Standard Normal Distribution Calculator

A Z-score (also called a standard score) measures how many standard deviations a data point is from the mean of a distribution. It is calculated as Z = (X − μ) / σ, where X is the raw score, μ is the mean, and σ is the standard deviation. Positive Z-scores indicate values above the mean; negative Z-scores indicate values below.

The standard normal distribution is a special case of the normal distribution with a mean of 0 and a standard deviation of 1. Any normal distribution can be converted to the standard normal by computing Z-scores. Z-tables and this calculator are built around the standard normal distribution.

The normal distribution appears naturally in many real-world phenomena — test scores, heights, measurement errors — and is central to the Central Limit Theorem, which states that sample means tend toward a normal distribution regardless of the underlying population distribution. This makes it foundational for hypothesis testing, confidence intervals, and statistical inference.

A cumulative probability is the probability that a random variable takes a value less than or equal to a specific point. For a normal distribution, the left-tail probability P(X < z) represents the area under the curve to the left of z. Values range from 0 to 1 (or 0% to 100%).

The left-tail probability P(X < z) is the area under the normal curve to the left of z — the probability of observing a value less than z. The right-tail probability P(X > z) is the complement: 1 − P(X < z). Together they always sum to 1.

Use the formula Z = (X − μ) / σ, where X is your raw score, μ is the population mean, and σ is the population standard deviation. For example, if a test score is 80, the mean is 70, and the standard deviation is 10, then Z = (80 − 70) / 10 = 1.0. Select 'Raw Score → Probability' mode in this calculator to do this automatically.

The two-tailed probability 2·P(X > |z|) represents the combined area in both tails of the normal distribution beyond ±|z|. It is used in hypothesis testing when you're testing for any deviation from the mean, regardless of direction. For z = 1.96, the two-tailed probability is approximately 0.05 (5%).

This is the inverse normal problem. Given a cumulative probability p, you want the Z-score z such that P(X < z) = p. For example, P(X < z) = 0.975 gives z ≈ 1.96. Select 'Probability → Z-score' mode in this calculator and enter your probability value.