Free online z-score and normal distribution calculator

How to use

1. Select a mode — convert a raw value to a z-score, look up a z-score’s percentile, or find the z-score for a given percentile.
2. Fill in the required fields.
3. Click Calculate to see the full output.

Formulas

Raw value to z-score:

z = (x − μ) / σ

where x is the raw value, μ is the mean, σ is the standard deviation.

Cumulative probability (CDF):

P(Z ≤ z) = 0.5 × (1 + erf(z / √2))

Error function approximation (Hart):

erf(x) ≈ 1 − (a₁t + a₂t² + a₃t³) × exp(−x²)
where t = 1 / (1 + 0.47047|x|)
a₁ = 0.3480242, a₂ = −0.0958798, a₃ = 0.7478556

Two-tailed p-value:

p = 2 × min(P(Z ≤ z), 1 − P(Z ≤ z))

Worked example

A student scores 75 on a test with mean 70 and standard deviation 10:

z = (75 − 70) / 10 = 0.50
P(Z ≤ 0.50) ≈ 0.6915 → 69.15th percentile
two-tailed p = 2 × 0.3085 ≈ 0.617

Standard critical values:

z = 1.645 → 95th percentile (one-tailed 5%)
z = 1.960 → 97.5th percentile (two-tailed 5%)
z = 2.576 → 99.5th percentile (two-tailed 1%)