Free normal distribution calculator. Enter mean, standard deviation and x for the z-score, pdf, and probabilities P(X<=x), P(X>=x), and P(x1<=X<=x2).
The normal distribution — the famous bell curve — describes data that clusters symmetrically around a central mean, with values becoming rarer the farther they are from that center. Heights, measurement errors, test scores, and many natural quantities follow it closely. Enter the mean μ, the standard deviation σ, and a value x (plus an optional second value x₂ for a range), and this calculator returns the z-score, the probability density, and the tail and range probabilities.
The first step is to standardize x into a z-score, which counts how many standard deviations x sits from the mean:
z = (x − μ) / σ
A z-score of 0 lands exactly on the mean; z = +1 is one standard deviation above the mean, z = −2 is two below, and so on. Standardizing lets any normal distribution be compared on a single universal scale, which is why z-scores appear throughout statistics, grading curves, and quality control.
The height of the curve at x is the probability density function:
f(x) = 1 / (σ√(2π)) · e−z²/2
This is a density, not a probability — for a continuous variable the chance of landing on any exact point is zero, so probability is measured as area under the curve. The cumulative probability P(X ≤ x), written Φ(z), is the area to the left of x; P(X ≥ x) is 1 − Φ(z); and the probability of a range P(x₁ ≤ X ≤ x₂) is Φ(x₂) − Φ(x₁). This tool computes Φ using a high-accuracy error-function approximation.
For any normal distribution, about 68% of values fall within one standard deviation of the mean, about 95% within two, and about 99.7% within three. This empirical rule is a fast sanity check: an IQ score of 130 (μ = 100, σ = 15) is two standard deviations above average, so it sits at roughly the 97.7th percentile — only about 2.3% of people score higher.
With μ = 100, σ = 15, and x = 115, the z-score is (115 − 100) / 15 = 1. The cumulative probability P(X ≤ 115) is about 0.8413, so P(X ≥ 115) is about 0.1587. To turn a real data set into μ and σ first, use the standard deviation calculator. Everything runs in your browser.
A z-score measures how many standard deviations a value lies from the mean. It is calculated as z = (x − μ) / σ. A z-score of 0 is exactly average, +1 is one standard deviation above the mean, and −2 is two below. Z-scores put any normal distribution onto a common scale so values from different data sets can be compared directly.
Also called the empirical rule, it states that for a normal distribution about 68% of values fall within one standard deviation of the mean, about 95% within two standard deviations, and about 99.7% within three. It is a quick way to judge how unusual a value is without a full calculation.
The probability density function (pdf) gives the height of the bell curve at a point — it shows where values are concentrated but is not itself a probability. The cumulative distribution function (cdf), Φ(x), gives the probability that the variable is less than or equal to x, which equals the area under the curve to the left of x. Probabilities come from areas, not heights.
Compute the cumulative probability at each endpoint and subtract: P(x1 ≤ X ≤ x2) = Φ(x2) − Φ(x1). Enter your lower value as x and your upper value as x₂ in this calculator and it returns the area between them automatically.
The normal distribution is continuous, so probability is spread over a range as area under the curve rather than concentrated at points. The area above a single exact point has zero width and therefore zero probability. That is why meaningful probabilities are always stated over intervals, such as P(X ≤ x) or P(a ≤ X ≤ b).