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z = (x − μ) ÷ σ

Z-score calculator

A value, a mean and a standard deviation give the standard score and the share of the normal curve below and above it.

Short answer: a value of 700 against a mean of 500 and a standard deviation of 100 has z = 2, and 97.72% of a normal distribution lies below it. Swap in 400 and z = −1, below which sits 15.87%.

Reading the score

Three z-scores are worth remembering: 1.0 is about the 84th percentile, 1.96 about the 97.5th, and 2.58 about the 99.5th. Those land at 0.8413, 0.9750 and 0.9951, which is why the table rows people actually use are so few.

Worked example

A test with mean 62 and standard deviation 9: a score of 71 has z = 1.0 and sits at the 84.13th percentile, so about 16% of candidates scored higher. Reporting the raw 71 hides which of two papers was actually harder, which is the reason standardising exists.

Z-score questions

What is a z-score?

The number of standard deviations a value sits from the mean: z = (x − μ) ÷ σ. Positive means above the mean, negative below, and 2 means far above for most real data.

Where does the percentile come from?

The normal curve’s area, evaluated with the Abramowitz–Stegun 7.1.26 approximation — accurate to about 1.5 × 10⁻⁷, which is well beyond any table. The approximation is stated on the page rather than presented as exact.

Does this work for any data set?

It assumes the data is roughly normal. For skewed data — incomes, waiting times — the percentile from this method is misleading, and the empirical rank of the value in the sample is the honest answer instead.

What does the two-tailed figure mean?

The share of values at least as far from the mean in either direction. It is the number hypothesis tests use: z = 1.96 puts 5% in both tails combined, which is why 1.96 shows up as the 95% threshold.

Related tools

Statistics calculatorMean and standard deviation from data.Frequency table calculatorThe same figures from counts.Percentage calculatorShare of total and change.