Z-score calculator

How do I calculate a z-score?

z = (value − mean) ÷ standard deviation. A test score of 85 in a class with mean 70 and SD 10 has z = (85 − 70) ÷ 10 = 1.5 — one and a half standard deviations above average. OpenStax (Introductory Statistics 2e, section 6.1) puts it the same way: a z-score is the number of standard deviations a value lies above or below the mean.

What does a z-score mean in percent?

For a normal distribution, z tells you the share of values below it. z = 1.5 is the 93.3rd percentile; z = 1.96 is 97.5%, and z = 0 is exactly 50%.

What is the probability between two z-scores?

Subtract their cumulative probabilities: between −1.96 and 1.96 lies 97.5% − 2.5% = 95% of a normal distribution.

Can a z-score be negative?

Yes — negative means below the mean. z = −1 is the 15.9th percentile.

How accurate are the probabilities?

They are computed from the normal distribution function to about 10 decimal places, so you do not need a printed z-table.

Worked example: comparing scores on different scales

Anna scores 85 on a test with mean 70 and SD 10; Ben scores 640 on a test with mean 500 and SD 100. The raw numbers cannot be compared, but the z-scores can: Anna’s is 1.50 and Ben’s is 1.40, so Anna did slightly better relative to her group. Assuming normal scores, those are percentile ranks of about 93.3% and 91.9%. OpenStax’s exam example has scores with mean 63 and SD 5: a 65 gives z = 0.4, a percentile rank of 65.5%, while the 90th percentile needs z = 1.2816, a score of 63 + 1.2816 × 5 = 69.4. And X = 17 in a distribution with mean 5 and SD 6 sits at z = (17 − 5) ÷ 6 = 2.

What if I am working with a sample mean?

Then the standard deviation to use is the standard error, σ ÷ √n, not σ. If scores have mean 100 and SD 15, a sample of 25 people averaging 103 has z = (103 − 100) ÷ (15 ÷ √25) = 1.0. The same 103 for a single person would have z = 0.2. Averages vary less than individuals, which is why z-tests use the standard error. The standard error calculator finds it from raw data.

Common mistakes with z-scores

First, reading the percentile off a z-score when the data are not roughly normal: the z-score itself is always valid, but the “93.3rd percentile” only holds for a bell-shaped distribution. Second, mixing the population SD with a sample SD; pick the one that matches your data, as the standard deviation calculator shows. Third, using a z-score on a sample of two or three values, where the mean and the SD are themselves unreliable. Fourth, dropping the sign: a z-score of −2 is two standard deviations below the mean, not above. Fifth, confusing a z-score with a probability; z = 1.96 is a distance, and 97.5% is the area below it.

From a z-score back to a raw value

The formula can be run in reverse: x = mean + z × SD. On the OpenStax exam example (mean 63, SD 5), a score two standard deviations above average is 63 + 2 × 5 = 73, and one standard deviation below is 63 − 5 = 58. This is how a percentile becomes a cutoff score: find the z for that percentile, as with 1.2816 for the 90th, then convert it back to the original scale.

How far is far? The 68-95-99.7 rule

For a normal distribution, 68.27% of values lie within one standard deviation of the mean (|z| ≤ 1), 95.45% within two and 99.73% within three. OpenStax gives these as the empirical rule, roughly 68%, 95% and 99.7%. That is why a z-score beyond ±2 is often called unusual and beyond ±3 very unusual. These cutoffs are conventions, not tests: in a data set of a thousand values, a few beyond ±3 are expected from chance alone, and the rule says nothing when the data are far from bell-shaped.

Sources: OpenStax, Introductory Statistics 2e, sections 6.1 (The Standard Normal Distribution) and 6.2 (Using the Normal Distribution). Related: normal distribution calculator, p-value calculator, empirical rule calculator, standard error calculator and percentile calculator.