Z-Score Calculator

What Is a Z-Score?

A z-score measures position relative to the mean in units of standard deviation. A z-score of 2 means the value is two standard deviations above the mean; −0.5 means half a standard deviation below it; 0 means it equals the mean.

Because it removes the original units, a z-score lets you compare values from different scales. A score of 85 on a test with mean 70 and standard deviation 10 (z = 1.5) is relatively better than 80 on a test with mean 65 and standard deviation 12 (z = 1.25), even though the raw scores look different.

How to Calculate a Z-Score

1. Identify the value x.

2. Identify the mean μ.

3. Identify the standard deviation σ.

4. Subtract the mean from the value: x − μ.

5. Divide the result by the standard deviation.

6. Read the sign (above or below the mean) and the size (how far away).

Example with x = 85, μ = 70 and σ = 10: 85 − 70 = 15, and 15 ÷ 10 = 1.5, so z = 1.5. The value is 1.5 standard deviations above the mean.

Positive vs Negative Z-Scores

Positive z-score: the value is above the mean. z = 1.2 means 1.2 standard deviations above.

Negative z-score: the value is below the mean. z = −1.2 means 1.2 standard deviations below.

Zero: the value equals the mean.

The sign only gives the direction. Whether above or below the mean is better depends on the context: a high exam score may be good, but a high blood pressure reading or a long delivery time usually is not.

What the Size of a Z-Score Means

The absolute value |z| is the distance from the mean in standard deviations. z = 0.5 is half a standard deviation from the mean, z = 2 is two standard deviations above it, and z = −3 is three standard deviations below it.

For data that is roughly normal, about 68% of values lie within 1 standard deviation of the mean, about 95% within 2, and about 99.7% within 3 (the 68-95-99.7 rule). So values with |z| above 3 are rare in normal data and are often checked as possible outliers or errors. For other distributions these proportions can be quite different.

Z-Score to Percentile

If the values follow a normal distribution, a z-score can be converted to a percentile using the standard normal distribution (mean 0, standard deviation 1). The cumulative probability Φ(z) is the share of values below the z-score.

Under a standard normal distribution, z = 0 is the 50th percentile, z = 1 about the 84th (84.13% below), z = −1 about the 16th (15.87% below), z = 1.5 about the 93rd (93.32% below) and z = 2 about the 98th (97.72% below).

This only holds for normally distributed data. A z-score of 2 does not always mean the 97.72nd percentile: in a skewed dataset, the share of values below z = 2 can be noticeably different. Always check the shape of your data before quoting a percentile.

When Z-Scores Are Useful

Comparing values from different distributions, such as scores on two tests with different averages and spreads.

Standardizing observations, so that variables measured in different units can be combined or compared; many statistical and machine-learning methods standardize data first.

Finding probabilities with the normal distribution, for example the chance that a measurement exceeds a limit.

Describing relative position and flagging unusual values, such as measurements far from the mean in quality control.

Hypothesis tests: a z-test compares a sample result with what would be expected, in standard-error units.

Z-Score vs Standard Score

"Z-score" and "standard score" usually mean the same thing: a value expressed as the number of standard deviations from the mean, z = (x − μ) / σ. Standardizing a whole dataset this way gives values with mean 0 and standard deviation 1.

Z-Score vs T-Score

A z-score standardizes a value using a mean and standard deviation, with mean 0 and standard deviation 1.

A T-score in psychometrics and education rescales a z-score to mean 50 and standard deviation 10: T = 50 + 10z, so z = 1.5 becomes T = 65. It avoids negative numbers and decimals.

A t-statistic in hypothesis testing is different again: it compares a sample mean with a hypothesized mean using the standard deviation estimated from the sample, t = (x̄ − μ) / (s / √n), and is read from a t-distribution, which has heavier tails than the normal distribution for small samples.