What is the normal distribution?

The normal (Gaussian) distribution is a symmetric, bell-shaped probability distribution defined by its mean (μ) and standard deviation (σ). It appears throughout nature and statistics due to the central limit theorem.

A z-score (standard score) measures how many standard deviations x is from the mean: z = (x − μ) / σ. A z-score of 0 means x equals the mean; ±1 means one standard deviation away.

What does CDF P(X ≤ x) mean?

The cumulative distribution function gives the probability that a randomly drawn value from the distribution is less than or equal to x. For example, P(X ≤ μ) = 0.5 because the normal distribution is symmetric.

What is the PDF value?

The probability density function (PDF) gives the relative likelihood of x. It is the height of the bell curve at that point. The PDF itself is not a probability — it must be integrated over a range to get a probability.

Why must σ be greater than zero?

Standard deviation represents spread. A value of zero would collapse the distribution to a single point (a Dirac delta), which is not a proper probability distribution.

How accurate is this calculator?

CDF values are computed using a rational approximation to the error function (erf) that is accurate to roughly 7 significant figures — more than sufficient for most practical uses.

Normal Distribution Calculator

Enter mean, standard deviation, and x to get the PDF, CDF (P(X ≤ x)), and z-score instantly.

Parameters

Mean (μ)

Standard Deviation (σ)

x Value

Presets

Bell Curve — Shaded Area = P(X ≤ x)

Probability Summary

Expression	Probability	Percentage

Summary

This calculator computes key values for the normal (Gaussian) distribution: the probability density function (PDF), the cumulative distribution function (CDF, i.e., P(X ≤ x)), and the standardized z-score. Results update instantly as you type. The bell-curve chart gives a visual sense of where your x value falls relative to the distribution.

How it works

Enter the population mean (μ) — the center of the distribution.
Enter the standard deviation (σ) — controls the spread; must be greater than zero.
Enter your x value — the point you want to evaluate.
The calculator returns the PDF (height of the curve at x), the CDF P(X ≤ x), and the z-score.
The tail probabilities P(X > x), P(X < x), and the two-tailed P(|X - μ| ≤ |x - μ|) are also shown.
Use the chart to see the shaded area representing the cumulative probability.

Use cases

Statistics courses: check homework answers for normal probability problems.
Quality control: find the probability a measurement falls within spec limits.
Finance: estimate the probability of a return exceeding a threshold.
Psychology/research: convert raw scores to percentile ranks.
Machine learning: evaluate how far a data point is from the mean in standard deviations.
Standardized testing: convert raw scores to z-scores and percentiles.

Frequently Asked Questions

Last updated: 2026-06-04 · Reviewed by Nham Vu