Interactive normal distribution calculator
Free online normal distribution calculator (Gaussian, bell curve). It computes probabilities, quantiles (inverse normal), the density and z-scores for any normal distribution N(μ, σ²), with an interactive chart, a step-by-step calculation, the equivalent R code and Excel formulas.
What you can calculate
- P(X ≤ x) — cumulative probability up to the score (CDF).
- P(X ≥ x) — upper-tail probability (1 − CDF).
- P(x₁ ≤ X ≤ x₂) — probability that X lies between x₁ and x₂.
- P(X ≤ x₁) + P(X ≥ x₂) — probability that X lies outside the range (two tails).
- Density f(x) — value of the probability density function at x.
- x from P(X ≤ x) — the score for a given cumulative probability (inverse CDF, quantile, percentile).
- x from P(X ≥ x) — the score for a given upper-tail probability.
- Tail values x₁, x₂ — both tail scores from two tail probabilities.
- Z score — the standard score, z = (x − μ)/σ.
How to use
- Enter the mean (μ) and the standard deviation (σ). Leave empty for the standard normal distribution (μ=0, σ=1).
- Pick what to calculate — the small icons show which region of the curve will be shaded.
- Enter the score(s) or probability and press Calculate.
- Explore the chart: hover it to read x, the density f(x) and the cumulative probability at any point; drag the round handles to move the boundaries — the results update as you drag.
The normal density function
f(x) = 1σ√(2π) exp(−(x − μ)²2σ²) and the z-score: z = x − μσ
What is the normal distribution?
The normal distribution, also known as the Gaussian distribution, is the most used distribution in statistical analysis. Many natural measurements follow it or resemble it — for example height, weight and measurement errors. It has a symmetrical “bell curve” shape: most of the data centers around the mean, and the frequency decreases as values move farther away from the center.
According to the Central Limit Theorem, the distribution of the sum or average of many independent random variables tends toward a normal distribution, even when the variables themselves are not normal — the larger the sample size, the closer the resemblance.
The standard normal distribution is the special case with μ = 0 and σ = 1. When X follows N(μ, σ²), the standardized value z = (x − μ)/σ follows the standard normal distribution, so any normal probability can be computed from the standard normal CDF: P(X ≤ x) = Φ(z). This calculator does exactly that — and shows the standardization in the step-by-step section.
Looking for other distributions (binomial, t, chi-square, F, Poisson…)? Use the all-distributions calculator.