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Z-Score to Percentile (Normal CDF Approximation) Calculator

Φ(z) via the Abramowitz-Hastings 7.1.26 polynomial.

Convert a standard-normal z-score to a percentile (cumulative probability) via the Abramowitz-Hastings rational approximation of the normal CDF (max error ≈ 7.5 × 10⁻⁸). Reports Φ(z), 1 − Φ(z), and the percentile as a percent.

Daniel Oh By Daniel Oh
Published Last reviewed 1 min read

Inputs

Results

Enter values and click Calculate to see results.
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How to use this calculator

  1. Fill in the inputs above using the units you already have.
  2. Values update automatically as you type — no submit button needed.
  3. Hover any result row for the underlying formula and intermediate values.

Formula

Φ(z) ≈ 1 − φ(z)·(b₁t + b₂t² + b₃t³ + b₄t⁴ + b₅t⁵), t = 1/(1 + 0.2316419 |z|).

In depth

Convert a standard-normal z-score to a percentile (cumulative probability) via the Abramowitz-Hastings rational approximation of the normal CDF (max error ≈ 7.5 × 10⁻⁸). Reports Φ(z), 1 − Φ(z), and the percentile as a percent.