (b x c).

Does the associative property work for subtraction and division?

No. Subtraction and division are not associative. For example (10 - 4) - 2 is 4, but 10 - (4 - 2) is 8, so regrouping changes the answer. The property holds only for addition and multiplication.

How is the associative property different from the commutative property?

The associative property is about grouping (where the brackets go), while the commutative property is about order (which number comes first). Addition and multiplication satisfy both, so you can reorder and regroup freely without changing the total.

AST/ALT Ratio Calculator

What is the AST/ALT ratio and what does it show?

The AST/ALT ratio (also called the De Ritis ratio) is the value of AST (aspartate aminotransferase) divided by ALT (alanine aminotransferase). Both are liver enzymes measured in U/L on a standard blood test. The ratio is used as one indicator of the likely cause of liver injury: a ratio below 1 often suggests viral hepatitis or non-alcoholic fatty liver disease (NAFLD); a ratio between 1 and 2 is non-specific and may indicate non-alcoholic liver disease; a ratio above 2 is classically associated with alcoholic liver disease. The ratio is a screening pattern and must always be interpreted alongside other clinical findings by a doctor.

What are normal AST and ALT levels?

Normal reference ranges vary slightly between laboratories, but typical adult ranges are: AST 10 to 40 U/L and ALT 7 to 56 U/L. Men tend to have slightly higher upper limits than women. In healthy individuals with both values within the normal range, the ratio is typically close to 1. Elevated enzymes with a ratio pattern outside the normal range prompt further investigation, but many factors including muscle injury (AST is also found in muscle) can affect results.

Can a high AST/ALT ratio always mean alcoholic liver disease?

No. While an AST/ALT ratio above 2 is a classic finding in alcoholic hepatitis, the ratio is not diagnostic on its own. AST is also released from muscle, heart, and red blood cells, so conditions like myocardial infarction, rhabdomyolysis (severe muscle breakdown), or haemolysis can raise the ratio without any liver disease present. Cirrhosis from any cause can also produce an elevated ratio as the liver architecture is disrupted. A ratio above 2 warrants further evaluation by a healthcare professional, including additional tests such as gamma-GT, bilirubin, and imaging.

Average Percentage Calculator

How do you find the average of multiple percentages?

To find the simple average of multiple percentages, add all the percentage values together and divide by the count. For example, the average of 40%, 60%, and 80% is (40 + 60 + 80) / 3 = 60%. This works when every percentage represents an equal-sized group or observation. If the groups are different sizes, use a weighted average instead: multiply each percentage by its weight (such as the size of the group), sum those products, then divide by the total weight.

When should I use a weighted average percentage instead of a simple average?

Use a weighted average when the groups or samples behind each percentage are different sizes. For example, if Class A has 10 students with 70% passing and Class B has 30 students with 90% passing, the simple average (80%) is misleading because Class B is three times larger. The weighted average is (10 x 70 + 30 x 90) / (10 + 30) = (700 + 2,700) / 40 = 85%. Use simple averaging only when all groups are the same size or you are averaging independent rates with equal importance.

Can you average percentages that add up to more than 100%?

Yes. Percentages used in averaging are treated as ordinary numbers. They do not have to sum to 100% and can be above 100% (for example, growth rates or markup percentages). The arithmetic mean formula applies regardless: sum the values and divide by the count. For example, if three products have markups of 50%, 120%, and 200%, the average markup is (50 + 120 + 200) / 3 = 123.3%. The restriction that percentages must sum to 100% applies only when percentages represent shares of a whole (like a pie chart), not when they are independent measurements.

Rate of Change Calculator

What is the average rate of change?

The average rate of change of a function f over the interval [x1, x2] is (f(x2) minus f(x1)) divided by (x2 minus x1). It measures how much the function output changes per unit change in x across the whole interval, and equals the slope of the secant line connecting the two points (x1, f(x1)) and (x2, f(x2)) on the graph.

What is the difference between average rate of change and instantaneous rate of change?

The average rate of change looks at the overall change across an interval, like average speed during a journey. The instantaneous rate of change is the derivative at a single point, like the speedometer reading at one moment. As the interval shrinks to zero the average rate approaches the instantaneous rate, which is the foundation of differential calculus.

What is the secant line?

The secant line is the straight line that passes through the two points (x1, f(x1)) and (x2, f(x2)) on the curve. Its slope is the average rate of change. Using point-slope form, the equation is y minus f(x1) equals the slope times (x minus x1), which can be rearranged into y = mx + c form.

Baker's Percentage Calculator

What is baker's percentage?

Baker's percentage (also called baker's math or baker's ratio) expresses each ingredient's weight as a percentage of the total flour weight in a recipe. Flour is always 100%. If a recipe uses 500 g of flour and 350 g of water, the water is 70% (350 divided by 500, multiplied by 100). This system makes it easy to scale recipes up or down, compare different formulas, and understand the hydration level of a dough at a glance.

How do I use baker's percentage to scale a recipe?

To scale a recipe using baker's percentage, decide on your target flour weight, then multiply each ingredient's percentage by the new flour weight and divide by 100. For example, if your recipe has water at 70% and you want to use 1,000 g of flour, you need 700 g of water (1,000 x 70 / 100). The percentages stay the same regardless of batch size, so you only need to change the flour weight to rescale everything.

What is a good hydration percentage for bread?

Hydration refers to the water percentage relative to flour. A beginner-friendly sandwich loaf typically sits around 65 to 70% hydration. Sourdough often ranges from 70 to 80%, producing an open crumb with a chewier texture. High-hydration loaves (85% and above) are sticky and difficult to handle but can produce very open, glossy crumbs. Pizza dough is often 60 to 65% for a firm, workable dough. Enriched doughs (with butter, eggs, or milk) use less water, often 55 to 65%.

Bearing Calculator

How is a bearing measured?

In degrees clockwise from north, from 0 to 360. North is 0, east 90, south 180, west 270.

Why east over north, not north over east?

Bearings are measured from north, clockwise, so the formula is atan2(east, north). That differs from a standard maths angle, which is measured from the east axis anticlockwise.

What if my bearing comes out negative?

Add 360 degrees. The calculator already shifts the result into the 0 to 360 range for you.

Benford's Law Calculator

What is Benford's law?

Benford's law says that in many real-world sets of numbers the leading digit is not evenly spread. Smaller digits lead far more often: a 1 appears as the first digit about 30.1 percent of the time, while a 9 leads only about 4.6 percent of the time. The probability that the first digit is d is log base 10 of (1 + 1 / d).

What is the formula for Benford's law?

The probability that the leading digit equals d, for d from 1 to 9, is P(d) = log10(1 + 1 / d). So P(1) = log10(2) which is about 0.301, P(2) = log10(1.5) which is about 0.176, and so on down to P(9) = log10(10 / 9) which is about 0.046. The nine probabilities add up to 1.

Where does Benford's law apply?

It tends to hold for figures that span several orders of magnitude and are not artificially bounded, such as populations, river lengths, physical constants, financial transactions and accounting figures. It is used in fraud detection and auditing, because fabricated numbers often fail to follow the expected first-digit distribution. It does not apply to numbers with a fixed range, like heights in centimetres.

Bisection Method Calculator

What is the bisection method?

The bisection method finds a root of a function by repeatedly halving an interval that is known to contain one. At each step it keeps the half where the function changes sign, so the root is trapped in an interval that shrinks by half every time.

What do you need for the bisection method to work?

You need an interval whose endpoints give function values of opposite sign, one positive and one negative. That guarantees a root lies between them for a continuous function, which is what the method relies on to converge.

How accurate is the bisection method?

The interval halves each step, so accuracy improves steadily and predictably. About ten steps shrink the interval by a factor of a thousand. It is slower than some methods but very reliable, always converging when the starting interval brackets a root.

Bonferroni Correction Calculator

How does the Bonferroni correction work?

Divide your family-wise significance level by the number of comparisons. With an alpha of 0.05 and 10 tests, the corrected threshold is 0.05 divided by 10, which is 0.005. A result counts as significant only if its p-value is below that stricter threshold.

What is a Bonferroni adjusted p-value?

An adjusted p-value multiplies each raw p-value by the number of comparisons, capped at 1, so you can compare it directly against your original alpha. A raw p-value of 0.008 across 10 tests becomes an adjusted p-value of 0.08, which is not significant at 0.05.

When should I use the Bonferroni correction?

Use it when you run several tests on the same data and want to control the chance of any false positive across the whole family. It is simple and conservative, so with many comparisons it can be strict; methods like Holm or Benjamini-Hochberg trade some of that strictness for more power.

Box Method Calculator

What is the box method in multiplication?

The box method, also called the area model or grid method, breaks each number into its place value parts, multiplies every pair in a grid, then adds all the partial products. For 23 x 47 you split into 20 + 3 and 40 + 7, giving partial products 800, 140, 120 and 21 that add to 1,081.

Why use the box method instead of long multiplication?

The box method makes the distributive property visible, so you can see exactly where each partial product comes from. It is a common check on long multiplication and a helpful bridge for students who lose track of carries in the standard algorithm.

Does the box method work for larger numbers?

Yes. Each number is split into as many place value parts as it has non-zero digits, so a three digit number by a two digit number simply produces a larger grid with more partial products, all added the same way.

Box Plot Calculator

What does a box plot show?

A box plot, or box and whisker plot, displays the five-number summary: the minimum, first quartile, median, third quartile and maximum. The box spans the middle 50% of the data from Q1 to Q3, a line marks the median, and the whiskers reach the smallest and largest values that are not outliers.

How are the quartiles worked out?

This calculator uses the exclusive method: sort the data, find the median, then take the median of the lower half for Q1 and the median of the upper half for Q3. When the count is odd the middle value is left out of both halves.

How are outliers found on a box plot?

The interquartile range is Q3 minus Q1. Any value below Q1 minus 1.5 times the IQR, or above Q3 plus 1.5 times the IQR, is flagged as an outlier. The whiskers then stop at the most extreme values that fall inside those fences.

Calibration Curve Calculator

What is a calibration curve?

A calibration curve is a graphical relationship between a known independent variable (such as concentration) and a measured dependent variable (such as absorbance or signal intensity). You prepare a series of standards with known concentrations, measure their signals, then fit a straight line through the data using least-squares linear regression. Once you have the equation y = mx + b, you can measure the signal of an unknown sample and rearrange the equation to find its concentration: x = (y - b) / m.

What does R² mean for a calibration curve?

R² (the coefficient of determination) measures how well the straight line fits your data. A value of 1.0 means a perfect fit, while 0 means no linear relationship. For analytical calibration curves, R² should typically be 0.999 or higher to be considered acceptable. Values below 0.99 suggest poor linearity, possible outliers in your standards, reagent problems, or that your signal-concentration relationship is not truly linear over the range you are using.

How many standard points do I need for a calibration curve?

Most analytical methods require a minimum of five standard points (including a blank or zero standard) to establish a reliable calibration curve. More points improve the statistical confidence of the line. Standards should bracket the expected concentration range of your unknowns. Including a blank (zero concentration) anchors the line at the origin and helps identify any baseline signal. Regulatory methods (such as EPA or ISO methods) often specify the minimum number of standards required.

Coefficient of Variation Calculator

What is the coefficient of variation?

The coefficient of variation (CV) is the standard deviation divided by the mean, expressed as a percentage. It measures relative variability: how large the spread is compared to the typical value. A CV of 20% means the standard deviation is 20% of the mean, making it easy to compare the variability of data sets measured in different units or at different scales.

When is the coefficient of variation useful?

The CV is particularly useful when comparing the spread of two data sets with different means or different units. For example, comparing the variability of house prices in two cities (which have very different average prices) using just the standard deviation would be misleading. The CV scales the spread by the mean so the comparison is fair.

What is a high coefficient of variation?

There is no universal threshold, but in many fields a CV below 10% is considered low variability, 10-20% moderate, and above 20% high. In financial markets, a high CV indicates greater risk relative to expected return. The CV is not meaningful when the mean is zero or negative, since dividing by zero or a negative number produces undefined or misleading results.

Cofactor Calculator

What is a minor?

The determinant of the smaller matrix left after deleting the row and column of a chosen entry.

What is a cofactor?

The minor multiplied by a sign, (minus 1) to the power (row plus column), giving the checkerboard pattern of signs.

What are cofactors used for?

Expanding determinants, building the adjugate, and computing the inverse of a matrix.

Cofactor Expansion Calculator

What is cofactor expansion?

Cofactor expansion, also called Laplace expansion, is a method for computing the determinant of a square matrix by expanding along a chosen row or column. For each element in that row or column, you multiply the element by its cofactor (the signed minor formed by deleting that element's row and column). The determinant equals the sum of these products. For a 3x3 matrix expanded along row 1: det(A) = a11*C11 + a12*C12 + a13*C13, where each cofactor Cij = (-1)^(i+j) * Mij and Mij is the 2x2 minor determinant.

Does it matter which row or column I expand along?

No, the result is the same regardless of which row or column you choose for the expansion. The determinant is a property of the matrix, not of the expansion path. Choosing a row or column that contains zeros simplifies the calculation, since any term with a zero element contributes nothing to the sum. For example, if a row has two zeros, only one minor needs to be computed.

What is the difference between a minor and a cofactor?

The minor M_ij is the determinant of the submatrix formed by deleting row i and column j from the original matrix. The cofactor C_ij is the signed minor: C_ij = (-1)^(i+j) * M_ij. The sign factor (-1)^(i+j) produces a checkerboard pattern of plus and minus signs across the matrix. For a 3x3 matrix the sign pattern is: + - + / - + - / + - +. The cofactor matrix (also called the matrix of cofactors) is used in computing the adjugate (classical adjoint) and the matrix inverse.

Coffee Ratio Calculator

What is the standard coffee-to-water ratio?

The Specialty Coffee Association (SCA) recommends 60 g of coffee per litre of water, which is a 1:16.7 ratio by weight. In practical terms, this means about 1 g of coffee for every 16 to 17 g (or ml) of water. This ratio is a starting point; personal taste, grind size, brew method, and coffee origin all affect the ideal ratio for a given cup.

How do I measure coffee and water accurately?

The most accurate method is to weigh both coffee and water in grams using a kitchen scale. Water weighs 1 g per ml at room temperature, so 200 ml of water weighs 200 g. Measuring coffee by weight (rather than scoops or tablespoons) accounts for variation in grind size and bean density. A medium-coarse grind for filter coffee weighs roughly 7 to 8 g per tablespoon, but this varies significantly.

Why does espresso use a different ratio to filter coffee?

Espresso extracts coffee under high pressure (around 9 bar) in 25 to 30 seconds, which produces a concentrated beverage. A standard espresso uses a ratio of about 1:2 (e.g., 18 g of coffee yields 36 g of espresso liquid). Filter methods such as pour over or French press use a much higher water-to-coffee ratio (1:15 to 1:17) because extraction relies on gravity and time rather than pressure, and the result is consumed as a full cup rather than a concentrated shot.

Cofunction Calculator

What are cofunction identities in trigonometry?

Cofunction identities state that each trigonometric function of an angle equals its paired cofunction evaluated at the complementary angle (90 degrees minus the angle). The three pairs are: sine and cosine (sin theta = cos(90 minus theta)), tangent and cotangent (tan theta = cot(90 minus theta)), and secant and cosecant (sec theta = csc(90 minus theta)).

What is the cofunction of sin(30)?

The cofunction of sin(30 degrees) is cos(60 degrees). Since 30 and 60 are complementary angles (they add to 90), sin(30) = cos(60) = 0.5. This is the cofunction identity for sine and cosine: sin(theta) = cos(90 minus theta).

Why are cofunctions called co-functions?

The prefix co- in cosine, cotangent and cosecant comes from the word complement. Cosine literally means complement of sine: it gives the sine of the complementary angle. The same logic applies to cotangent (complement of tangent) and cosecant (complement of secant). This naming reflects the fundamental cofunction identities.

Cohen's d Calculator

What is Cohen's d?

Cohen's d is a standardised effect size: the difference between two group means divided by the pooled standard deviation. Because it is expressed in standard-deviation units, it does not depend on the measurement scale and can be compared across studies and variables.

What is a small, medium or large effect?

Cohen's conventional benchmarks are around 0.2 for a small effect, 0.5 for medium and 0.8 or above for large. These are rough guides; what counts as a meaningful effect depends on the field and the practical context of the comparison.

Why report effect size as well as significance?

A significance test only says whether a difference is likely real, not how big it is. With large samples, even tiny differences can be significant. Effect size like Cohen's d shows the magnitude, which is what determines whether the difference actually matters.

Cold Brew Ratio Calculator

What is the best cold brew ratio?

The most common cold brew ratios are 1:5 (coffee to water by weight) for a concentrate, and 1:8 for ready-to-drink cold brew. A 1:5 concentrate is diluted 1:1 with water or milk when served, giving an effective ratio of 1:10. Specialty coffee guides often suggest 1:4 to 1:5 for concentrate and 1:7 to 1:8 for ready-to-drink. Ratios are measured by weight (grams) because volume measurements vary depending on how finely the coffee is ground.

How long should cold brew steep?

Cold brew steeped in the refrigerator (around 4 degrees Celsius) typically takes 18 to 24 hours. Room temperature steeping (around 20 degrees Celsius) is faster, usually 12 to 14 hours, but carries a slightly higher risk of bacterial growth if left too long. Most home brewers find 20 hours in the fridge produces a smooth, low-acidity result. Once brewed, cold brew concentrate keeps in the refrigerator for up to two weeks.

Should I measure cold brew coffee by weight or volume?

Weight (grams) is the most accurate method because the volume of ground coffee varies significantly depending on grind size and how tightly it is packed. A coarse cold brew grind, which is recommended, is much less dense than an espresso grind. Measuring by weight gives consistent results every batch. Water volume (millilitres) is fine to measure by volume since 1 ml of water weighs approximately 1 gram.

Collatz Conjecture Calculator

What is the Collatz conjecture?

The claim that the 3n+1 process reaches 1 from every positive starting number. It is unproven but verified for very large ranges.

Why are these called hailstone numbers?

Because the values rise and fall repeatedly, like hailstones in a cloud, before finally settling to 1.

Does every number really reach 1?

It is conjectured to, and checked far beyond a billion, but there is no general proof.

Complex Argument Calculator

What is the argument of a complex number?

The angle it makes with the positive real axis on the complex plane, found with atan2 of the imaginary over the real part.

Why use atan2?

It returns the correct quadrant and works when the real part is zero, unlike plain arctan.

What range is it in?

The principal value runs from minus 180 to 180 degrees, or minus pi to pi radians.

Compression Ratio to PSI Calculator

How do you convert compression ratio to PSI?

The standard method uses the polytropic compression formula: Cylinder PSI = Atmospheric Pressure x Compression Ratio^n, where atmospheric pressure is approximately 14.7 PSI at sea level and n is the polytropic exponent (typically 1.3 for real engines). For example, a 10:1 compression ratio gives approximately 14.7 x 10^1.3 = 293 PSI. A purely isentropic calculation would use n = 1.4, while a simple linear estimate uses n = 1.0 (giving 147 PSI for a 10:1 ratio), but 1.3 is the most widely used approximation for cranking compression tests.

What is a normal compression ratio and PSI for a petrol engine?

Most naturally aspirated petrol engines have compression ratios between 8:1 and 12:1, producing estimated cranking pressures of roughly 220 to 370 PSI (1,515 to 2,551 kPa) using the standard polytropic formula (n = 1.3). Performance and modern direct-injection engines may run 11:1 to 13:1, giving around 330 to 415 PSI. Diesel engines use much higher compression ratios of 14:1 to 23:1 to achieve auto-ignition, producing 450 to 870 PSI or more. Turbocharged engines may have lower static compression ratios (8:1 to 9.5:1) to account for boost pressure added before the cylinder.

Why does my cranking compression test read lower than the calculated PSI?

Several factors cause real cranking pressure to read lower than the theoretical calculation: engine speed during cranking (lower rpm means less air ingested), valve timing and overlap (some air escapes back past the intake valve before it closes), leakage past worn rings or valve seats, restricted air filter, altitude reducing atmospheric pressure, and engine temperature affecting gas behaviour. A healthy engine typically reads within 10 to 15 percent of the theoretical value. If your measured PSI is more than 20 percent below the theoretical figure, it may indicate ring wear, valve sealing issues, or a head gasket problem.

Contrast Ratio Calculator

What contrast ratio do I need to pass WCAG AA?

To pass WCAG 2.1 Level AA, normal body text (below 18pt or 14pt bold) requires a minimum contrast ratio of 4.5:1 against its background. Large text (18pt or larger, or 14pt bold or larger) requires a minimum of 3:1. User interface components and graphical objects also require a minimum of 3:1 against adjacent colours. Level AAA is stricter: normal text needs 7:1 and large text needs 4.5:1. These requirements apply to all web content covered by accessibility guidelines and most public-sector websites in New Zealand under the Web Accessibility Standard 1.1.

How is contrast ratio calculated?

Contrast ratio is calculated using the WCAG relative luminance formula. First, each RGB channel value (0 to 255) is divided by 255 to get a value between 0 and 1. Values at or below 0.04045 are divided by 12.92; values above 0.04045 use the formula ((channel + 0.055) / 1.055) raised to the power of 2.4. The relative luminance L is then: L = 0.2126 * R_linear + 0.7152 * G_linear + 0.0722 * B_linear. The contrast ratio is (L1 + 0.05) / (L2 + 0.05), where L1 is the higher of the two luminance values. The 0.05 offset represents the luminance of the display surface. The result is expressed as a ratio such as 4.5:1 or 21:1.

What is the highest possible contrast ratio?

The maximum possible contrast ratio is 21:1, which occurs between pure white (#FFFFFF, relative luminance 1.0) and pure black (#000000, relative luminance 0.0). The formula gives (1.0 + 0.05) / (0.0 + 0.05) = 1.05 / 0.05 = 21. The minimum possible contrast ratio is 1:1, which occurs when both colours are identical. Most accessible colour combinations for body text fall between 4.5:1 and 15:1.

Cosecant Calculator

How do I find the cosecant of an angle?

Enter the angle and pick degrees or radians. The calculator converts to radians and evaluates the cosecant.

Degrees or radians?

Either. Choose the unit in the dropdown. The angle is shown in both units beside the result so you can check.

Why is the result the same every 360 degrees?

The trig functions are periodic. Sine and cosine repeat every 360 degrees (2 pi radians), so equivalent angles give the same value.

Cotangent Calculator

How do I find the cotangent of an angle?

Enter the angle and pick degrees or radians. The calculator converts to radians and evaluates the cotangent.

Degrees or radians?

Either. Choose the unit in the dropdown. The angle is shown in both units beside the result so you can check.

Why is the result the same every 180 degrees?

Cotangent is periodic with a period of 180 degrees (pi radians), so cot(x) = cot(x + 180 degrees). Although sine and cosine repeat every 360 degrees, the sign changes cancel in cos/sin, giving cotangent a 180-degree period.

Covariance Calculator

What is covariance?

Covariance measures how two variables change together. A positive covariance means they tend to increase together; a negative covariance means one tends to increase when the other decreases. Unlike correlation, covariance is not scaled, so its magnitude depends on the units of the variables.

What is the difference between sample and population covariance?

Sample covariance divides by n minus 1 (Bessel's correction) to give an unbiased estimate of the population covariance when you only have a sample of data. Population covariance divides by n and is used when you have data for the entire population.

Pearson correlation r is simply covariance divided by the product of the standard deviations of x and y. This standardisation removes the effect of the units and scales r to always lie between -1 and +1, which makes it easier to compare relationships across different data sets.

Cramer's Rule Calculator

What is Cramer's rule?

A method that solves a linear system using ratios of determinants, one for each unknown.

When does it work?

When the coefficient determinant D is non-zero, giving exactly one solution. If D is zero, the system has no unique solution.

Does it scale to bigger systems?

Yes, in principle, but it becomes slow for large systems, where elimination methods are preferred.

Crawl Ratio Calculator

What is a good crawl ratio for off-road driving?

A crawl ratio of around 30:1 is considered adequate for most light off-road use, such as fire roads and farm tracks. A ratio of 50:1 to 70:1 suits serious off-road terrain including steep rocky climbs and deep ruts. Dedicated rock crawlers often target 70:1 or higher, which allows the vehicle to creep forward at very low speed with full engine torque, giving the driver precise throttle control without stalling. Many stock 4WDs leave the factory with crawl ratios in the 30:1 to 50:1 range.

How do I increase my crawl ratio?

There are three main ways to increase crawl ratio: fit a transfer case with a lower low-range ratio (for example, swapping a 2.72:1 transfer case for a 4.70:1 aftermarket unit), re-gear the axles to a higher (numerically larger) ring and pinion ratio such as moving from 3.73:1 to 4.88:1 or 5.38:1, or fit a transmission with a lower first gear. Many off-road enthusiasts combine axle re-gearing with a transfer case upgrade to achieve a significant improvement in crawl ratio without engine or transmission changes. Re-gearing the axles also helps restore power and fuel economy when running larger tyres.

Does crawl ratio include the differential?

Yes. The standard crawl ratio formula multiplies the transmission first gear ratio by the transfer case low-range ratio and then by the axle ring and pinion (final drive) ratio. All three stages of gear reduction are included. Some calculations also factor in tyre diameter, because a larger tyre effectively reduces the mechanical advantage at the wheel, similar to raising the final drive ratio. This calculator uses the three-gear-ratio formula, which is the most widely used definition of crawl ratio.

Critical Points Calculator

What is a critical point?

A critical point of a function f(x) is any x value where the derivative f'(x) equals zero or is undefined. At a critical point the function is neither increasing nor decreasing, so it may have a local maximum, a local minimum, or neither (an inflection point). Finding critical points is the first step in optimisation: locating the highest or lowest values a function can reach.

How do you classify a critical point?

Use the second derivative test. If f''(x) is negative at the critical point, the function is concave down there and you have a local maximum. If f''(x) is positive, the function is concave up and you have a local minimum. If f''(x) is zero the test is inconclusive and the point could be an inflection point or a higher-order extremum, so you check the sign of f' on either side instead.

How does the calculator find critical points numerically?

The calculator scans the search interval by evaluating f'(x) at many closely spaced x values using central difference numerical differentiation. Wherever f' changes sign between two adjacent sample points, the calculator uses a bisection search to pinpoint the zero crossing to high precision. This approach finds all critical points in the interval without needing a symbolic derivative.

Cross Product Calculator NZ

What is the cross product used for?

It produces a vector perpendicular to two given vectors, which is essential for finding surface normals in 3D graphics, calculating torque and angular momentum in physics, and the moment of a force. Its magnitude also gives the area of the parallelogram the two vectors span.

Why does the order of the vectors matter?

The cross product is anti-commutative: a cross b equals the negative of b cross a. Swapping the two vectors reverses the direction of the result, flipping every sign, even though the magnitude stays the same. The right-hand rule fixes the direction for a given order.

What does a zero cross product mean?

If the cross product is the zero vector, the two input vectors are parallel or anti-parallel, pointing along the same line. This is because the parallelogram they form has zero area. The cross product is therefore a quick test for whether two vectors are parallel.

Damping Ratio Calculator

What is the formula for damping ratio?

The damping ratio (zeta, ζ) is calculated as ζ = c / (2√(km)), where c is the damping coefficient (Ns/m), k is the spring stiffness (N/m), and m is the mass (kg). The term 2√(km) is known as the critical damping coefficient, c_c. The damping ratio is a dimensionless number: ζ = 0 means no damping, 0 < ζ < 1 means underdamped, ζ = 1 means critically damped, and ζ > 1 means overdamped.

What does an underdamped, critically damped and overdamped system mean?

An underdamped system (ζ < 1) oscillates back and forth with the amplitude gradually decreasing over time, such as a car suspension that bounces a few times after hitting a bump. A critically damped system (ζ = 1) returns to equilibrium as fast as possible without any oscillation. An overdamped system (ζ > 1) returns to equilibrium without oscillating, but more slowly than the critically damped case. Most engineered systems (suspensions, door closers, control loops) are designed close to critical damping or slightly underdamped for a fast, stable response.

How do you find the damping ratio from natural and damped frequency?

If you know the undamped natural angular frequency (ωn) and the damped angular frequency (ωd) measured from a real oscillating system, the damping ratio is ζ = √(1 - (ωd/ωn)²). This relationship comes from ωd = ωn√(1 - ζ²), which only applies to underdamped systems (ζ < 1). Alternatively, the damping ratio can be estimated from the logarithmic decrement of successive oscillation peaks in a measured response.

Degree of Unsaturation Calculator

What is degree of unsaturation?

Degree of unsaturation (also called index of hydrogen deficiency, or IHD) is the total number of rings and pi bonds (double or triple bonds) in a molecule. Each ring or double bond counts as one degree, and a triple bond counts as two degrees. It is calculated directly from a molecular formula without needing to draw the structure, which makes it a fast first check on possible structures in organic chemistry.

What is the formula for degree of unsaturation?

Degree of unsaturation equals C minus (H plus X) divided by 2, plus N divided by 2, plus 1, where C is the number of carbon atoms, H is the number of hydrogen atoms, X is the number of halogen atoms (F, Cl, Br, I), and N is the number of nitrogen atoms. Oxygen and sulfur do not appear in the formula because they are divalent and do not change the hydrogen count needed for a saturated structure.

How do you use degree of unsaturation to find a structure?

Once you have the degree of unsaturation, you know how many rings and pi bonds the structure must contain in total. A value of 0 means a fully saturated acyclic molecule (an alkane or similar), 1 means one ring or one double bond, 2 could mean two double bonds, one ring plus one double bond, or two rings, and 4 is the typical value for a benzene ring (one ring plus three double bonds, though it is often treated as a single aromatic unit). You then use other evidence, such as IR or NMR spectra, to decide which arrangement is correct.

Diffraction Grating Calculator

What is the diffraction grating equation?

The diffraction grating equation is d sin(theta) = m lambda, where d is the distance between adjacent slits (grating spacing), theta is the angle of diffraction measured from the normal, m is the diffraction order (an integer: 0, 1, 2, 3...), and lambda is the wavelength of light. This equation describes the angles at which constructive interference produces bright fringes (maxima) when light passes through a diffraction grating.

How do you find grating spacing from lines per mm?

Grating spacing d is the reciprocal of the number of lines per unit length. If a grating has N lines per millimetre, then d = 1 / N millimetres, which you then convert to metres or nanometres for use in the diffraction equation. For example, a grating with 600 lines/mm has d = 1/600 mm = 1666.7 nm.

What is the maximum diffraction order possible?

The maximum order m occurs when sin(theta) reaches its limit of 1 (theta = 90 degrees). This gives m_max = d / lambda, rounded down to the nearest whole number. Beyond this order, no further diffraction maxima are physically possible because the equation would require sin(theta) greater than 1, which has no real solution.

Direct Variation Calculator

What is direct variation?

A relationship where y = kx, so y is a constant multiple of x and the two are proportional.

How do I find the constant k?

Divide a known y by its matching x. That ratio is the same for every pair in a direct variation.

What does the graph look like?

A straight line through the origin with slope k.

Discriminant Calculator

What does the discriminant tell you?

The discriminant of a quadratic ax squared plus bx plus c is the expression b squared minus 4ac. When it is positive the equation has two distinct real roots; when it is zero it has exactly one repeated real root; when it is negative it has no real roots (the roots are complex numbers). It tells you the number and type of solutions before you solve the equation.

What is the formula for the discriminant?

For the quadratic equation ax squared plus bx plus c equals zero, the discriminant is Delta = b squared minus 4ac. It appears under the square root sign in the quadratic formula x = (-b plus or minus square root of Delta) divided by 2a, which is why its sign determines how many real solutions exist.

Can the discriminant be used for non-quadratic equations?

The classical discriminant formula b squared minus 4ac applies specifically to quadratics. Cubic and higher-degree polynomials have their own discriminant expressions, which are more complex. For cubics you can use the Polynomial Roots Calculator on this site, which evaluates the cubic discriminant internally.

Domain and Range Calculator

What is the domain of a function?

The set of all input values for which the function is defined.

What is the range?

The set of all output values the function actually produces.

Why is the domain sometimes restricted?

To avoid dividing by zero or taking the square root of a negative number, which are undefined for real numbers.

Dot Product Calculator NZ

What is the dot product used for?

The dot product measures how much two vectors point the same way. It is used to find the work done by a force, to project one vector onto another, to test for perpendicularity, and to measure similarity in data and machine learning. It returns a single number, not a vector.

What does a dot product of zero mean?

A dot product of zero means the two vectors are perpendicular, at right angles to each other. This is the quickest test for orthogonality, since the dot product equals the product of the magnitudes times the cosine of the angle, and the cosine of 90 degrees is zero.

How is the dot product different from the cross product?

The dot product returns a single number (a scalar) measuring alignment, while the cross product returns a new vector perpendicular to both inputs, whose length measures the area they span. The dot product works in any dimension; the cross product is specific to three dimensions.

Effect Size Calculator

What is Cohen's d?

Cohen's d is the standardised difference between two group means, calculated as (mean1 minus mean2) divided by the pooled standard deviation. It expresses the difference in units of standard deviations. By convention: d less than 0.2 is negligible, 0.2 to 0.5 is small, 0.5 to 0.8 is medium, and above 0.8 is large. A d of 1.0 means the means are one standard deviation apart.

What is eta squared in statistics?

Eta squared is the proportion of total variance explained by a factor in ANOVA. It is calculated as F times df_between divided by (F times df_between plus df_within). By convention: eta squared less than 0.01 is small, 0.01 to 0.06 is medium, and above 0.14 is large. It is analogous to R squared in regression.

What is the difference between statistical significance and effect size?

Statistical significance (a p-value) tells you whether an observed difference is likely due to chance. Effect size tells you how large or practically meaningful the difference is. A very large sample can make a tiny, practically irrelevant difference statistically significant. Reporting effect size alongside p-values gives a fuller picture of the result.

Effect Size (Cohen's d) Calculator

What is a good Cohen's d effect size?

Jacob Cohen's original conventions classify d = 0.2 as a small effect, d = 0.5 as a medium effect, and d = 0.8 as a large effect. These benchmarks are widely used in psychology, education and medicine. A d of 0.5 means the two group means are half a standard deviation apart, which is noticeable in most practical settings. In medical research, even a small effect (d around 0.2) can be clinically meaningful if the outcome is serious. The appropriate interpretation always depends on the research context.

What is the difference between Cohen's d and Hedges' g?

Cohen's d uses the pooled standard deviation of the two samples. Hedges' g applies a small-sample correction factor (sometimes called J) that reduces slight upward bias when sample sizes are small (typically below 20 per group). For large samples the two measures are nearly identical. This calculator computes Cohen's d. If your groups have fewer than 20 observations each, multiplying d by the correction factor J = 1 - (3 / (4*(n1+n2-2) - 1)) gives Hedges' g.

Can Cohen's d be negative and what does that mean?

Yes. Cohen's d is negative when Group 2 has a higher mean than Group 1 (mean1 minus mean2 is negative). The sign simply indicates the direction of the difference: which group scored higher. The magnitude (absolute value) tells you how large the difference is. In most research reports the convention is to subtract in the direction that makes the effect positive, but this calculator preserves the sign so you can see the direction directly. The interpretation labels (small, medium, large) use the absolute value of d.

Eigenvalue Calculator NZ

What is an eigenvalue?

An eigenvalue is a scaling factor for a special direction, called an eigenvector, that a matrix stretches without rotating. Eigenvalues reveal how a matrix transforms space and underpin vibration analysis, stability, principal component analysis and quantum mechanics.

How do you find eigenvalues of a 2x2 matrix?

Solve the characteristic equation: lambda squared minus the trace times lambda plus the determinant equals zero. This quadratic gives the two eigenvalues. They always sum to the trace and multiply to the determinant, which is a handy way to check your answer.

What do complex eigenvalues mean?

When the discriminant is negative, the eigenvalues are a complex conjugate pair. Geometrically this means the matrix involves a rotation, characteristic of oscillating or spiralling systems, rather than a simple stretch along fixed directions, which is what real eigenvalues represent.

Euler Totient Calculator

What does the totient count?

The whole numbers from 1 to n that are coprime to n, meaning their greatest common factor with n is 1.

What is phi of a prime p?

p minus 1, because every number from 1 to p minus 1 is coprime to a prime.

Why does RSA use the totient?

The totient of the modulus sets the order of the multiplicative group, which determines the encryption and decryption exponents.

Expected Value Calculator

What is expected value?

Expected value E(X) is the long-run average outcome of a random variable if you repeated the experiment many times. It is calculated as the sum of each possible outcome multiplied by its probability. For a fair six-sided die, E(X) = (1+2+3+4+5+6)/6 = 3.50.

What does it mean if probabilities do not sum to 1?

A valid probability distribution must have all probabilities sum to exactly 1. If your values do not sum to 1, the calculator will warn you and the expected value calculation is not valid. Check that each probability is between 0 and 1 and that you have not missed any outcomes.

Variance is E(X squared) minus [E(X)] squared, which measures how spread out the outcomes are around the expected value. Standard deviation is the square root of variance and is expressed in the same units as the outcomes, making it easier to interpret.

Linear Extrapolation Calculator

What is extrapolation?

Estimating a value beyond the range of known data by extending an established trend, here a straight line.

How is it different from interpolation?

Interpolation estimates between known points; extrapolation estimates outside them. Both use the same line here.

Is extrapolation reliable?

Use it with care. It assumes the straight line trend continues, which may not hold far from the data.

Fibonacci Calculator

What is the Fibonacci sequence?

The Fibonacci sequence is a series of numbers where each term is the sum of the two preceding terms. Starting from 1 and 1, the sequence is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, and so on. It was described by the Italian mathematician Fibonacci in the 13th century and appears in nature in the spiral arrangements of seeds, petals, and shells.

What is the formula for the nth Fibonacci number?

The direct (closed-form) formula is known as Binet's formula: F(n) = (phi to the n minus psi to the n) divided by the square root of 5, where phi = (1 plus square root of 5) divided by 2 (approximately 1.618) and psi = (1 minus square root of 5) divided by 2 (approximately minus 0.618). Because psi has absolute value less than 1, its contribution vanishes for large n and F(n) is very close to phi to the n divided by the square root of 5.

What is the connection between Fibonacci numbers and the golden ratio?

The ratio of consecutive Fibonacci numbers, such as 8 divided by 5 or 55 divided by 34, converges to the golden ratio phi as the terms get larger. By the time you reach F(20) divided by F(19) the ratio is accurate to more than six decimal places. This is why the Fibonacci sequence and the golden ratio are so closely linked in both mathematics and design.

Final Exam Mark Needed Calculator NZ 2026

How do I work out what I need on my final exam?

Subtract the grade you have already secured, your current grade times the weight of completed work, from your target, then divide by the exam's weight. This calculator does it for you and tells you if the required mark is achievable or already secured.

What if I need more than 100 percent?

If the required mark is over 100 percent, your target is no longer reachable with the exam alone, so you would need to lower the target or seek extra credit if available. The calculator flags this so you can set a realistic goal.

What if the result is zero or negative?

That means you have already done enough on earlier assessment that even a zero on the final would meet your target. You have secured the grade, though it is still worth sitting the exam well in case weightings or marks differ from your estimate.

Floor and Ceiling Calculator

What is the floor of a number?

The greatest integer that is less than or equal to it. The floor of 3.7 is 3, and the floor of minus 3.7 is minus 4.

What is the ceiling?

The smallest integer greater than or equal to the number. The ceiling of 3.2 is 4.

How do they behave for negatives?

For minus 2.5 the floor is minus 3 (further from zero) and the ceiling is minus 2 (towards zero).

Frustum Calculator

What is a frustum?

A frustum is the solid formed when a cone is cut by a plane parallel to its base, removing the top. The result has a large circular base, a smaller circular top, and a slanted lateral surface connecting them. Everyday examples include plant pots, drinking cups, traffic cones, and bucket shapes. The word comes from the Latin for fragment or piece.

What is the volume formula for a frustum?

The volume of a frustum with large base radius R, small top radius r, and perpendicular height h is pi times h divided by 3 times (R squared plus Rr plus r squared). This formula is derived by subtracting the volume of the small cone removed from the full cone, and simplifies to the cone formula when r equals zero.

What is the lateral surface area of a frustum?

The lateral surface area (LSA) is the curved slanted surface of the frustum, not including the two circular faces. LSA equals pi times (R plus r) times the slant height l, where l equals the square root of (h squared plus (R minus r) squared). The slant height is the straight-line distance along the sloped surface from the edge of the base to the edge of the top.

Gaussian Elimination Calculator

What is Gaussian elimination?

Gaussian elimination is a systematic method for solving a system of linear equations. It uses row operations to turn the equations into a triangular form, then solves for the unknowns one at a time by back-substitution from the bottom up.

How does Gaussian elimination work?

You write the equations as an augmented matrix, then add multiples of one row to another to create zeros below the diagonal, producing an upper triangular system. The last equation gives one unknown directly, and you substitute back up to find the rest.

When does a system have no unique solution?

If elimination produces a row of zeros equal to a non-zero number, there is no solution. If it produces a row of all zeros, there are infinitely many solutions. A unique solution exists only when the coefficient matrix has full rank.

Golden Ratio Calculator

What is the golden ratio?

The golden ratio, denoted phi, is the irrational number (1 plus the square root of 5) divided by 2, approximately equal to 1.6180339887. It describes a division of a line segment where the ratio of the whole to the longer part equals the ratio of the longer part to the shorter part. This self-similar property makes phi unique in mathematics.

Where does the golden ratio appear in real life?

The golden ratio appears in art, architecture, design, and nature. It is said to produce aesthetically pleasing proportions and is used in logo design, photography composition, and typography. In nature it is related to the spiral growth patterns of shells, sunflower seeds, and pine cones. It is also closely connected to the Fibonacci sequence: the ratio of consecutive Fibonacci numbers converges to phi.

How do you calculate the golden ratio split of a length?

Given a total length T, the longer segment is T divided by phi, which equals T times phi divided by (phi plus 1), or equivalently T times (phi minus 1). The shorter segment is the total minus the longer segment. The simpler formulas are: longer = T divided by phi (approximately T times 0.618), and shorter = T divided by phi squared (approximately T times 0.382).

Gray Code Calculator

What is Gray code?

A binary numbering where each step changes only one bit, also called reflected binary code.

Why is it useful?

Single bit changes avoid glitches in position encoders and simplify Karnaugh maps in logic design.

How is it computed?

Gray code equals the number XOR the number shifted right by one bit.

Harmonic Mean Calculator NZ

When should I use the harmonic mean?

Use it for averaging rates, speeds and ratios, especially over equal amounts of the denominator. The classic case is average speed over equal distances, where the harmonic mean, not the arithmetic mean, gives the correct round-trip average because more time is spent at the slower speed.

How do I calculate the harmonic mean?

Divide the number of values by the sum of their reciprocals. So for two speeds, it is two divided by one over the first plus one over the second. This calculator does it for any list of positive numbers and shows the other means for comparison.

Why is the harmonic mean smaller than the arithmetic mean?

Because it weights smaller values more heavily through their reciprocals. For any set of positive numbers that are not all equal, the harmonic mean is the smallest of the three classical means, the arithmetic mean is the largest, and the geometric mean sits in between.

Hemisphere Calculator

What is the volume formula for a hemisphere?

The volume of a hemisphere is two-thirds pi r cubed, which is exactly half the volume of a full sphere (four-thirds pi r cubed). For a radius r, volume = (2/3) times pi times r cubed. This formula applies to a solid hemisphere. For a hollow hemispherical shell, you would subtract the volume of the inner hemisphere from the outer.

What is the difference between curved and total surface area of a hemisphere?

A hemisphere has two surfaces: the curved surface (the bowl-shaped outer surface) and the flat circular base (the cut face). The curved surface area equals 2 pi r squared, which is half the full sphere's surface area of 4 pi r squared. The flat base area is pi r squared. The total surface area is the sum of both: 3 pi r squared.

Where do hemisphere calculations come up in real life?

Hemisphere formulas appear in architecture for dome design, in engineering for pressure vessel end caps, in cookware for bowl capacity, in astronomy for describing half-sphere sky surveys, and in medicine for calculating the volume of brain hemispheres. The flat base area is used when the hemisphere sits on a flat surface and that contact area is needed.

Heron's Formula Calculator

What is Heron's formula?

Heron's formula calculates the area of a triangle from its three side lengths without needing to know any angles. You first compute the semi-perimeter s = (a + b + c) / 2, then the area = square root of s times (s minus a) times (s minus b) times (s minus c). It was described by the ancient Greek mathematician Heron of Alexandria.

When should I use Heron's formula instead of the base times height formula?

Use Heron's formula whenever you know all three side lengths but do not know the height of the triangle. The standard area formula (half times base times height) requires the perpendicular height, which can be hard to measure directly. Heron's formula works from side lengths alone, which are usually easier to obtain.

What happens if the sides cannot form a triangle?

For three lengths to form a valid triangle, each side must be less than the sum of the other two. This is called the triangle inequality. If you enter values that violate this rule, the expression inside the square root becomes negative and has no real solution, so the calculator will display an error.

Hypotenuse Calculator

What is Pythagoras's theorem?

Pythagoras's theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides: a squared plus b squared equals c squared. You can rearrange this to find any side if you know the other two.

How do you find a missing leg of a right triangle?

If you know the hypotenuse c and one leg a, you can find the missing leg b by rearranging Pythagoras's theorem: b = square root of (c squared minus a squared). Make sure c is larger than a, otherwise the value under the square root is negative and no real solution exists.

What is the difference between the hypotenuse and a leg?

In a right-angled triangle, the hypotenuse is the longest side and is always opposite the right angle (90 degrees). The other two shorter sides are called the legs or catheti. The hypotenuse is always strictly longer than either leg.

Image Ratio Calculator

How do I find the aspect ratio of an image?

Divide both the width and the height by their greatest common divisor (GCD). For example, a 1920x1080 image has a GCD of 120, giving 1920/120 : 1080/120 = 16:9. If you only know one ratio component, you can multiply by any common factor to get equivalent pixel sizes. This calculator does the division automatically for any width and height you enter.

How do I scale an image to a new size without distortion?

To scale while keeping the aspect ratio, fix one dimension and calculate the other using the ratio. If your original image is 1920x1080 (16:9) and you want a width of 1280, multiply: height = 1280 x (1080 / 1920) = 720. This calculator lets you enter a target width or target height and it computes the missing dimension for you.

What are the most common image aspect ratios?

The most common aspect ratios are: 16:9 (widescreen video, most modern monitors and TVs, YouTube), 4:3 (older monitors, standard definition TV, iPad), 1:1 (square, Instagram posts), 3:2 (35mm film, most DSLR cameras, standard prints), 2:1 (wide format, some smartphones in panorama mode), and 21:9 (ultrawide monitors). Print sizes like A4 use an irrational ratio close to 1:1.414.

Joint Variation Calculator

What is joint variation?

A relationship where one quantity varies as the product of two others, z = kxy.

How is it different from direct variation?

Direct variation has one driving variable; joint variation has two, multiplied together.

What is a real example?

The area of a triangle varies jointly with its base and height, with constant one half.

Kurtosis Calculator

What is kurtosis?

A measure of how heavy a distribution’s tails are. Excess kurtosis compares them with the normal distribution, which scores 0.

What do positive and negative values mean?

Positive (leptokurtic) means heavier tails and more outliers; negative (platykurtic) means lighter tails and a flatter peak.

Why subtract 3?

So that the normal distribution, which has a raw kurtosis of 3, reads as an excess kurtosis of 0.

Linear Interpolation Calculator NZ

What is linear interpolation?

It estimates a value between two known data points by assuming a straight line connects them. You read off the value at your point of interest along that line. It is a quick, widely used way to fill gaps in data or read between table entries.

What is the difference between interpolation and extrapolation?

Interpolation estimates a value between known points, which is generally reliable. Extrapolation projects beyond the known points, which is riskier because the trend may not continue. This calculator works for both but flags whether your x falls within the two points.

When is linear interpolation accurate?

When the two points are close together or the underlying relationship is nearly linear. For strongly curved data, a straight line between distant points can be a poor approximation, and methods like polynomial or spline interpolation would do better. Keep the points close for best accuracy.

Log Calculator

What is a logarithm?

A logarithm answers the question: to what power must a base be raised to produce a given number? Written log_b(x) = y, it means b^y = x. For example, log_10(1000) = 3 because 10^3 = 1000. The base must be a positive number other than 1. The most common bases are 10 (common logarithm, written 'log'), e (natural logarithm, written 'ln'), and 2 (binary logarithm, written 'lb' or 'log2').

How do you calculate a log in a different base?

Use the change-of-base formula: log_b(x) = ln(x) / ln(b), where ln is the natural logarithm. Most calculators only have log base 10 and ln built in, so this formula lets you calculate any base. For example, log_2(8) = ln(8) / ln(2) = 2.0794... / 0.6931... = 3, which confirms that 2^3 = 8.

What is the difference between log and ln?

'log' without a stated base almost always means log base 10 (the common logarithm) in everyday maths, engineering, and science. 'ln' always means the natural logarithm, which uses base e (approximately 2.71828). Mathematicians sometimes use 'log' to mean ln, but in calculator and computing contexts, log base 10 is the convention. Log base 2 is common in computer science and information theory.

Logistic Growth Calculator NZ

What is logistic growth?

Logistic growth is an S-shaped model where a quantity grows almost exponentially at first, then slows as it approaches a carrying capacity, the maximum its environment can sustain, and finally levels off. It is more realistic than exponential growth wherever resources are limited.

What is carrying capacity?

The carrying capacity is the largest value the environment can support long term. In the logistic model the quantity approaches this ceiling but never exceeds it. For a population it might be set by food or space; for a product, by the size of the market.

When is growth fastest?

In the logistic model, growth is fastest at the midpoint, when the quantity is half the carrying capacity. Before that, few individuals or adopters limit the rate; after it, the shrinking room left to grow slows things down, producing the characteristic S-curve.

Long Division Calculator

What is long division?

Long division is a method for dividing a large number (the dividend) by another number (the divisor) digit by digit. At each step you ask how many times the divisor goes into the current partial dividend, write the result in the quotient, subtract the multiple from the partial dividend, and bring down the next digit. You repeat this until all digits of the dividend have been used, at which point any remaining value is the remainder.

What is the remainder in division?

The remainder is the amount left over after you have divided as evenly as possible. If you divide 17 by 5, the quotient is 3 (because 5 goes into 17 three times for a product of 15) and the remainder is 2 (because 17 minus 15 equals 2). A remainder of zero means the divisor divides the dividend exactly.

How do I check a long division answer?

Multiply the quotient by the divisor, then add the remainder. The result should equal the original dividend. For example, if 1547 divided by 7 gives a quotient of 221 and a remainder of 0, then 221 times 7 equals 1547. If the check does not match, there is an error somewhere in the working.

Luhn Algorithm Calculator

What is the Luhn algorithm?

A simple checksum that doubles alternate digits and checks the total is a multiple of ten. Credit cards and IMEIs use it.

Does passing Luhn mean a card is real?

No. It only catches typing errors. A number can pass Luhn yet not be a genuine, active card.

What errors does it catch?

All single digit mistakes and most swaps of two adjacent digits.

Mann-Whitney U Test Calculator

What is the Mann-Whitney U test?

It is a non-parametric test that compares two independent groups using ranks rather than raw values, so it makes no assumption that the data is normally distributed. It tests whether one group tends to have higher values than the other.

When should I use it instead of a t-test?

Use it when the data is not normally distributed, the samples are small, the data is ordinal, or there are outliers that would distort a t-test. In those cases the rank-based Mann-Whitney U gives a more reliable comparison of the two groups.

What does the U statistic mean?

U measures the separation between the groups based on ranks. The reported U is the smaller of the two group values. A small U indicates the groups are well separated; you compare it against critical tables or convert it to a p-value to judge significance.

Measurement Uncertainty Calculator

What is relative uncertainty?

The absolute uncertainty divided by the measured value, usually as a percentage. It shows precision relative to size.

What is the difference from absolute uncertainty?

Absolute uncertainty is the plus or minus in the same units. Relative uncertainty expresses that as a fraction of the value.

How do I report a measurement?

As value plus or minus uncertainty, for example 10.0 plus or minus 0.2, ideally with the relative uncertainty too.

Midpoint Calculator

What is the midpoint formula?

The midpoint of a line segment with endpoints (x1, y1) and (x2, y2) is the point M = ((x1+x2)/2, (y1+y2)/2). You simply average the x-coordinates and average the y-coordinates to get the centre point of the segment.

How do you verify that a point is the midpoint of a segment?

Calculate the distance from the candidate midpoint to each endpoint. If both distances are equal and together they add up to the total length of the segment, the point is the true midpoint. You can also verify by checking that the midpoint's coordinates are exactly the averages of the two endpoint coordinates.

Can the midpoint have negative coordinates?

Yes. If one or both endpoints have negative x or y coordinates, the midpoint may also have negative coordinates. The formula works the same way regardless of the signs: average the two x values and average the two y values.

Mixture Calculator

How do I find a mixture concentration?

Take the weighted average: add the solute from each solution and divide by the total amount.

Is the result halfway between the two?

Only if you use equal amounts. Otherwise it leans toward the concentration you use more of.

Does it work for any units?

Yes, as long as both amounts use the same unit and both concentrations use the same percentage basis.

Moving Average Calculator

What is a moving average?

It replaces each point in a series with the average of a window of neighbouring points, sliding the window along. This smooths short-term fluctuations and noise, making the underlying trend in the data easier to see. It is widely used in finance and time-series analysis.

How do I choose the window size?

It is a trade-off. A larger window smooths more, flattening fluctuations but responding more slowly to genuine changes. A smaller window stays closer to the raw data and reacts faster but smooths less. Choose based on how much noise you want to remove.

What is the difference from a weighted moving average?

A simple moving average weights every point in the window equally. A weighted or exponential moving average gives more weight to recent points, so it responds faster to recent changes. This calculator computes the simple, equal-weight version, the most common starting point.

Newton's Method Calculator

What is Newton's method?

Newton's method (also called Newton-Raphson) is an iterative algorithm for finding roots of a function, that is, values of x where f(x) = 0. Starting from an initial guess x0, each step computes x1 = x0 minus f(x0) divided by f'(x0), where f'(x0) is the derivative. This is geometrically equivalent to following the tangent line at the current point to where it crosses the x-axis.

How fast does Newton's method converge?

Newton's method converges quadratically when it converges, meaning the number of correct decimal places roughly doubles with each iteration. Starting near a simple root, you might get 1 correct digit after one iteration, 2 after two, 4 after three, and 8 after four. This very rapid convergence makes it one of the most efficient root-finding methods.

When does Newton's method fail?

Newton's method can fail or diverge if the derivative f'(x) is zero or near zero at an iteration point, if the starting guess is too far from the root, or if the function has a repeated root (which slows convergence). It also converges to different roots depending on the starting point for functions with multiple roots.

Number to Words Calculator

How high can it go?

Up to the quadrillions, that is numbers below one thousand trillion.

Does it use the short scale?

Yes, the short scale standard in New Zealand, where a billion is a thousand million.

Are negative numbers handled?

Yes, they are prefixed with the word negative.

Numerical Integration Calculator NZ

What is Simpson's rule?

Simpson's rule is a numerical integration method that approximates the area under a curve by fitting parabolas through groups of three points, rather than straight lines. This captures curvature and gives high accuracy, and it is exact for polynomials up to cubic degree.

When should I use numerical integration?

Use it when a function has no neat antiderivative, or when you only have data points rather than a formula. Many real integrals in physics, engineering and statistics cannot be solved symbolically, so a numerical value from Simpson's rule is the practical answer.

How many intervals should I use?

More intervals generally give more accuracy. For smooth functions, 100 intervals is usually plenty. For functions that wiggle a lot over the range, increase the number until the result stops changing. The calculator makes the count even, as Simpson's rule requires.

Outlier Calculator

How does the IQR method detect outliers?

The IQR method calculates the interquartile range (Q3 minus Q1) and then sets fences at Q1 minus 1.5 times IQR (lower fence) and Q3 plus 1.5 times IQR (upper fence). Any value below the lower fence or above the upper fence is flagged as a potential outlier. This approach is robust because the fences are based on the middle 50% of the data rather than the extreme values.

Is the IQR outlier method the same as the Z-score method?

No. The Z-score method flags values that are more than a certain number of standard deviations from the mean (typically 2 or 3). The IQR method uses quartiles and is more resistant to outliers themselves distorting the calculation, which makes it better suited to skewed data or distributions with heavy tails.

Should I always remove outliers?

Not necessarily. Outliers can represent genuine extreme values that are part of the true distribution, or they can indicate data entry errors, measurement mistakes, or a genuinely unusual observation. Always investigate the cause of an outlier before deciding whether to remove it or treat it separately in your analysis.

P-value Calculator

What does a p-value mean?

The p-value is the probability of observing a test statistic at least as extreme as the one you calculated, assuming the null hypothesis is true. A small p-value (typically below 0.05) suggests the observed result is unlikely under the null hypothesis, and you reject it in favour of the alternative hypothesis.

What is the difference between a one-tailed and two-tailed test?

A two-tailed test checks whether the result is extreme in either direction (higher or lower than expected). A one-tailed test checks only one direction. For the same test statistic, a one-tailed p-value is half the two-tailed p-value. Use a one-tailed test only when you had a directional hypothesis before collecting data.

When do I use a t-score instead of a z-score?

Use a t-score when your sample is small (typically n below 30) or when the population standard deviation is unknown and you are estimating it from the sample. The t-distribution has heavier tails than the normal distribution, producing larger p-values for the same test statistic, which accounts for the extra uncertainty.

Parity Bit Calculator

What is a parity bit?

An extra bit added to make the number of 1s either even or odd, used to detect single-bit errors.

What is the difference between even and odd parity?

Even parity makes the total count of 1s even; odd parity makes it odd.

What errors can parity catch?

Any odd number of flipped bits, most usefully a single bit error. It cannot catch two flips that cancel out.

Percent Error Calculator

What is the percent error formula?

Percent Error = (|Experimental Value minus Theoretical Value| divided by |Theoretical Value|) times 100. The absolute value signs mean the result is always a positive number regardless of whether the measurement came in above or below the true value.

Can percent error be negative?

The standard percent error formula uses absolute values, so the result is always reported as a positive number. Whether the measurement was high or low is shown separately as the direction of the error, not by making the percentage negative.

What is a good percent error?

It depends on the context. In a typical school science experiment, under 5% is often considered good. In engineering and manufacturing, tolerances of 1 to 2% or tighter are common. In precise scientific work, errors below 0.1% may be required. Lower is always better, since it means the measured value is closer to the true value.

Percent Off Calculator

How do you calculate a percent off a price?

To calculate a percent-off discount, multiply the original price by the percentage divided by 100, then subtract that amount from the original price. For example, 30% off $120: saving = $120 x 0.30 = $36; sale price = $120 - $36 = $84. Alternatively, multiply the original price by (1 minus the decimal percentage): $120 x 0.70 = $84.

What is the formula for the final price after a percent-off discount?

The formula is: Sale Price = Original Price x (1 - Discount% / 100). The dollar saving is: Saving = Original Price x (Discount% / 100). For example, 25% off $200: Sale Price = $200 x (1 - 0.25) = $200 x 0.75 = $150. Saving = $200 x 0.25 = $50.

How do I work out what percentage off was applied if I know the original and sale prices?

Divide the saving (original price minus sale price) by the original price, then multiply by 100. For example, if an item was $80 and is now $60: saving = $20; discount% = ($20 / $80) x 100 = 25%. This calculator's reverse mode lets you enter any two known values to find the third.

Percentage Calculator

How do I find a percentage of a number?

Multiply the number by the percentage and divide by 100. For example, 15 percent of 80 is 80 times 15 divided by 100, which is 12. The first calculator here does this for you as you type.

How do I work out what percentage one number is of another?

Divide the first number by the second and multiply by 100. For example, 45 out of 60 is 45 divided by 60 times 100, which is 75 percent. The second calculator does this.

How do I calculate percentage change?

Subtract the old value from the new value, divide by the old value, and multiply by 100. A positive answer is an increase and a negative answer is a decrease. The third calculator does this.

Percentage Decrease Calculator

How do you calculate percentage decrease?

Percentage decrease = ((Original value - New value) / Original value) x 100. For example, if a price drops from 200 to 150, the decrease is 50, and the percentage decrease is (50 / 200) x 100 = 25%. The original value must be greater than zero, and the new value must be less than the original for a true decrease.

What is the difference between percentage decrease and percentage change?

Percentage change can be either positive (an increase) or negative (a decrease). Percentage decrease specifically measures how much a value has fallen as a proportion of the original, and is expressed as a positive number (e.g. '25% decrease' rather than '-25% change'). Use this calculator when you know the value has gone down and you want to express that fall as a percentage.

How do I find the new value after a percentage decrease?

New value = Original value x (1 - Percentage decrease / 100). For example, a 20% decrease from 500 gives 500 x (1 - 0.20) = 500 x 0.80 = 400. This calculator lets you enter either the new value directly or a known percentage decrease to find the new value.

Percentage Discount Calculator

How do I calculate the sale price after a percentage discount?

Multiply the original price by (1 minus the discount rate). For example, a $120 item with a 25% discount has a sale price of $120 x (1 - 0.25) = $120 x 0.75 = $90. The dollar saving is $120 - $90 = $30.

What is the combined discount when two discounts are applied in sequence?

Two sequential discounts are not simply added together. A 20% discount followed by a 10% discount gives a combined saving of 1 - (0.80 x 0.90) = 1 - 0.72 = 28%, not 30%. The second discount is applied to the already-reduced price. This calculator handles both the first and an optional second stacked discount.

How do I work backwards to find the original price from a sale price?

Divide the sale price by (1 minus the discount rate). If an item now costs $75 after a 25% discount, the original price was $75 / 0.75 = $100. This is often called reverse percentage or finding the original from the discounted value.

Polar to Complex Calculator

How do I convert polar to rectangular?

Multiply the modulus by the cosine of the angle for the real part, and by the sine for the imaginary part.

Why convert to rectangular?

Rectangular form a + bi is needed to add or subtract complex numbers; polar form is better for multiplying and powers.

What units is the angle in?

This calculator takes the angle in degrees and converts internally to radians.

Power Tower Calculator

What is tetration?

Repeated exponentiation: a power tower of a, the next operation after multiplication and exponentiation.

Why evaluate from the top?

Stacked exponents are right associative, so a^a^a means a^(a^a), not (a^a)^a.

Why does it say too large?

Power towers explode in size. Even a modest height exceeds what a computer can store, so it is flagged rather than shown.

Product to Sum Calculator

What does product to sum mean?

It rewrites a product of two trig functions as a sum or difference, for example sin A sin B becomes one half of [cos(A minus B) minus cos(A plus B)].

Where are these identities used?

In signal processing, acoustics and Fourier analysis, and historically as a multiplication shortcut called prosthaphaeresis.

Is there a reverse?

Yes, the sum to product identities go the other way, turning a sum of sines or cosines into a product.

Proportion Calculator

What is a proportion?

A proportion is a statement that two ratios are equal: a divided by b equals c divided by d. It means that a and b have the same relationship to each other as c and d. Proportions appear in scale drawings, recipe scaling, unit conversions, probability, and many other practical situations.

How do you solve a proportion for a missing value?

You use cross-multiplication. If a divided by b equals x divided by d, then a times d equals b times x, so x equals a times d divided by b. This works because multiplying both sides of a divided by b equals x divided by d by the product b times d clears both denominators.

What are cross-products and why do they matter?

In a proportion a divided by b equals c divided by d, the cross-products are a times d and b times c. A proportion is valid if and only if the cross-products are equal. Cross-products give you a quick way to verify that a proportion is correct without doing any division.

Rationalise the Denominator Calculator

Why rationalise the denominator?

A surd in the denominator is hard to work with. Rewriting it with a whole number denominator makes the expression easier to add, compare and evaluate.

How does it work?

Multiply the top and bottom by the root in the denominator, which turns root b times root b into b, a whole number.

Does the value change?

No. You are multiplying by a form of 1, so the value is unchanged, only the way it is written.

Rectangular to Polar Calculator

How do I convert (x, y) to polar?

Radius r = sqrt(x^2 + y^2), angle theta = atan2(y, x). The angle is measured anticlockwise from the positive x axis.

Why use atan2 instead of arctan?

Plain arctan(y/x) loses the quadrant and fails when x is zero. atan2(y, x) returns the correct angle for any point.

What range is the angle in?

atan2 returns an angle between -180 and 180 degrees (-pi and pi radians).

Recurrence Relation Calculator

What is a recurrence relation?

A recurrence relation defines each term of a sequence from earlier terms. A second-order linear one says the next term is p times the previous term plus q times the one before that, starting from two given values, which generates the whole sequence.

Is the Fibonacci sequence a recurrence relation?

Yes. The Fibonacci sequence is the recurrence where each term is the sum of the two before it, so p and q are both 1, starting from 0 and 1. It gives 0, 1, 1, 2, 3, 5, 8, 13 and so on.

How do you compute a term of a recurrence?

Start from the two given terms and apply the rule repeatedly, each time combining the two most recent terms to get the next one, until you reach the term you want. This calculator iterates the rule and lists the sequence for you.

Relative Error Calculator

What is relative error?

The absolute error divided by the true value, usually written as a percentage. It shows the error in proportion to the quantity.

Why use relative error?

It lets you compare the accuracy of measurements on different scales, which raw absolute error cannot do.

What if the true value is zero?

Relative error is undefined when the true value is zero, since you cannot divide by zero.

Remainder Calculator

What is a remainder in division?

The remainder is the amount left over after dividing one integer by another as evenly as possible. When you divide 23 by 4, the divisor 4 goes into 23 five times (the quotient) with 3 left over (the remainder), giving the equation 23 = 4 times 5 plus 3. The remainder is always a non-negative integer less than the divisor.

What is the difference between remainder and modulo?

For positive integers, remainder and modulo give the same result. They differ when one of the numbers is negative, depending on the programming language or convention used. The truncated remainder follows the sign of the dividend, while the modulo result follows the sign of the divisor. This calculator uses truncated integer division, which matches the behaviour of most calculators and the JavaScript % operator.

How do you find the remainder without a calculator?

Divide the dividend by the divisor using long division, keeping only the whole-number (integer) part of the quotient. Multiply the quotient by the divisor and subtract from the dividend. What is left is the remainder. For example, 23 divided by 4 is 5 remainder 3 because 4 times 5 is 20, and 23 minus 20 is 3.

Residual Calculator

What is a residual?

The difference between an observed value and the value a model predicts: observed minus predicted.

What does the sign mean?

A positive residual means the model under-predicted; a negative residual means it over-predicted.

Why square residuals?

Squaring removes the sign and penalises large errors more. Least squares regression minimises the sum of squared residuals.

Riemann Sum Calculator

What is a Riemann sum?

A Riemann sum approximates the area under a curve by dividing the interval into n equal subintervals, drawing a rectangle in each subinterval, and summing the rectangle areas. The height of each rectangle depends on which endpoint or midpoint you use to sample the function. As n increases the approximation approaches the definite integral.

What is the difference between left, right, and midpoint Riemann sums?

The left sum uses the function value at the left edge of each subinterval as the rectangle height. The right sum uses the right edge. The midpoint sum uses the function value at the centre of each subinterval, which is generally the most accurate of the three for the same number of rectangles because errors on either side partially cancel.

How many rectangles should I use?

More rectangles give a more accurate approximation. For a smooth function like x squared, 6 rectangles gives a rough estimate and you can see the left sum undershooting and the right sum overshooting. Use 100 or more rectangles to get an approximation close to the true integral value. The midpoint rule converges to the true value roughly four times faster than the left or right rules.

Roman to Arabic Numerals Calculator

How do Roman numerals work?

Letters stand for values that you add when written in descending order. A smaller letter before a larger one is subtracted, like IV for 4.

Why only up to 3999?

Standard Roman numerals have no symbol above M (1000), so the usual range without special bars is 1 to 3999.

Is there a zero in Roman numerals?

No. The Roman system has no symbol for zero.

Rounding Calculator

How do you round to decimal places?

Look at the digit immediately after the decimal place you are rounding to. If it is 5 or more, round the last kept digit up by one. If it is 4 or less, leave the last kept digit as it is and drop everything after it. For example, 3.14159 rounded to two decimal places is 3.14 because the third decimal digit is 1.

What is the difference between round, round up, and round down?

Standard rounding uses the half-up rule: digits of 5 or more round up, digits below 5 round down. Round up, also called ceiling, always moves to the next value above regardless of the following digit. Round down, also called floor, always moves to the next value below. The choice matters when you cannot afford to err on one side, such as rounding up quantities when ordering materials to ensure you have enough.

How do you round to the nearest 10 or 100?

Divide the number by 10 or 100, round to the nearest whole number using the standard rule, then multiply back. So 47 divided by 10 is 4.7, which rounds to 5, multiplied back by 10 gives 50. The calculator does this automatically for any multiple you choose.

Rule of Three Calculator

What is the rule of three?

The rule of three is a simple way to solve a proportion: given three values in a relationship and one unknown, you cross-multiply to find the missing value. It answers questions like, if 2 items cost a certain amount, how much do 5 cost.

How do you use the direct rule of three?

Set up the proportion a is to b as c is to x. Then x equals b times c divided by a. For example, if 2 corresponds to 10, then 5 corresponds to 10 times 5 over 2, which is 25. More of one means more of the other.

What is the inverse rule of three?

In an inverse proportion, more of one means less of the other, so the products stay constant. Then x equals a times b divided by c. For example, if 4 workers take 6 hours, 8 workers take 4 times 6 over 8, which is 3 hours.

Scientific Notation Calculator

What is scientific notation?

Scientific notation expresses a number as a coefficient between 1 and 10 multiplied by a power of 10. For example, 0.000045 becomes 4.5 times 10 to the power of negative 5. It makes very large or very small numbers easier to read, compare, and use in calculations.

What is the difference between scientific and engineering notation?

Scientific notation always places the decimal so the coefficient is between 1 and 10. Engineering notation adjusts the exponent to always be a multiple of three (such as 0, 3, 6, -3, -6), which aligns with metric prefixes like milli, kilo, and mega, making it more convenient for electronics and physics.

What is order of magnitude?

Order of magnitude is the power of 10 in the scientific notation of a number. A number with an order of magnitude of 5 is roughly 100,000 times larger than one with an order of magnitude of 0. It is used to compare quantities that differ by enormous factors without needing exact values.

Secant Calculator

How do I find the secant of an angle?

Enter the angle and pick degrees or radians. The calculator converts to radians and evaluates the secant.

Degrees or radians?

Either. Choose the unit in the dropdown. The angle is shown in both units beside the result so you can check.

Why is the result the same every 360 degrees?

The trig functions are periodic. Sine and cosine repeat every 360 degrees (2 pi radians), so equivalent angles give the same value.

Set Intersection Calculator

What is the intersection of two sets?

The elements that are in both sets. {1,2,3} intersect {2,3,4} is {2,3}.

The intersection is the empty set, sometimes written with the symbol for nothing, and its size is zero.

Can elements be words?

Yes. Any comma separated values work; matching is by exact value.

Set Union Calculator

What is the union of two sets?

Every element that is in either set, listed once. {1,2} union {2,3} is {1,2,3}.

Are duplicates removed?

Yes. A set holds each distinct value once, so repeats within or across the two inputs collapse to a single element.

Can I use words, not just numbers?

Yes. Enter any comma separated values; the union works the same way.

Significant Figures Calculator

What are significant figures?

Significant figures are the digits in a number that carry meaning about its precision. All non-zero digits count, zeros trapped between non-zero digits count, and trailing zeros after a decimal point count. Leading zeros, such as those in 0.0042, never count because they only set the decimal place.

How do you round to significant figures?

Start from the first non-zero digit and count to your target number of significant figures. Look at the next digit: if it is 5 or more, round the last kept digit up; if it is 4 or less, leave it. For example, 3.14159 rounded to 3 significant figures is 3.14, because the fourth digit is 1.

Why do significant figures matter?

They communicate how precise a measurement or calculation is. Reporting more figures than your instrument or method can support overstates the precision, which is misleading in science, engineering, and laboratory work. Sig fig rules ensure that reported results are honest about the limits of accuracy.

Sin Cos Tan Calculator

What are sine, cosine and tangent?

Sine, cosine and tangent are the three primary trigonometric functions. For a right-angled triangle, sine of an angle equals the opposite side divided by the hypotenuse, cosine equals the adjacent side divided by the hypotenuse, and tangent equals the opposite side divided by the adjacent side. Together they relate angles to side ratios in any right triangle.

What are cosecant, secant and cotangent?

Cosecant, secant and cotangent are the reciprocal trigonometric functions. Cosecant is 1 divided by sine, secant is 1 divided by cosine, and cotangent is 1 divided by tangent. They are defined for all angles where the denominator function is not zero.

Why is tan undefined at 90 degrees?

Tangent equals sine divided by cosine. At 90 degrees, cosine equals zero, so the division is undefined. This means the tangent function has a vertical asymptote at 90 degrees and at every subsequent 180-degree interval. The calculator shows undefined for these angles.

Skewness Calculator

What does skewness measure?

The asymmetry of a distribution. Positive skew has a long right tail, negative skew a long left tail.

What does zero skewness mean?

A symmetric distribution, where values are balanced either side of the mean, like the normal distribution.

Why does skewness matter?

It tells you whether the mean is pulled by a tail, and whether the median might describe the centre better.

Slope Calculator

What is the slope formula?

The slope (m) of a line through two points (x1, y1) and (x2, y2) is m = (y2 minus y1) divided by (x2 minus x1). This is often described as rise over run: how much the line rises (or falls) for each unit it moves to the right.

What does a negative slope mean?

A negative slope means the line falls as it moves from left to right. For every unit you move to the right along the x-axis, the y-value decreases by the magnitude of the slope. A slope of zero means the line is perfectly horizontal.

How do you write the equation of a line from two points?

First calculate the slope m = (y2 minus y1) / (x2 minus x1). Then find the y-intercept b by substituting one point: b = y1 minus m times x1. The equation of the line is then y = mx + b, which is called slope-intercept form.

SOHCAHTOA Calculator

What does SOHCAHTOA mean?

SOHCAHTOA is a memory aid for the three right-triangle ratios. SOH: sine equals opposite over hypotenuse. CAH: cosine equals adjacent over hypotenuse. TOA: tangent equals opposite over adjacent. These three ratios let you find any missing side or angle in a right triangle.

How do you use SOHCAHTOA to find a missing side?

Choose the ratio that links your known angle, your known side and the side you want. If you know the angle and hypotenuse and want the opposite side: opposite = hypotenuse x sin(angle). For angle 30 and hypotenuse 10: opposite = 10 x sin(30) = 10 x 0.5 = 5.

Which sides are opposite, adjacent and hypotenuse?

The hypotenuse is the longest side, always opposite the right angle. For your chosen angle, the opposite side is across from it and the adjacent side is next to it but not the hypotenuse.

Standard Error Calculator

What is the standard error?

The standard error (SE) measures how much your sample statistic (such as the mean or a proportion) is likely to vary from sample to sample. It equals the standard deviation divided by the square root of the sample size for means, or the square root of p(1-p)/n for proportions. A smaller SE means the estimate is more precise.

How is standard error different from standard deviation?

Standard deviation describes the spread of individual data points around the sample mean. Standard error describes how precisely the sample mean estimates the true population mean. SE shrinks as sample size grows (SE = SD divided by sqrt(n)), while SD is a property of the data set itself and does not depend on n.

How do I use SE to build a confidence interval?

A 95% confidence interval for a mean is approximately the sample mean plus or minus 1.96 times the standard error. This means that if you repeated the sampling process many times, about 95% of the resulting intervals would contain the true population mean.

Stars and Bars Calculator

What is the stars and bars method?

Stars and bars is a counting technique for working out how many ways identical items can be shared among distinct groups. The items are the stars and the dividers between groups are the bars, and arranging them in a row gives every possible distribution.

What is the stars and bars formula?

To put n identical items into k groups with empty groups allowed, the number of ways is the combination of n plus k minus 1, choose k minus 1. If every group must have at least one item, it is the combination of n minus 1, choose k minus 1.

When do you use stars and bars?

Use it whenever you split identical things among labelled groups, such as handing out identical lollies to children, finding non-negative integer solutions to an equation, or counting the terms in a polynomial expansion. The items must be identical and the groups distinct.

Sum and Difference Identities Calculator

What are the sum and difference identities?

They expand a function of A plus or minus B: sin(A+B) = sin A cos B + cos A sin B, cos(A+B) = cos A cos B - sin A sin B, and the matching tan formula.

Why are they useful?

They give exact values for angles like 15 or 75 degrees, simplify expressions, and are used throughout calculus and physics.

Degrees or radians?

Either. Pick the unit in the dropdown. The calculator converts to radians to evaluate.

Summation Calculator

What is sigma notation?

Sigma notation (using the Greek capital letter sigma) is a compact way to write the sum of many terms that follow a pattern. It specifies a formula in a variable i, a lower bound where i starts, and an upper bound where i finishes. For example, sigma from i equals 1 to 5 of i squared means 1 squared plus 2 squared plus 3 squared plus 4 squared plus 5 squared, which equals 55.

What are the formulas for common summations?

Three classic summation formulas are: the sum of i from 1 to n equals n times (n plus 1) divided by 2; the sum of i squared from 1 to n equals n times (n plus 1) times (2n plus 1) divided by 6; and the sum of i cubed from 1 to n equals the square of n times (n plus 1) divided by 2. These are known as the power sum formulas and are useful in calculus and discrete mathematics.

Can I use the summation calculator with a custom expression?

Yes. Select the custom option and type any mathematical expression in terms of i, using standard operators: i*2 for 2i, i^2 for i squared, sqrt(i) for the square root of i, 1/i for a harmonic sum, and so on. The calculator evaluates the expression for each integer value of i from the lower to the upper bound and sums the results.

Summation (Sigma) Calculator

What is sigma (summation) notation?

Sigma notation, written with the Greek capital letter sigma (Σ), is a compact way to express the sum of a sequence of terms. The notation Σ(i=a to b) f(i) means: evaluate the expression f(i) for each integer value of i from a to b, then add all the results together. For example, Σ(i=1 to 5) i = 1 + 2 + 3 + 4 + 5 = 15. The lower bound is the starting index, the upper bound is the ending index, and the expression defines the general term.

What is the formula for the sum of an arithmetic series?

For an arithmetic series where the general term is f(i) = a + (i-1)*d (with first term a and common difference d), the sum from i=1 to n is: S = n/2 * (2a + (n-1)*d), which simplifies to S = n*(first + last)/2. As a specific example, the sum of the first n natural numbers Σ(i=1 to n) i = n*(n+1)/2. For n=10, this equals 55.

Can this calculator handle geometric series and powers of n?

Yes. The calculator evaluates any safe mathematical expression in n, including powers (n^2, n^3, 2^n), products (3*n), and combinations (n^2 + n). Geometric series such as Σ(i=1 to 10) 2^i are evaluated by substituting each integer value of i into the expression. The calculator works by direct term-by-term summation, so it handles any expression that can be evaluated numerically, rather than being restricted to specific formula types.

Surds Simplification Calculator

What is a surd?

An irrational root, such as the square root of 2, that cannot be written exactly as a fraction or whole number.

How do I simplify a square root?

Take out the largest perfect square factor. The square root of 50 is 5 root 2 because 50 = 25 times 2.

What if the number is a perfect square?

Then the root is a whole number and there is no surd left, for example the square root of 49 is 7.

Synthetic Division Calculator

When can I use synthetic division?

When dividing by a linear factor x minus r. For higher degree divisors use polynomial long division.

What does the remainder tell me?

It equals P(r). If it is zero, then x minus r is a factor of the polynomial.

What is r for x + 3?

r is minus 3, because x + 3 is x minus (minus 3).

T-Score Calculator

What is a t-score?

The number of standard errors a sample mean is from a hypothesised mean, used in t tests for small samples.

What are the degrees of freedom?

For a one-sample t test, n minus 1, where n is the sample size.

When do I use t instead of z?

Use the t statistic when the population standard deviation is unknown and estimated from a small sample.

Test Score Percentage Calculator NZ 2026

How do I convert a test score to a percentage?

Divide the marks you scored by the total marks available, then multiply by 100. For example, 42 out of 50 is 84 percent. This calculator does it instantly and adds an indicative grade band so you can see where the result sits.

What grade is my percentage?

On the common general scale used here, 80 percent and above is an A, 65 to 79 a B, 50 to 64 a C, 40 to 49 a D and below 40 an E. Your school, university or NCEA may use different boundaries, so check the scheme for your test.

Why use percentages instead of raw marks?

Percentages let you compare tests of different sizes on the same scale, so 18 out of 25 and 36 out of 50 are both 72 percent. They make it easy to track progress over time and compare your results across subjects.

Tetrahedron Calculator

What is the volume formula for a regular tetrahedron?

The volume of a regular tetrahedron with edge length a is V = (a cubed) divided by (6 times the square root of 2). This is approximately 0.11785 times a cubed.

What is the height of a regular tetrahedron?

The height of a regular tetrahedron (from base face to apex) is h = a times the square root of (2/3), which is approximately 0.8165 times a. All four faces are equilateral triangles of the same size.

What is the difference between circumradius and inradius of a tetrahedron?

The circumradius is the radius of the sphere that passes through all four vertices: R = a times the square root of 6 divided by 4, or about 0.6124 a. The inradius is the radius of the sphere tangent to all four faces: r = a divided by (2 times the square root of 6), or about 0.2041 a. The circumradius is always exactly three times the inradius for a regular tetrahedron.

Trimmed Mean Calculator

What is a trimmed mean?

It is an average calculated after removing a set percentage of the highest and lowest values from the sorted data. By discarding the extremes, it reduces the influence of outliers, giving a central value that better represents the bulk of the data.

When should I use a trimmed mean?

Use it when your data contains outliers or is skewed, and you want an average that is not dragged around by a few extreme values. It is common in sports scoring, economics and any analysis where extreme readings should not dominate the result.

How much should I trim?

Commonly 5 to 25 percent from each end. More trimming gives a more robust result but discards more data; a 50 percent trim converges on the median. Comparing the trimmed mean with the ordinary mean shows how much the outliers are affecting your data.

Two’s Complement Calculator

What is two’s complement?

A way to store signed integers in a fixed number of bits, where a negative number n is held as 2 to the width plus n. It makes addition and subtraction uniform.

What range fits in 8 bits?

Minus 128 to 127. In general minus 2 to the (width minus 1) up to 2 to the (width minus 1) minus 1.

What is the sign bit?

The leftmost bit. If it is 1 the number is negative, if 0 it is zero or positive.

Unit Rate Calculator

What is a unit rate?

A rate with a denominator of one, such as price per kilogram or kilometres per hour. Divide the amount by the quantity.

How does it help me compare prices?

Putting different pack sizes onto the same per unit basis shows which is genuinely cheaper.

What is the inverse rate?

The quantity per single unit of amount, found by dividing the quantity by the amount.

Vector Calculator NZ

How do I find the magnitude of a vector?

Take the square root of the sum of the squares of its components. For a 3D vector (x, y, z), the magnitude is the square root of x squared plus y squared plus z squared. This is Pythagoras extended to three dimensions.

What does the dot product tell me?

The dot product measures how much two vectors point the same way. It is positive when they point in similar directions, zero when they are perpendicular, and negative when they oppose. It is used to find angles, projections and the work done by a force.

How do I find the angle between two vectors?

Divide the dot product by the product of the two magnitudes, then take the inverse cosine. The result is the angle between them. This calculator does it for you and reports the angle in degrees.

Vector Projection Calculator NZ

What is vector projection?

It is the shadow one vector casts onto the direction of another, showing how much of the first vector points along the second. The vector projection is itself a vector along the second; the scalar projection is its signed length.

What is the difference between scalar and vector projection?

The scalar projection is a single number, the signed length of the projection, found as the dot product divided by the magnitude of the vector you project onto. The vector projection is that length given direction, a vector pointing along the second vector.

What does a negative scalar projection mean?

It means the first vector points generally in the opposite direction to the second, so its shadow falls on the negative side. A positive scalar projection means it points broadly the same way, and zero means the two vectors are perpendicular.

Vertex Form Calculator

What is vertex form?

y = a(x minus h)^2 + k, which shows the vertex (h, k) directly and is got by completing the square.

How do I find the vertex?

The x coordinate is minus b over 2a; put that back into the quadratic for the y coordinate.

What does a tell me?

If a is positive the parabola opens upward with a minimum at the vertex; if negative it opens downward with a maximum.

Vieta’s Formulas Calculator

What are Vieta’s formulas?

Relations between a polynomial’s coefficients and its roots. For a quadratic, the roots sum to minus b/a and multiply to c/a.

Do I need to solve the quadratic first?

No. That is the point: Vieta gives the sum and product straight from the coefficients.

What does the discriminant tell me?

b squared minus 4ac: positive means two real roots, zero a repeated root, negative a complex conjugate pair.

Weight Loss Percentage Calculator

How do you calculate weight loss percentage?

Weight loss percentage is calculated by dividing the amount of weight you have lost by your starting weight, then multiplying by 100. The formula is: Weight Loss % = ((Starting Weight - Current Weight) / Starting Weight) x 100. For example, if you started at 90 kg and now weigh 75 kg, you have lost 15 kg. Dividing 15 by 90 gives 0.1667, and multiplying by 100 gives 16.67%. Your starting weight is always the denominator, not your goal weight.

Is a weight loss percentage of 5% significant?

Yes. A weight loss of 5 to 10 percent of your starting body weight is widely recognised as clinically meaningful. Research consistently shows that losing just 5% of body weight can improve blood pressure, blood sugar levels, cholesterol, and reduce the risk of type 2 diabetes and cardiovascular disease. For a person starting at 90 kg, a 5% loss is 4.5 kg, bringing them to 85.5 kg. While the absolute number may seem modest, the health benefits at this threshold are well established.

What is a healthy rate of weight loss?

Health authorities generally recommend losing no more than 0.5 to 1 kg per week (roughly 1 to 2 lbs per week). Faster loss is possible in the early weeks of a new diet as the body sheds water weight, but sustained rapid loss typically requires very severe calorie restriction and can result in muscle loss, nutrient deficiencies, and difficulty maintaining the weight loss long term. A slower, steady rate of around 0.5 kg per week represents a calorie deficit of approximately 500 kcal per day, which is achievable through a combination of reduced intake and increased activity.

Weighted Average Calculator NZ

How do I calculate a weighted average?

Multiply each value by its weight, add up those products, then divide by the sum of the weights. Values with bigger weights have more influence on the result. This calculator does it for you from your lists of values and matching weights.

What can I use as weights?

Any positive numbers that reflect relative importance: percentages, counts, dollar amounts, credit points or arbitrary scores. Only the relative sizes matter, so weights of 1, 2, 3 give the same result as 10, 20, 30. Just keep them in the same order as the values.

How is a weighted average different from a simple average?

A simple average treats every value equally, while a weighted average lets some values count for more. When the weights are all equal, the two give the same answer. When they differ, the weighted average shifts toward the values carrying the most weight.

Work Rate Calculator

Why not just average the times?

Because rates add, not times. You add the fractions of the job each does per unit time, then take the reciprocal.

Will the joint time always be shorter?

Yes. Two workers together finish faster than either alone, so the combined time is less than the smaller of the two.

Can I extend it to three workers?

Yes, add all three rates (1/a + 1/b + 1/c) and take the reciprocal.

Z-Test for a Proportion Calculator

What is a one-proportion z-test?

It tests whether an observed proportion from a sample differs significantly from a hypothesized value. The z statistic compares the difference against the standard error expected under the hypothesis, indicating whether the observed proportion is likely real or due to chance.

How do I judge significance?

Compare the magnitude of the z statistic against the critical value for your significance level: about 1.96 for a two-sided test at 95 percent confidence, or 1.645 for a one-sided test. If the magnitude exceeds the critical value, the difference is statistically significant.

When is the test reliable?

It relies on the normal approximation to the binomial, which is accurate when both the expected number of successes and failures are reasonably large, generally at least about ten each. For very small samples or extreme proportions, exact binomial methods are better.