Standard Normal Distribution Calculator

Last reviewed: June 2026 by the InvNorm Calculator Editorial Team. Report an issue

This Standard Normal Distribution Calculator computes probabilities and Z scores for the standard normal distribution, which has a mean of 0 and a standard deviation of 1. Choose a calculation mode to find the cumulative probability for a Z score, the right-tail probability, the probability between two Z values, or the Z score corresponding to a given probability.

What Is the Standard Normal Distribution?

The standard normal distribution is a special case of the normal distribution with mean μ = 0 and standard deviation σ = 1. It is denoted by Z and serves as the reference distribution for all normal distributions. Any normal distribution can be converted to the standard normal by applying the Z score transformation z = (x − μ) / σ. This standardization means that probabilities computed from the standard normal table or function apply universally to any normal distribution after the appropriate transformation.

The probability density function (PDF) of the standard normal distribution reaches its maximum of approximately 0.3989 at z = 0 and decreases symmetrically in both directions. The total area under the curve equals 1. The distribution is symmetric about zero, so P(Z ≤ −z) = P(Z ≥ z) for any value z. This symmetry property simplifies many calculations and is the reason standard normal tables often list only positive Z values.

Key reference points include: approximately 68.27% of the distribution falls within one standard deviation of the mean (between z = −1 and z = 1), about 95.45% within two standard deviations, and 99.73% within three standard deviations. Values beyond z = 3 or below z = −3 are extremely rare, occurring less than 0.27% of the time combined.

Standard Normal Distribution Formulas

The CDF (cumulative distribution function) of the standard normal distribution is:

Φ(z) = P(Z ≤ z) = ½[1 + erf(z / √2)]

The inverse CDF (quantile function) is:

z = Φ⁻¹(p)

For the probability between two values:

P(a ≤ Z ≤ b) = Φ(b) − Φ(a)

Worked Example

Find the probability that a standard normal variable falls between z = −1.5 and z = 2.0.

1. Find the upper CDF: Φ(2.0) ≈ 0.97725
2. Find the lower CDF: Φ(−1.5) ≈ 0.06681
3. Subtract: P(−1.5 ≤ Z ≤ 2.0) = 0.97725 − 0.06681 = 0.91044
4. Interpret: About 91.04% of values from a standard normal distribution fall between −1.5 and 2.0.

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