Z-score Calculator

What a z-score measures

A z-score of 0 means the value equals the mean exactly. A z-score of +1 means the value is one standard deviation above the mean; −1 means one standard deviation below. Because the calculation uses the population standard deviation and mean, z-scores are unit-free — you can directly compare a student's exam score to a patient's blood pressure reading once both are converted to z-scores.

The sign of the z-score shows direction: positive values are above average, negative values are below. The magnitude shows how unusual the value is relative to the spread of the distribution.

Z-score formula

The formula is straightforward: subtract the population mean from the raw value, then divide by the standard deviation. The result is a dimensionless number on the standard normal scale.

Converting z-scores to percentiles

The cumulative distribution function (CDF) of the standard normal distribution, often written Φ(z), gives the probability that a randomly chosen value from the population is less than or equal to x. Multiplied by 100, this becomes the percentile rank.

For example, a z-score of 0.70 corresponds to roughly the 75.8th percentile — meaning about 75.8% of the population scores below that value and 24.2% score above it. Z-scores between −3 and +3 cover approximately 99.7% of a normal distribution (the empirical rule).

Worked example

A student scores 72 on an exam where the class mean is 65 and the standard deviation is 10. The z-score is (72 − 65) / 10 = 0.70. Looking up Φ(0.70) gives approximately 0.758, so the student is at the 75.8th percentile — scoring higher than about 75.8% of the class.

If another exam had a mean of 500 and a standard deviation of 100, a score of 570 gives z = 0.70 — the same relative standing, even though the raw scores look completely different.

Using z-scores to detect outliers

A common rule of thumb treats any value with |z| > 3 as a potential outlier, since only about 0.3% of values in a normal distribution fall beyond three standard deviations from the mean. Some fields use |z| > 2 or |z| > 2.5 as a more conservative threshold.

Z-score based outlier detection works best when the underlying data is approximately normally distributed. For skewed distributions, alternative methods such as the IQR fence or Grubbs' test may be more appropriate.

Population versus sample z-scores

The standard z-score formula uses the population mean (μ) and population standard deviation (σ). When you only have sample statistics, you can substitute the sample mean (x̄) and sample standard deviation (s) to get an approximate standardised score — but strictly speaking, that produces a t-score rather than a true z-score, and the t-distribution should be used for probability lookups when the sample is small.

This calculator uses the values you enter as μ and σ; if you are working with sample estimates and a small dataset, treat the percentile as approximate.