Normal Distribution Calculator
The normal distribution calculator finds probabilities for the standard normal distribution or converts raw scores to z-scores given a mean and standard deviation. The standard normal distribution has a mean of 0 and standard deviation of 1. Given a z-score, this calculator finds P(Z < z), the cumulative distribution function (CDF): the probability that a random value is less than z. You can also enter a raw value, mean, and standard deviation to convert to a z-score first. For example, if test scores have mean 100 and standard deviation 15, a score of 115 has z-score (115 - 100) / 15 = 1, and P(Z < 1) is approximately 0.8413, meaning about 84 percent of test-takers score below 115.
Formulas
Z-score: z = (x - μ) / σ
CDF: P(Z < z) uses error function approximation
Standard normal reference table
| Z-score | P(Z < z) | Percentile |
|---|---|---|
| -3.0 | 0.0013 | 0.13% |
| -2.0 | 0.0228 | 2.28% |
| -1.0 | 0.1587 | 15.87% |
| 0.0 | 0.5000 | 50.00% |
| 1.0 | 0.8413 | 84.13% |
| 2.0 | 0.9772 | 97.72% |
| 3.0 | 0.9987 | 99.87% |
Normal distribution calculator: frequently asked questions
What is the normal distribution?
The normal distribution (Gaussian distribution) is a bell-shaped probability distribution symmetric around its mean. It is the most common distribution in statistics and nature. About 68 percent of data falls within 1 standard deviation of the mean, 95 percent within 2, and 99.7 percent within 3.
What is a z-score?
A z-score measures how many standard deviations a value is from the mean. The formula is z = (x - mu) / sigma, where x is the value, mu is the mean, and sigma is the standard deviation. A z-score of 0 means the value equals the mean.
What does P(Z < z) mean?
P(Z < z) is the cumulative distribution function (CDF): the probability that a random value from the distribution is less than z. For example, P(Z < 0) = 0.5 because the mean divides the distribution in half.
How do you use the normal distribution in practice?
The normal distribution is used in quality control, medical testing, standardized tests (like SAT scores), and many other fields. If data is normally distributed, you can use z-scores to find percentiles and probabilities.
What is the standard normal distribution?
The standard normal distribution has mean 0 and standard deviation 1. Any normal distribution can be converted to standard normal using z-scores, making calculations easier.
Official sources
- Wolfram MathWorld: Normal Distribution.
- Wikipedia: Normal Distribution.
Reviewed by the CalculatorHub team, edited by James Graham, 14 June 2026. See our methodology.