Statistics

Normal Distribution Calculator

Convert raw scores to z-scores and compute tail or interval probabilities for any normal distribution.

normal distributionz-scoreGaussian

Normal Distribution

Calculate z-scores and tail probabilities for the normal distribution.

Mean (μ)Std deviation (σ)Mode

Value a

Z-score

1.2

Probability

0.88493

PDF at value

0.194186

Normal CDF

Φ(z) = ½ [1 + erf(z / √2)]

The calculator leverages an error-function approximation to deliver accurate probabilities and densities.

Enter the mean μ and standard deviation σ.
Provide one or two values depending on whether you want a single tail or a between probability.
Read the z-score, PDF value, and probability.

Example

Input: μ = 0, σ = 1, mode = P(X ≤ a), a = 1.2

Output: z ≈ 1.2, P ≈ 0.884

Can it compute two-tailed probabilities?

Use the between mode with symmetric bounds (e.g., ±z) to obtain central or tail probabilities.

What happens if σ = 0?

A zero or negative standard deviation is invalid for a normal distribution, so the calculator flags the input.