Z-Score Calculator

How is the z-score and the percentile calculated?

z = (x − μ) ÷ σ. The percentile is the standard normal cumulative probability Φ(z) × 100, computed with the Abramowitz and Stegun polynomial approximation to the error function, whose error is about 1.5 × 10⁻⁷ in erf, far below the two decimals displayed. Checks: z = 1.96 gives 97.50%, z = 1 gives 84.13%, and z = −3 gives 0.13%.

Does the percentile assume a normal distribution?

Yes, always. The z-score itself is valid for any data, but the percentile and "Above this value" rows read the value off a bell curve. For skewed data such as incomes or waiting times, those percentages can be far from the actual share of observations. Use the percentile calculator on the raw data in that case.

What do the interpretation labels mean?

They come from fixed cut-offs: z above 2 is "Unusually high", above 1 up to 2 is "Above average", above −1 up to 1 is "Average range", above −2 up to −1 is "Below average", and −2 or lower is "Unusually low". The comparisons are strict, so z = 2 exactly (for example 130 on a scale with mean 100 and SD 15) is "Above average", and z = −2 falls in "Unusually low". Treat them as a rough guide, not a test.

Which standard deviation should I enter?

The standard deviation of the reference group the value is being compared with. For individual scores that is the population or norm-group σ. If you only have a sample, you may enter s, but the result is then an estimate. To standardise a sample mean against a hypothesised mean, enter the standard error σ ÷ √n in the SD box rather than σ.

Is the "Above this value" figure a p-value?

It is the upper-tail probability, which is the one-tailed p-value for a z test in the upward direction. For a two-tailed p-value double the smaller of the two tail percentages. For z = 1.96 the upper tail is 2.50%, so the two-tailed p is 5%. This page does not use the t distribution, so small-sample tests need the t-test calculator.

What input is rejected?

A standard deviation of zero or below gives an error, since a zero spread makes z undefined. Every other numeric value is accepted, including negative means and scores, and an empty box counts as 0.

Can I compare scores from different tests with z-scores?

Yes, that is their main use. A score of 130 on a test with mean 100 and SD 15 has z = 2.0000, while 560 on a test with mean 500 and SD 100 has z = 0.6000. The first is far above its group (97.72nd percentile) and the second is modestly above (72.57th), so the first is the stronger result relative to peers, provided both groups are roughly normal.