Z-Score Probability Calculator

How it is calculated

The tool gives the area under the standard normal curve (mean 0, standard deviation 1) to the left of your z-score: Φ(z) = P(Z ≤ z). The right tail is the rest, 1 − Φ(z). It does not call an exact function but evaluates a well-known polynomial approximation of the normal curve, accurate to a few parts in ten million, then shows the results as percentages with two decimals. For a negative z it computes Φ(|z|) and flips it, so Φ(−z) = 1 − Φ(z). A z of 0 gives 50 %. Very large |z| round to 100.00 % and 0.00 %. Only one z is entered, and the tool returns one-sided areas only; for a two-sided p-value double the smaller tail yourself.

Common mistakes

• Reading the cumulative figure as the p-value. The 97.50 % for z = 1.96 is the area to the left; the one-sided p-value is the right tail, 2.50 %, and a two-sided p-value is double that, 5 %.
• Entering a raw score or a percentage instead of a z-score. Convert first with z = (x − mean) ÷ standard deviation.
• Forgetting the sign. A negative z gives a small cumulative probability and a large right tail, the mirror image of the positive case.
• Using the result for data that is not roughly normal. The conversion assumes a bell-shaped distribution, so for heavily skewed data the percentile will be wrong.
• Expecting an exact answer at extreme z. From about |z| = 4 the two percentages show as 0.00 % and 100.00 %, so the tool cannot tell you how small a tail probability really is.