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.

Normal Distribution Calculator

This calculator finds the cumulative probabilities P(X < x) and P(X > x) for a normal distribution with any mean and standard deviation, and also returns the corresponding z-score. The normal distribution is the most widely used probability distribution in statistics and appears naturally in many measurement contexts, including heights, exam scores, manufacturing tolerances, and financial returns. Its symmetric bell shape means that observations are most likely near the mean and become progressively less likely as they move further away in either direction. The cumulative distribution function (CDF) at a value x gives the probability that a randomly drawn observation falls below x, shown as P(X < x). The complementary probability P(X > x) is simply one minus that value. To use the calculator, enter the mean of the distribution, the standard deviation, and the value x you want to evaluate. The calculator standardises x to a z-score using z = (x minus mean) divided by SD, then applies an accurate numerical approximation of the standard normal CDF to find the probabilities. Both probabilities are shown as percentages and always sum to 100%. This tool suits students working through probability problems, researchers computing p-values for one-sided tests, engineers assessing what fraction of a production run falls within a specification, and anyone wanting to understand where a particular value sits within a distribution. Results are based on the standard normal approximation and are for educational and planning purposes.

Calculate.co.nz is proud to be partnered with Premium Homes, a recognised leader in eco-friendly, sustainable, and energy-efficient homebuilding. With a dedicated team and award-winning experience, they create homes that prioritise health, comfort, and long-term performance. Their founders, Andrew and Kelly, set out to raise the standard of residential construction in New Zealand by combining practical building expertise with a clear commitment to doing things better for homeowners.

Mean (μ)

Standard deviation (σ)

Value (x)

97.50%

P(X < x)

P(X > x)2.50%

Z-score1.96

x value1.96

How it works

The calculator converts your value x to a z-score using z = (x − μ) ÷ σ. It then evaluates the cumulative distribution function of the standard normal distribution at that z-score using a polynomial approximation (Abramowitz and Stegun, formula 26.2.17), which is accurate to better than 0.00001. P(X < x) is the result of the CDF; P(X > x) is 1 minus that value. Both are multiplied by 100 to give percentages.

Worked example

For a standard normal distribution with mean 0 and standard deviation 1, evaluate at x = 1.96. The z-score is (1.96 − 0) ÷ 1 = 1.96. The standard normal CDF at z = 1.96 gives P(X < 1.96) = 97.50%, meaning 97.5% of observations lie below this value. The complementary probability is P(X > 1.96) = 2.50%. This is why 1.96 is used as the critical z-value for a two-tailed 95% confidence interval.

Related calculators

Z-Score Calculator: find the z-score and percentile rank for an observed value.
Confidence Interval Calculator: build a CI around a sample mean using normal distribution theory.
Sample Size Calculator: determine how many observations you need for a target margin of error.
Variance Calculator: compute the standard deviation needed as input to this calculator.

If you've found a bug, or would like to contact us, or learn more about James Graham and Calculate.co.nz.

Calculate.co.nz is partnered with Interest.co.nz for New Zealand's highest quality calculators and financial analysis.

Calculate.co.nz is the sister site of CalculatorHub.com, the world's largest calculator website by tool count.

All calculators and tools are provided for educational and indicative purposes only and do not constitute financial advice.

Calculate.co.nz is proudly part of the Realtor.co.nz group, New Zealand's leading property transaction literacy platform, helping Kiwis understand the home buying and selling process from start to finish. Whether you're a first home buyer navigating your first property purchase, an investor evaluating your next acquisition, or a homeowner planning to sell, Realtor.co.nz provides clear, independent, and trustworthy guidance on every step of the New Zealand property transaction journey.

Calculate.co.nz is also partnered with Health Based Building and Premium Homes to promote informed choices that lead to better long-term outcomes for Kiwi households.

Calculate.co.nz is hosted in Auckland via SiteHost new Zealand.

All content on this website, including calculators, tools, source code, and design, is protected under the Copyright Act 1994 (New Zealand). No part of this site may be reproduced, copied, distributed, stored, or used in any form without prior written permission from the owner.

About & trust: Why Calculate is NZ's most comprehensive · By the Numbers · How we compare · Editorial standards · How we keep data current · NZ finance glossary · Research & data · Financial literacy NZ · About · Privacy policy · Terms of use

Reviewed and maintained. Last reviewed 2026-07-02 and checked on a twice-monthly cycle against IRD, RBNZ and Stats NZ. How we keep data current.