Statistics and probability questions, answered

Altman Z-Score Calculator NZ 2026/27

What is the Altman Z-Score?

It is a formula developed by Edward Altman in 1968 that combines several balance sheet and profit and loss ratios into a single score predicting the likelihood of bankruptcy within about two years. It was built by statistically comparing companies that failed with companies that did not, and identifying which ratios best separated them. It remains one of the most widely used measures of financial distress, more than fifty years after publication.

Which Altman Z-Score model should a New Zealand small business use?

The Z-double-prime model, which is why it is the default here. The original 1968 Z requires a market value of equity, which a private company does not have, and it was calibrated on listed manufacturers. Z-double-prime was developed for private and non-manufacturing firms, drops the sales to assets ratio that varies enormously between industries, and uses book equity instead of market value. That describes almost every New Zealand SME.

What is the Z-double-prime formula?

Z-double-prime equals 6.56 times working capital over total assets, plus 3.26 times retained earnings over total assets, plus 6.72 times EBIT over total assets, plus 1.05 times book value of equity over total liabilities. The zone thresholds are: above 2.6 safe, 1.1 to 2.6 grey, below 1.1 distress. On the default figures the score is 3.05, which sits in the safe zone.

Why does retained earnings matter so much in the score?

Because it measures cumulative profitability and, indirectly, age. A company that has been profitable for years and retained those profits has a buffer and a track record. A young company, or one that has distributed everything it earned, has neither. This is why the Z-Score is systematically harsh on new businesses: a two year old company can be trading profitably and still score poorly simply because it has not had time to accumulate anything. It is a limitation to be aware of rather than a fault in the model.

Should I use net profit or EBIT in the Z-Score?

EBIT, meaning earnings before interest and tax. The model deliberately measures how well the assets generate operating profit, independently of how the business is financed or taxed. Substituting net profit double counts the effect of leverage, because debt already appears in the equity to liabilities ratio, and it will understate the score for any business carrying borrowings.

Is the Altman Z-Score reliable for small businesses?

Treat it as a screening indicator, not a verdict. It was developed on larger firms with audited accounts, and small company balance sheets can distort it in both directions: a shareholder current account can be classified as either a liability or quasi-equity, revaluations can inflate assets, and an owner taking income as drawings rather than salary changes EBIT. A poor score is a reason to look closely at liquidity and leverage, and a good score is not a guarantee of anything. Use it alongside cash flow, not instead of it.

How is the Altman Z-Score different from a statistical z-score?

They are unrelated despite the shared name. A statistical z-score expresses how many standard deviations a value sits from the mean of a distribution, and is used throughout statistics. The Altman Z-Score is a weighted combination of financial ratios used to predict corporate failure. If you are looking for the statistical measure, use our separate z-score calculator instead.

What should I do if the score is in the grey or distress zone?

Look at which component is dragging it down, which is why this calculator itemises the contributions. A low working capital term points at short-term liquidity, so examine debtor days, stock and the overdraft. A low EBIT term points at trading performance rather than the balance sheet. A low equity to liabilities term points at leverage, and the answer is usually equity rather than more debt. Note also that in New Zealand a poor score is worth reading alongside the section 4 solvency test, which is what directors are actually required to satisfy before making any distribution.

ANOVA Calculator

What is the ANOVA Calculator?

Calculate analysis of variance (ANOVA) for multiple groups. Enter your data to get F-statistic, p-value, and between/within group variance. Free online ANOVA tool.

Is the ANOVA Calculator free to use?

Yes. The ANOVA Calculator is free to use on Calculate.co.nz, with no sign-up, paywall or account required.

What assumptions does one-way ANOVA make?

One-way ANOVA assumes your samples are independent, each group is roughly normally distributed and the groups have similar variances. It uses the standard F-test, comparing between-group variance with within-group variance, so the maths is the same everywhere and runs in your browser. A significant result shows only that some group means differ, not which ones, so follow up with a post-hoc test.

Barn Pole Paradox Calculator

What is the barn pole paradox?

A pole longer than a barn is run through it at near light speed. In the barn frame the pole is length contracted and briefly fits inside with both doors momentarily shut. In the pole frame the barn is contracted and the pole never fits. Both are correct because simultaneity is relative.

How is length contraction calculated?

A moving length is its rest length divided by the Lorentz factor gamma, where gamma is 1 divided by the square root of 1 minus the speed squared over the speed of light squared. At 0.9 c, gamma is about 2.294, so a 20 m pole measures about 8.72 m.

Does the pole really fit in the barn?

In the barn's frame, yes, at the instant both ends are inside. In the pole's own frame, no, because the barn is the thing that contracts. There is no true contradiction: the two frames disagree about whether the two doors shut at the same time.

Bertrand's Box Paradox Calculator

What is Bertrand's box paradox?

Three boxes each hold two coins: one has two gold, one has two silver, and one has a gold and a silver. You pick a box at random and draw one coin, which turns out to be gold. The paradox is that the chance the other coin in that box is also gold is two thirds, not one half as intuition suggests, because a gold coin is twice as likely to have come from the two-gold box.

Why is the answer two thirds, not one half?

Count the gold coins, not the boxes. The two-gold box supplies two gold coins and the mixed box supplies one, so three gold coins could have been drawn. Two of those three sit in the two-gold box, where the partner is also gold. So the chance the other coin is gold is two out of three. Reasoning by boxes hides that the two-gold box has twice the chance of producing a gold draw.

How does this relate to Bayes' theorem?

It is a textbook Bayes update. Before drawing, each box has a one third prior. The likelihood of drawing gold is 1 from the two-gold box, one half from the mixed box, and 0 from the two-silver box. Bayes' theorem gives the posterior for the two-gold box as (1/3 x 1) divided by (1/3 x 1 + 1/3 x 1/2 + 1/3 x 0), which is two thirds.

Bertrand's Paradox Calculator

What is Bertrand's paradox?

Bertrand's paradox asks for the probability that a random chord of a circle is longer than the side of the inscribed equilateral triangle, which for a unit circle is the square root of three. Depending on how you define a random chord, the answer comes out as one half, one third, or one quarter. The paradox shows that random is not well defined until you say exactly how the chord is chosen.

Why are there three different answers?

Each method picks a chord by a different uniform rule. The random endpoints method fixes one end and picks the other uniformly on the circle, giving one third. The random radius method picks a point uniformly along a radius and draws the perpendicular chord, giving one half. The random midpoint method picks the chord midpoint uniformly over the disk, giving one quarter. All are valid; they just define randomness differently.

Which answer is correct?

None is uniquely correct, which is the point. Each answer is right for its own definition of a random chord. Some argue the random radius method of one half is the most natural because it is the only one invariant under scaling, rotation and translation, an argument made by Edwin Jaynes, but the paradox stands as a warning to define your probability space precisely.

Beta Distribution Calculator

What is the mean of a beta distribution?

The mean of a beta distribution equals alpha divided by (alpha plus beta). For example, with alpha of 2 and beta of 5 the mean is 2 / 7, which is about 0.2857. Because the distribution is bounded between 0 and 1, the mean is always a value in that range.

How do you calculate the variance of a beta distribution?

The variance is (alpha times beta) divided by ((alpha plus beta) squared times (alpha plus beta plus one)). With alpha of 2 and beta of 5 that is 10 / (49 times 8), which is 10 / 392, or about 0.0255, giving a standard deviation of about 0.1597.

What is the beta distribution used for?

The beta distribution models a proportion or probability that must lie between 0 and 1, such as a conversion rate, a success rate, or the reliability of a component. It is widely used in Bayesian statistics as the conjugate prior for the binomial distribution, where alpha and beta act as counts of prior successes and failures.

Binomial Confidence Interval Calculator

What is the Wilson score interval?

It is a method for finding a confidence interval for a proportion that is accurate even for small samples and extreme proportions. Unlike the simple Wald interval, it always stays within zero and one hundred percent and has better coverage, so statisticians recommend it.

Why not use the simple proportion plus or minus a margin?

That Wald interval performs poorly with small samples or proportions near zero or one, sometimes giving impossible bounds below zero or above one hundred percent. The Wilson score interval avoids these problems and is more trustworthy across the board.

How does the confidence level affect the interval?

A higher confidence level, such as 99 percent, produces a wider interval because you are demanding greater certainty that it contains the true proportion. A lower level gives a narrower interval. The calculator lets you choose 90, 95 or 99 percent.

Binomial Distribution Calculator

What is the Binomial Distribution Calculator?

Calculate binomial distribution probabilities for any number of trials, successes, and probability. Returns exact probability, cumulative probability, mean, and standard deviation.

Is the Binomial Distribution Calculator free to use?

Yes. The Binomial Distribution Calculator is free to use on Calculate.co.nz, with no sign-up, paywall or account required.

When can I use the binomial distribution?

Use the binomial distribution when you have a fixed number of independent trials, each trial has only two outcomes (success or failure), and the probability of success is the same on every trial. The calculator then gives the probability of x successes in n trials as P(X = x) = nCx p^x (1 - p)^(n - x), along with the mean (np) and variance (np(1 - p)).

Binomial Expansion Calculator

What is the binomial theorem?

The binomial theorem states that (a + b) to the power n equals the sum from k=0 to n of C(n,k) times a to the power (n minus k) times b to the power k, where C(n,k) is the binomial coefficient n choose k, equal to n factorial divided by k factorial times (n minus k) factorial. These coefficients appear as rows of Pascal's triangle.

What is (x+y)^4 expanded?

Using the binomial theorem with n=4: (x+y)^4 = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4. The coefficients 1, 4, 6, 4, 1 are the fifth row of Pascal's triangle (row index 4). Each term has total degree 4 (the exponents of x and y sum to 4 in every term).

How do Pascal's triangle and the binomial theorem relate?

Each row of Pascal's triangle (starting from row 0 at the top) gives the binomial coefficients for expanding (a+b) to the corresponding power. Row 4 is 1, 4, 6, 4, 1, which are the coefficients of (a+b)^4. Each entry is the sum of the two entries directly above it in the previous row.

Binomial Theorem Calculator

What is the binomial theorem?

A formula for expanding (a + b)^n as a sum of terms (n choose k) a^(n-k) b^k.

What are the coefficients?

The binomial coefficients n choose k, which are exactly row n of Pascal’s triangle.

How many terms are there?

n plus 1 terms, from k equals 0 up to k equals n.

Birth Weight Percentile Calculator

What does birth weight percentile mean?

A birth weight percentile tells you how your baby's weight compares to other babies born at the same gestational age and of the same sex. A baby at the 50th percentile weighs exactly the median for their gestational age. A baby at the 10th percentile weighs more than 10% of babies at the same gestational age. Percentile bands below 10 (small for gestational age, SGA) or above 90 (large for gestational age, LGA) may be reviewed by your midwife or doctor, though many babies in these ranges are perfectly healthy.

Which reference chart does this calculator use?

This calculator uses the INTERGROWTH-21st international newborn size standards published by researchers at the University of Oxford (Villar et al., The Lancet, 2014). These are the most widely used international standards for birth weight by gestational age and sex, covering 24 to 42 completed weeks. The method uses a Box-Cox z-score transformation with published median (mu), coefficient of variation (sigma), and skewness (lambda) parameters, converted to a percentile via the standard normal distribution.

What is considered a normal birth weight in New Zealand?

For full-term babies (37 to 42 weeks gestation), a normal birth weight in New Zealand is generally between 2,500 g and 4,500 g, with the average around 3,300 to 3,450 g for males and slightly less for females. The percentile system accounts for gestational age, making it a more accurate measure than absolute weight alone. A premature baby at 28 weeks would be expected to weigh around 1,100 to 1,300 g at the median, while a term baby at 40 weeks would be expected to weigh around 3,300 to 3,450 g.

Bug and Rivet Paradox Calculator

What is the bug and rivet paradox?

A rivet with a shaft shorter than a hole is deep is fired at the hole at near light speed, with a bug sitting at the bottom. In the hole's frame the moving shaft is length contracted and looks too short to reach the bug, but in the rivet's frame the hole is contracted and too shallow, so the shaft reaches the bug. Both frames must agree the bug is crushed.

How is length contraction calculated?

A moving object's length is its rest length divided by the Lorentz factor gamma, where gamma is one divided by the square root of one minus the speed squared over the speed of light squared. At 0.9 times light speed gamma is about 2.29, so lengths along the direction of motion shrink to roughly 44 percent of their rest value.

How is the paradox resolved?

Rigidity is not instant. When the rivet head slams to a stop, that news can only travel down the shaft at the speed of light or slower, so the tip keeps moving after the head stops and drives deeper than the rest length suggests. Working through the timing in either frame, the tip reaches the bug, so both frames agree the bug is crushed.

Coin Flip Probability Calculator

What is the probability of getting exactly 5 heads in 10 coin flips?

The probability of getting exactly 5 heads in 10 fair coin flips is approximately 24.61%. This is calculated using the binomial formula: P = C(10,5) x 0.5^10 = 252 / 1024 = 0.2461. Although 5 heads out of 10 is the single most likely outcome, it still only happens about one in four times because there are so many other possible results.

How do you calculate the probability of getting at least k heads in n flips?

To find the probability of at least k heads, sum the binomial probabilities for every outcome from k heads up to n heads. For example, the probability of at least 7 heads in 10 flips equals P(7) + P(8) + P(9) + P(10), where each P(x) = C(10,x) x 0.5^10. This gives approximately 17.19%. The at-most probability works the same way, summing from 0 heads up to k heads.

Does the coin need to be fair for this calculator to apply?

No. This calculator lets you set any probability of heads between 0% and 100%, so you can model biased coins as well as fair ones. A fair coin has a 50% probability of heads on each flip. A biased coin might have a 60% or 30% probability. The binomial formula P(X=k) = C(n,k) x p^k x (1-p)^(n-k) works for any fixed probability p as long as each flip is independent.

Combination Calculator (nCr)

What is a combination?

A combination is a selection where order does not matter. nCr counts how many different groups of r items you can choose from a set of n. Picking any three people from a group of ten for a committee is a combination, because the group is the same regardless of the order in which the members were selected.

What is the nCr formula?

nCr equals n factorial divided by r factorial times the factorial of n minus r. It equals nPr (the permutation count) divided by r factorial, which removes the different orderings of the same group. For example, 5C3 is 5P3 divided by 3 factorial, which is 60 divided by 6, giving 10.

How is nCr different from nPr?

nPr counts ordered arrangements where the sequence matters, while nCr counts unordered selections where only the membership of the group matters. nPr is always greater than or equal to nCr for the same n and r, and the ratio nPr divided by nCr always equals r factorial.

Combination (nCr) Calculator

What is the difference between a combination and a permutation?

A combination counts the number of ways to choose r items from n items when order does not matter. A permutation counts the number of ways to arrange r items from n when order does matter. For example, choosing 3 players from a squad of 10 to start a game is a combination (C(10,3) = 120) because the order in which you pick them makes no difference to the result. Lining those same 3 players up in batting order is a permutation (P(10,3) = 720) because the first, second, and third positions are distinct. The relationship is nCr = nPr / r!.

How do you calculate nCr by hand?

Use the formula C(n,r) = n! / (r! x (n-r)!). For C(10,3): the numerator is 10! = 3,628,800; the denominator is 3! x 7! = 6 x 5,040 = 30,240. Dividing gives 120. In practice you can simplify by cancelling the larger factorial first: C(10,3) = (10 x 9 x 8) / (3 x 2 x 1) = 720 / 6 = 120. Always start from the top with r terms and divide by r! to keep the numbers manageable.

Why does C(n,r) equal C(n, n-r)?

Choosing r items to include is mathematically identical to choosing (n-r) items to exclude. For example, C(10,3) = 120 and C(10,7) = 120 because picking 3 items out of 10 leaves 7, and picking which 7 to leave out is the same choice. This symmetry is expressed by the formula because swapping r and (n-r) in n! / (r! x (n-r)!) gives the same denominator. It is also visible in Pascal's Triangle, where each row is symmetric.

Combination Without Repetition Calculator

What is the formula for combinations without repetition?

The formula is C(n, r) = n! / (r! x (n - r)!), where n is the total number of items to choose from and r is the number of items you select. The exclamation mark means factorial, so 5! = 5 x 4 x 3 x 2 x 1 = 120. This formula counts how many unique groups of r items can be formed from n distinct items when order does not matter and each item can only be selected once.

What is the difference between combinations with and without repetition?

Combinations without repetition (this calculator) means each item from the total set can appear at most once in any selection. For example, choosing 3 cards from a standard deck without putting any back. The formula is C(n, r) = n! / (r! x (n - r)!). Combinations with repetition allow an item to be chosen more than once. For example, choosing 3 scoops of ice cream from 5 flavours where you can pick the same flavour multiple times. That formula is C(n + r - 1, r) = (n + r - 1)! / (r! x (n - 1)!).

How does a combination differ from a permutation?

A combination counts selections where order does not matter, while a permutation counts arrangements where order does matter. For example, choosing the team {Alice, Bob, Carol} is the same combination regardless of which name you list first. But the arrangement Alice-Bob-Carol is a different permutation from Bob-Alice-Carol. For any given combination of r items from n, there are r! corresponding permutations. So the number of combinations C(n, r) always equals the number of permutations P(n, r) divided by r!.

Combinations with Repetition Calculator NZ

What are combinations with repetition?

They count the ways to choose r items from n types when you can pick the same type more than once and order does not matter. Choosing a dozen doughnuts from six flavours, taking several of some, is a classic example. The count uses the multiset formula.

What is the formula?

It is the binomial coefficient of n plus r minus 1, choose r, where n is the number of types and r the number chosen. This follows from the stars and bars method, which arranges the r choices as stars divided into n groups by n minus 1 bars.

How is this different from ordinary combinations?

Ordinary combinations do not allow repeats, so you cannot pick the same item twice. Combinations with repetition do allow repeats, giving a larger count for the same n and r. Both ignore order, unlike permutations, where order matters.

Conditional Probability Calculator

What is conditional probability?

Conditional probability is the probability of an event A occurring given that event B has already occurred. It is written P(A|B) and calculated as P(A and B) divided by P(B). For example, if a medical test is positive (B), the conditional probability P(A|B) tells you the chance the patient actually has the disease (A), taking into account how common the disease is and how accurate the test is.

What is Bayes' theorem and how is it used here?

Bayes' theorem provides a formula for computing conditional probability from three known values: the prior probability P(A), the true positive rate P(B|A), and the false positive rate P(B|not A). The formula is: P(A|B) = P(B|A) x P(A) / [P(B|A) x P(A) + P(B|not A) x P(not A)]. The denominator is the total probability of B, often written P(B). Bayes' theorem is widely used in medicine, spam filtering, machine learning, and forensics.

Why does a positive test result not always mean you have the condition?

Even a highly accurate test can produce a misleading result when the underlying condition is rare. If only 1 in 1,000 people have a disease, a test with a 99% true positive rate and a 5% false positive rate will still generate about 50 false positives for every true positive in that population. This is why doctors consider the prevalence of a condition alongside test accuracy when interpreting results. The conditional probability formula captures this relationship precisely.

Confidence Interval Calculator

What does a 95% confidence interval mean?

A 95% confidence interval means that if you repeated your sampling procedure many times and built a confidence interval each time, about 95% of those intervals would contain the true population mean. It does not mean there is a 95% chance the true mean lies in your specific interval. The interval gives you a range of plausible values for the true mean based on your sample.

What is the margin of error?

The margin of error is half the width of the confidence interval, equal to the critical z-value multiplied by the standard error. A larger sample size reduces the standard error and therefore narrows the margin of error, giving a more precise estimate of the true mean.

When should I use a z-interval versus a t-interval?

A z-interval is appropriate when the population standard deviation is known or when the sample size is large (typically n greater than 30), because the Central Limit Theorem ensures the sampling distribution is approximately normal. A t-interval is more appropriate for small samples with an unknown population standard deviation, using the t-distribution instead of the z-distribution.

Confidence Interval Calculator

What does a 95 percent confidence interval mean?

It means that if you repeated the sampling many times and built an interval each time, about 95 percent of those intervals would contain the true population mean. It does not mean there is a 95 percent chance the one interval you have contains the mean. It is a statement about the long run method.

Why does a bigger sample give a narrower interval?

The standard error is the standard deviation divided by the square root of the sample size. As the sample grows, that denominator grows, so the standard error and the margin of error both shrink. The result is a tighter, more precise interval.

When should I use a t interval instead?

Use a t distribution interval when the sample is small, roughly under 30, and the population standard deviation is unknown. The t multiplier is larger than the z multiplier for small samples, which widens the interval to account for the extra uncertainty. For large samples the two methods give very similar results.

Confidence Interval for a Proportion Calculator NZ

What is a confidence interval for a proportion?

It is a range, built from a sample, within which the true population proportion is likely to lie. It equals the sample proportion plus or minus a margin of error. A 95 percent interval means that, over many samples, about 95 percent of such intervals would contain the true proportion.

What is the margin of error?

The margin of error is the half-width of the confidence interval, the amount added to and subtracted from the sample proportion. It depends on the sample size and the confidence level. The figure quoted alongside opinion polls is the margin of error at the 95 percent level.

How can I make the interval narrower?

Increase the sample size, which reduces the standard error and tightens the interval, or accept a lower confidence level. A larger sample is the main lever: quadrupling the sample roughly halves the margin of error, because it depends on the square root of the sample size.

Confidence Level Calculator

What is the critical z for 95 percent?

1.96 for a two-tailed interval, and about 1.645 one-tailed.

What is alpha?

The significance level, equal to 1 minus the confidence level. For 95 percent confidence, alpha is 0.05.

Two-tailed or one-tailed?

Use two-tailed for a confidence interval or a non-directional test, one-tailed for a directional test.

Correlation Coefficient Calculator (Pearson r)

What does Pearson r measure?

Pearson r measures the strength and direction of the linear relationship between two continuous variables. A value of +1 means a perfect positive linear relationship, -1 means a perfect negative linear relationship, and 0 means no linear association. It does not detect curved or non-linear relationships.

What is a strong correlation?

Common thresholds treat r values from 0 to 0.19 as negligible, 0.20 to 0.39 as weak, 0.40 to 0.59 as moderate, 0.60 to 0.79 as strong, and 0.80 to 1.00 as very strong. These are conventions, and what counts as strong depends on the field of study.

What is r squared in correlation?

r squared (the coefficient of determination) tells you what proportion of the variance in one variable is explained by the linear relationship with the other. An r of 0.7746 gives r squared of 0.60, meaning 60% of the variance in y is explained by a linear relationship with x.

Dice Probability Calculator

What is the probability of rolling a 7 with two dice?

With two standard six-sided dice, a sum of 7 can be made in 6 ways: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). The total number of outcomes is 6 x 6 = 36. So the probability is 6/36 = 1/6, which is approximately 16.67%. Seven is the most likely single sum when rolling two dice.

How do you calculate dice probabilities?

For multiple identical dice, you can build the distribution by convolution. Start with a uniform distribution for one die (each face has probability 1/n). For each additional die, convolve the current distribution with the single-die distribution. The resulting counts for each possible sum, divided by the total number of outcomes (n raised to the power of the number of dice), give the probabilities.

Why is 7 the most common sum for two six-sided dice?

The sum of 7 can be formed in the most ways: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) = 6 combinations out of 36. Sums closer to the middle of the possible range always have more combinations than extreme sums, which is why the distribution of two-dice totals is roughly bell-shaped and peaks at 7.

Exponential Distribution Calculator

What is the exponential distribution?

The exponential distribution models the time between events in a Poisson process, where events occur continuously and independently at a constant average rate lambda. It is used to model waiting times, such as the time until the next customer arrives, the time until a component fails, or the time until a radioactive decay occurs. The probability density function is f(x) = lambda times e to the power minus lambda x, for x greater than or equal to zero.

What is the mean of the exponential distribution?

The mean of the exponential distribution is 1 divided by lambda. If events occur at rate lambda = 0.5 per unit time (meaning on average 0.5 events per unit), the mean time between events is 1/0.5 = 2 time units. The median is ln(2) divided by lambda, which is approximately 0.693 divided by lambda.

What is the memoryless property of the exponential distribution?

The memoryless property means that the probability of waiting an additional time t does not depend on how long you have already waited. Formally, P(X greater than s + t | X greater than s) = P(X greater than t) for all s, t greater than zero. The exponential distribution is the only continuous distribution with this property.

Frequency Distribution Calculator

What is a frequency distribution?

A frequency distribution is a table that shows how many times each unique value (or range of values) appears in a data set. It gives a quick picture of which values are common and which are rare. The relative frequency expresses each count as a percentage of the total, making it easy to compare distributions of different sizes.

What is relative frequency?

Relative frequency is the count for a value divided by the total number of observations, expressed as a percentage. If a value appears 3 times in a data set of 9 observations, its relative frequency is 3/9 = 33.33%. Relative frequencies always sum to 100%, making them useful when comparing groups of different sizes.

What is cumulative frequency?

Cumulative frequency is the running total of counts as you move through the sorted values from smallest to largest. After listing the first three unique values in order, the cumulative frequency at the third value is the total number of observations with a value less than or equal to that third value. Cumulative frequency is used to find percentiles and to build ogive charts.

Geometric Distribution Calculator

What is the geometric distribution?

The geometric distribution models the number of trials needed to get the first success in a sequence of independent Bernoulli trials, each with the same probability of success p. The probability that the first success occurs on the k-th trial is P(X = k) = (1 minus p) to the power (k minus 1) times p, for k = 1, 2, 3 and so on.

What is the mean of the geometric distribution?

The mean (expected value) of the geometric distribution is 1 divided by p. So if each trial has a 30 percent chance of success, you expect on average 1 divided by 0.3 = 3.33 trials until the first success. The variance is (1 minus p) divided by p squared.

What is the cumulative probability P(X less than or equal to k) for the geometric distribution?

The cumulative distribution function (CDF) for the geometric distribution is P(X less than or equal to k) = 1 minus (1 minus p) to the power k. This gives the probability that the first success occurs on or before the k-th trial.

Grouped Standard Deviation Calculator NZ

What is grouped data?

Grouped data is summarised into classes or categories with a frequency for each, rather than listed as individual values. A frequency table of test scores in bands is a common example. To find statistics from it, the midpoint of each class represents its values.

How do you find the standard deviation of grouped data?

Use each class midpoint, weighted by its frequency. Find the mean as the sum of midpoint times frequency over the total frequency, then take the frequency-weighted squared deviations from the mean, divide by one less than the count for a sample, and take the square root.

Why is it an approximation?

Because the actual values within each class are unknown, the midpoint stands in for all of them. This is a good approximation when classes are narrow and values are spread evenly within them, but it can differ slightly from the standard deviation you would get from the raw data.

Height Percentile Calculator

What does height percentile mean?

A height percentile tells you what percentage of people of the same age and sex are shorter than you. For example, the 75th percentile means 75% of people in that reference group are shorter and 25% are taller. The 50th percentile is exactly median height. Percentiles above 85 are considered tall for age and sex; below 15 is considered short. Neither extreme is necessarily cause for concern without other clinical signs.

What growth charts are used in this calculator?

For children aged 2 to 17, this calculator uses mean and standard deviation values derived from the CDC 2000 growth reference charts, which are widely used by health professionals. For adults aged 18 and over, it uses approximate population reference values. The percentile is calculated by converting height to a z-score (z = (height - mean) / SD) and then converting the z-score to a percentile using the standard normal cumulative distribution function.

What height percentile is considered normal?

The 5th to 95th percentile range is generally considered within the normal range for height. A child below the 3rd or 5th percentile or above the 97th or 95th percentile may be referred for further assessment, particularly if this represents a change from their previous growth pattern. For adults, percentile is used for comparison rather than clinical diagnosis. If you are concerned about a child's growth, speak with a paediatrician or GP.

Hypergeometric Distribution Calculator

When do I use the hypergeometric distribution?

For sampling without replacement from a finite population, where each draw changes the remaining proportions.

How is it different from the binomial?

The binomial assumes a constant success probability (sampling with replacement). The hypergeometric corrects for the shrinking population.

What is the expected number of successes?

n times K over N, the sample size times the proportion of successes in the population.

Linear Regression Analysis Calculator

What is the Linear Regression Analysis Calculator?

Calculate linear regression for any set of (x, y) data pairs. Get the line of best fit, slope, intercept, Pearson's r, and a scatter plot. Includes prediction table and residual analysis.

Is the Linear Regression Analysis Calculator free to use?

Yes. The Linear Regression Analysis Calculator is free to use on Calculate.co.nz, with no sign-up, paywall or account required.

How does the Linear Regression Analysis Calculator work?

It fits a line of best fit to your (x, y) data pairs using the least squares method, then reports the slope, intercept, Pearson's r, predicted values and residuals. It is a general statistics tool with no jurisdiction-specific rules or rates.

Linear Regression Calculator

What do the slope and intercept mean?

The slope is how much the predicted y changes for each one unit increase in x. The intercept is the predicted y when x is zero. Together they define the line y equals slope times x plus intercept.

How is the line chosen?

It uses least squares, which picks the slope and intercept that minimise the total of the squared vertical distances between the points and the line. This gives the slope as the sum of the cross products divided by the sum of squared x deviations. The intercept then follows from the means.

What does r-squared tell me?

It is the proportion of the variation in y that the line explains, between zero and one. A value of 0.6 means the line accounts for about 60 percent of the variation. Higher values mean a closer fit, but always check a plot too.

Logistic Regression Calculator

What is the Logistic Regression Calculator?

Calculate logistic regression for binary outcome data. Enter your observed data to get regression coefficients, predicted probabilities, and model fit statistics. Includes worked tutorial.

Is the Logistic Regression Calculator free to use?

Yes. The Logistic Regression Calculator is free to use on Calculate.co.nz, with no sign-up, paywall or account required.

How does this logistic regression calculator work?

It fits a weighted linear regression of the natural logarithm of the observed odds against your independent variable X, using the number of observations in each category as weights. This gives an equation of the form ln(odds) = intercept + slope * X, which is converted to a probability with probability = odds / (1 + odds).

Mean, Median and Mode Calculator

What is the difference between mean and median?

The mean is the arithmetic average, found by adding all values and dividing by the count. The median is the middle value when the data is sorted in order. In a symmetrical data set they are close together, but in a skewed set the median is usually the better measure of centre because it is not pulled by extreme values the way the mean is.

What does it mean when there is no mode?

If every value in the data set appears exactly once, there is no mode because no value occurs more frequently than any other. Some data sets have more than one mode (bimodal or multimodal) when two or more values share the highest frequency.

How is the range calculated?

The range is the difference between the largest and smallest values in the data set. It is the simplest measure of spread but is sensitive to outliers because a single extreme value can make the range much larger than the typical spread of the data.

Mean Median Mode Range Calculator

What is the difference between mean, median and mode?

The mean is the average, the sum divided by the count. The median is the middle value when the numbers are sorted. The mode is the value that appears most often. Together they describe the centre of a data set.

What is the range?

The range is the difference between the largest and smallest values, a simple measure of how spread out the data is.

Can there be more than one mode?

Yes. If two or more values tie for the most frequent, the data set has more than one mode. If every value appears once, there is no mode. The calculator lists all modes it finds.

Median Calculator

What is the median?

The median is the middle value when the numbers are sorted in order. If there is an even count, it is the average of the two middle values. Unlike the mean, the median is not pulled by very high or low outliers.

When should I use the median instead of the mean?

Use the median when the data has outliers or is skewed, such as house prices or incomes. The median better reflects a typical value, which is why it is widely reported for those figures.

What is the mode?

The mode is the value that appears most often. A set can have one mode, several, or none if every value is unique. The calculator shows the mode along with the median and mean.

Negative Binomial Distribution Calculator

What is the Negative Binomial Distribution Calculator?

Calculate negative binomial distribution probabilities. Enter the number of successes and probability of success to see the PDF, CDF, mean, variance, and interactive probability charts.

Is the Negative Binomial Distribution Calculator free to use?

Yes. The Negative Binomial Distribution Calculator is free to use on Calculate.co.nz, with no sign-up, paywall or account required.

What does the negative binomial distribution model?

It models the number of failures before a set number of successes in a run of independent trials that each share the same probability of success. The calculator uses the standard probability mass and cumulative formulas, which are identical everywhere, and works out the mean and variance for you. Everything runs in your browser, so your data stays on your device.

Normal Distribution Calculator

What is the normal distribution?

The normal distribution (also called the Gaussian distribution or bell curve) is a symmetrical, continuous probability distribution where most observations cluster around the mean and frequencies taper off equally in both directions. It is described by two parameters: the mean (which sets the centre) and the standard deviation (which sets the spread). Many natural measurements such as heights, weights and test scores follow an approximately normal distribution.

What does P(X less than x) mean?

P(X less than x) is the cumulative probability that a randomly chosen value from the distribution will be less than the value x you specified. It is the area under the normal curve to the left of x. For example, if the mean is 70 and SD is 10, then P(X less than 80) tells you the probability that a random observation falls below 80.

How is the normal CDF calculated?

The cumulative distribution function (CDF) of the normal distribution does not have a simple closed-form expression. In practice it is computed using numerical approximations of the error function or polynomial approximations of the standard normal CDF. This calculator uses an accurate rational approximation valid across the full range of z-scores.

One-Sample t-Test Calculator

What is a one-sample t-test?

It tests whether the mean of a single sample differs significantly from a specified value. The t statistic compares the gap between the sample mean and that value against the variability in the data, telling you whether the difference is likely real or due to chance.

How are the degrees of freedom found?

For a one-sample t-test, the degrees of freedom are the sample size minus one. You use them, along with your significance level, to find the critical t value or the p-value that determines whether the result is statistically significant.

What assumption does it make?

It assumes the sample is drawn from a roughly normal distribution. This matters most for small samples; with larger samples the test is fairly robust to departures from normality thanks to the central limit theorem. For very non-normal small samples, a rank-based test may be better.

One-Way ANOVA Calculator

What does the F statistic tell me?

It is the between groups mean square divided by the within groups mean square. A larger F means the group means are far apart relative to the variation inside the groups. Compare it with a critical value from an F table to judge significance.

What are the degrees of freedom?

Between groups degrees of freedom equal the number of groups minus one. Within groups degrees of freedom equal the total number of values minus the number of groups. You need both to look up the critical F value.

What if ANOVA is significant?

A significant result tells you at least one group mean differs, but not which one. Follow up with a post hoc test such as Tukey to compare specific pairs. Always report the group means alongside the F statistic.

Paired t-Test Calculator

What data do I enter?

Enter the difference for each pair, separated by commas. For a before and after study, subtract the before value from the after value for each person. The calculator does the rest from that single list.

How is the t statistic worked out?

It is the mean of the differences divided by the standard error of those differences. The standard error is the sample standard deviation of the differences divided by the square root of the number of pairs. A value far from zero suggests a real effect.

What are the degrees of freedom?

For a paired t-test the degrees of freedom equal the number of pairs minus one. With five pairs you have four degrees of freedom. You use this number to find the critical value in a t distribution table.

Pearson Correlation Calculator

What does r mean?

It measures the strength and direction of a straight line relationship between two variables. Values near plus one or minus one show a strong relationship, while values near zero show a weak one. The sign tells you whether the variables move in the same or opposite directions.

What is r-squared?

It is r multiplied by itself, and it tells you the proportion of the variation in one variable that is explained by the other. An r of 0.7746 gives an r-squared of about 0.6, so roughly 60 percent of the variation is shared. It is always between zero and one.

Does correlation prove causation?

No. A high correlation shows that two variables move together but not that one causes the other. There may be a third factor driving both, or the link may be coincidence, so treat correlation as a starting point for investigation.

Percentile Calculator

How is a percentile calculated?

To find the p-th percentile, sort the data, then compute the rank position as L = (p / 100) times (n + 1) where n is the number of values. If L is a whole number, the percentile is the value at that position. If not, linear interpolation blends the two neighbouring values in proportion to the fractional part of L.

What is the difference between a percentile and a quartile?

Quartiles are just specific percentiles: Q1 is the 25th percentile, Q2 (the median) is the 50th, and Q3 is the 75th. Any percentile from 1 to 99 can be calculated; quartiles are the three that divide the sorted data into four equal groups.

Why does the 90th percentile sometimes fall between two values?

When the rank position L falls between two data points, linear interpolation blends them. For a 10-value data set, P90 has position 0.90 times 11 = 9.9, so the result is the 9th value plus 0.9 times the difference between the 9th and 10th values.

Permutation Calculator (nPr)

What is a permutation?

A permutation is an arrangement where order matters. nPr counts how many different ways you can arrange r items chosen from a set of n. Awarding gold, silver, and bronze among ten athletes is a permutation problem because changing the order of the same three athletes produces a different outcome.

What is the nPr formula?

nPr equals n factorial divided by the factorial of n minus r, written n! divided by (n minus r)!. In practice it is the descending product of r terms starting at n: n times (n minus 1) times (n minus 2) down to (n minus r plus 1). For example, 5P3 is 5 times 4 times 3, which equals 60.

When should I use a permutation instead of a combination?

Use a permutation whenever the order of the chosen items matters and produces a distinct outcome. Arranging books on a shelf, ranking competitors, assigning different roles to people, and ordering a sequence of tasks are all permutation problems. If the order does not matter and you just want to know which items are selected, use a combination.

Permutation (nPr) Calculator

What is a permutation and how is nPr calculated?

A permutation is an ordered arrangement of items chosen from a larger set. The order in which items are selected matters: ABC and BAC are counted as two different permutations. The formula is nPr = n! / (n - r)!, where n is the total number of items in the set and r is how many you are choosing. For example, 10P3 = 10! / 7! = 10 x 9 x 8 = 720.

What is the difference between a permutation and a combination?

The difference is whether order matters. In a permutation (nPr), the sequence of selection matters, so choosing A then B is different from choosing B then A. In a combination (nCr), only the group matters and order is ignored. As a result, nPr is always greater than or equal to nCr for the same n and r. The relationship is nPr = nCr x r!. For example, 5P2 = 20 while 5C2 = 10.

Can r be greater than n in a permutation?

No. You cannot choose more items than are available, so r must be less than or equal to n. If r is greater than n the permutation is undefined (there are zero ways to arrange more items than exist). The special case of 0P0 and nP0 both equal 1, because there is exactly one way to arrange zero items: do nothing.

Permutations and Combinations Calculator

What is the difference between a permutation and a combination?

A permutation counts ordered arrangements, so picking A then B is different from B then A. A combination counts unordered selections, so A and B is the same group either way. Use permutations when order matters and combinations when it does not.

Why is nPr always larger than nCr?

Every combination of r items can be arranged in r factorial different orders, and each of those counts as a separate permutation. So the number of permutations is the number of combinations times r factorial. That makes nPr at least as large as nCr for the same n and r.

What happens if r is bigger than n?

You cannot choose more items than you have, so both nPr and nCr are zero. The calculator treats r greater than n as having no valid arrangements. Make sure r is between zero and n for a meaningful result.

Poisson Confidence Interval Calculator NZ

What is a Poisson confidence interval?

It is a range, based on an observed count of events, within which the true average rate is likely to lie. Because Poisson counts have a variance equal to their mean, the interval can be built directly from the observed count using the square root as the standard error.

When is the normal approximation reliable?

For counts that are not too small, roughly 20 or more, the normal approximation used here works well. For very small counts, the distribution is skewed and exact methods based on the chi-square distribution give more accurate intervals, so treat small-count results with caution.

Why does the interval narrow for larger counts?

The margin of error depends on the square root of the count, while the count itself grows linearly, so the margin shrinks relative to the estimate as counts increase. This means larger counts give proportionally tighter, more precise intervals around the true rate.

Poisson Distribution Calculator

What is the Poisson Distribution Calculator?

Calculate Poisson distribution probabilities for any expected event rate. Enter the mean number of occurrences to get the PDF, CDF, mean, variance, and interactive probability charts.

Is the Poisson Distribution Calculator free to use?

Yes. The Poisson Distribution Calculator is free to use on Calculate.co.nz, with no sign-up, paywall or account required.

What is the difference between the PMF and CDF in a Poisson distribution?

The PMF (probability mass function) gives the probability of exactly k events occurring, while the CDF (cumulative distribution function) gives the probability of k or fewer events. In a Poisson distribution the mean and the variance are both equal to the rate parameter lambda.

Poisson Probability Calculator

What is lambda in the Poisson distribution?

Lambda is the average number of events you expect in the chosen interval, such as an hour, a day or a square metre. It is both the mean and the variance of the distribution. The rate must stay roughly constant for the Poisson model to fit.

How do I find the chance of more than k events?

Work out the cumulative probability of k or fewer events, then subtract it from one. For example the chance of more than 2 events is one minus the chance of 0, 1 or 2 events. This tail probability is often what you need for planning capacity.

When does the Poisson model apply?

It applies when events occur independently, one at a time, at a steady average rate over time or space. Examples include call arrivals, equipment faults and rare events in a fixed region. If events cluster or the rate changes a lot, a different model is more suitable.

Poker Odds Calculator

What are outs in poker?

Outs are the unseen cards that improve your hand to a likely winner. A flush draw after the flop has 9 outs (13 of the suit minus the 4 you can see), an open-ended straight draw has 8, and a gutshot has 4. Only count cards that genuinely win the hand, and discount outs that could complete a better hand for your opponent.

How does the rule of 4 and 2 work?

Multiply your outs by 4 on the flop (two cards to come) or by 2 on the turn (one card to come) for a quick percentage estimate. Nine outs on the flop is roughly 36%, close to the exact 35.0%. The shortcut drifts high once you get past about 12 outs, but it is accurate enough for table decisions.

What pot odds do I need to call?

Divide the amount you must call by the pot size after your call. Calling $20 into a $100 pot means risking 20 to win 120, so you need at least 20/120, about 16.7% equity, for the call to break even. If your chance of hitting is higher than that number, calling shows a long-run profit.

Probability Calculator

What is the Probability Calculator NZ?

Comprehensive probability calculator covering all types: basic probability, combined events, conditional probability, Bayes theorem, permutations, combinations, binomial distribution, normal distribution, and Poisson distribution. Free, with worked examples.

Is the Probability Calculator NZ free to use?

Yes. The Probability Calculator NZ is free to use on Calculate.co.nz, with no sign-up, paywall or account required.

Is the Probability Calculator NZ made for New Zealand?

Yes. It is hosted on Calculate.co.nz for New Zealand students. The calculations use standard probability formulas that work the same way anywhere, so no country-specific rules or rates are involved.

Quartile and IQR Calculator NZ

What is the interquartile range?

The interquartile range, or IQR, is the difference between the third quartile (75th percentile) and the first quartile (25th percentile). It measures the spread of the middle half of the data and is robust against extreme values, unlike the full range.

How do I find outliers with the IQR?

The standard rule flags any value more than 1.5 times the IQR below Q1 or above Q3 as a potential outlier. These are the outlier fences, and they are the same boundaries used for the whiskers of a box plot. This calculator computes them for you.

How are quartiles calculated?

The data is sorted and the quartiles are found as the 25th, 50th and 75th percentiles, using linear interpolation between the nearest values. This is the standard method used in most statistical software, so the results match what you would get in a spreadsheet.

Quartile and IQR Calculator

What is the interquartile range?

The interquartile range (IQR) is the difference between the third quartile (Q3) and the first quartile (Q1). It represents the spread of the middle 50% of the data and is a robust measure of variability because it is not affected by extreme values or outliers the way the range is.

How are outlier fences calculated from the IQR?

The standard Tukey fences set the lower fence at Q1 minus 1.5 times IQR and the upper fence at Q3 plus 1.5 times IQR. Values beyond these fences are considered potential outliers. Some analyses use a factor of 3 instead of 1.5 for more extreme outliers.

What is the difference between range and IQR?

The range is simply the largest value minus the smallest. It is sensitive to a single extreme data point. The IQR measures the spread of the central half of the data (from Q1 to Q3) and is therefore much more resistant to the influence of outliers.

Sample Size Calculator

Why do I use p = 0.5 when I am not sure of the proportion?

A proportion of 0.5 produces the maximum possible variance for a binomial outcome, which gives the most conservative (largest) sample size estimate. Using 0.5 when you have no prior estimate ensures your sample will be large enough no matter what the true proportion turns out to be. If you have a reasonable prior estimate of the proportion, using it will generally give a smaller required sample size.

What happens if I increase the confidence level?

Increasing the confidence level from 95% to 99% raises the critical z-value from 1.96 to 2.576, which increases the required sample size. You need more data to be more certain your interval captures the true value. Conversely, accepting a lower confidence level or a larger margin of error reduces the required sample size.

Does population size affect the required sample size?

For large populations (typically above 10,000), the required sample size is nearly independent of the population size, which is why national polls of millions of people can be conducted with samples of just a few hundred to a few thousand. For smaller populations you can apply a finite population correction to reduce the required sample size.

Sample Size Calculator for a Proportion

How do I calculate the sample size for a proportion?

Use the z-value for your confidence level squared, times the estimated proportion times one minus it, divided by the margin of error squared. This gives the minimum sample needed for your target precision. A finite population correction reduces it for small populations.

What proportion should I assume if I don't know it?

Use 50 percent. The product of the proportion and one minus it is largest at 50 percent, so this gives the biggest, most conservative sample size. Choosing it ensures your sample is large enough whatever the true proportion turns out to be.

When does the population size matter?

When you are sampling a meaningful fraction of a small population, the finite population correction noticeably reduces the required sample. For large populations, say tens of thousands or more, the correction makes little difference and the base formula suffices.

Spearman's Rank Correlation Calculator

What is Spearman's rank correlation?

It measures how well the relationship between two variables follows a monotonic, consistently rising or falling, pattern, using the ranks of the data rather than the raw values. This makes it robust to outliers and suitable for skewed or ordinal data.

How is it different from Pearson's correlation?

Pearson measures linear association on the actual values and assumes roughly normal data. Spearman works on ranks, so it captures any monotonic relationship, not just straight lines, and is far less affected by outliers or non-normal distributions.

When should I use Spearman's rho?

Use it when your data is ordinal (rankings or ratings), skewed, contains outliers, or when the relationship is monotonic but not linear. In those cases it gives a more reliable measure of association than Pearson's correlation.

Standard Deviation Calculator

What is standard deviation?

Standard deviation measures how spread out a set of numbers is around the mean. A small value means the numbers are clustered close together, and a large value means they are widely spread. It is the square root of the variance.

What is the difference between population and sample?

Use population standard deviation when your numbers are the whole group you care about. Use sample when they are a sample drawn from a larger group; it divides by one less than the count to correct for the smaller data set. The calculator shows both.

What is variance?

Variance is the average of the squared differences from the mean. Standard deviation is its square root, which brings the figure back to the original units, making it easier to interpret.

T-Test Calculator

What is the T-Test Calculator?

Calculate t-test statistics for one-sample and two-sample hypothesis tests. Enter your data to get the t-statistic, degrees of freedom, p-value, and whether to reject the null hypothesis.

Is the T-Test Calculator free to use?

Yes. The T-Test Calculator is free to use on Calculate.co.nz, with no sign-up, paywall or account required.

Is the T-Test Calculator made for New Zealand?

Yes. It is hosted on Calculate.co.nz for New Zealand students and researchers. The calculations use standard statistical formulas that work the same way anywhere, so no country-specific rules or rates are involved.

Two-Sample t-Test Calculator NZ

What is a two-sample t-test?

It compares the means of two independent groups to test whether they differ more than chance would explain. The t statistic measures the gap between the means relative to the variability within the groups. A large magnitude suggests a real difference.

What is Welch's t-test?

Welch's t-test is a version that does not assume the two groups have equal variances, making it more robust and the recommended default. It adjusts the degrees of freedom accordingly, which is why they are usually not a whole number. This calculator uses Welch's method.

How do I judge significance?

Compare the magnitude of the t statistic against the critical value from a t table at your significance level and the calculated degrees of freedom, or find the p-value. If the t magnitude exceeds the critical value, the two means differ significantly.

Variance Calculator

What is the difference between population and sample variance?

Population variance divides the sum of squared deviations by N (the total count), and is used when your data set contains every member of the group you are studying. Sample variance divides by N minus 1, which is Bessel's correction, and is used when your data is a sample drawn from a larger population. Using N minus 1 produces an unbiased estimate of the true population variance.

What does standard deviation tell you?

Standard deviation is the square root of variance and is expressed in the same units as your original data, which makes it easier to interpret. A low standard deviation means the values are clustered tightly around the mean; a high standard deviation means they are spread widely. About 68% of values in a normal distribution fall within one standard deviation of the mean.

Can variance be negative?

No. Variance is calculated from squared deviations, so it is always zero or positive. A variance of zero means all values in the data set are identical. Standard deviation, as the square root of a non-negative number, is also always zero or positive.

Z-Score Calculator

What is a z-score?

A z-score, also called a standard score, measures how many standard deviations a value is above or below the mean of a distribution. A z-score of 0 means the value equals the mean exactly. A z-score of 1 means the value is one standard deviation above the mean, and a z-score of minus 1 means it is one standard deviation below the mean.

How do I convert a z-score to a percentile?

In a standard normal distribution, you look up the cumulative distribution function (CDF) at the z-score. A z-score of 0 corresponds to the 50th percentile, a z of 1.00 corresponds to roughly the 84th percentile, and a z of 1.96 corresponds to roughly the 97.5th percentile. This calculator uses a polynomial approximation of the normal CDF to convert z directly to a percentile.

What is a good or bad z-score?

There is no universally good or bad z-score; it depends on context. In hypothesis testing, a z-score beyond plus or minus 1.96 often indicates statistical significance at the 5% level. In standardised testing, a z-score above 2 means you scored in roughly the top 2.5% of participants. In quality control, values with z beyond plus or minus 3 are often flagged as outliers.

Z-Score and Percentile Calculator

What does a z-score tell me?

A z-score tells you how many standard deviations a value is above or below the mean. A z-score of 2 means the value is two standard deviations above the mean, while minus 1 means one standard deviation below. It puts values from different scales onto one common measure.

How is a z-score turned into a percentile?

The percentile is the area under the normal curve to the left of the z-score, found with the normal cumulative distribution. A z-score of zero gives the 50th percentile, and higher z-scores give higher percentiles. This calculator uses an accurate error function approximation to compute it.

Does the percentile work for any data?

It assumes the data follows a roughly normal, bell shaped distribution. For data that is heavily skewed or has long tails, a percentile read from the normal curve can be misleading. In those cases an empirical percentile from the actual data is more reliable.