Z-Score Calculator

How it works

A z-score rewrites a value in units of standard deviations from the mean: z = (x − μ) ÷ σ. A z of +2 means the value is two standard deviations above the mean, −1 means one below, 0 means it is exactly average. Because the units cancel, z-scores let you compare values from different scales, such as a score on one test against a score on another with a different mean and spread.

Choose the first question when the mean and standard deviation are published, as they are for IQ tests, standardised exams and growth charts. Choose the second when all you have is the raw data: the calculator finds the mean and standard deviation of the list first, then standardises your value against them. Choose the third to go backwards, from a z-score to the raw value that sits at that position.

On its own a z-score is a distance, not a probability. If the values follow a normal distribution, about 68% fall within one standard deviation of the mean, 95% within two and 99.7% within three, so a z-score beyond ±2 is unusual and beyond ±3 is rare.

Formula

z = (x − μ) ÷ σ
x = μ + z × σ
from a data list:  μ = Σd ÷ n,   σ = √( Σ(d − μ)² ÷ N )   or   √( Σ(d − μ)² ÷ (n − 1) ) for a sample

Example

IQ tests are scaled to a mean of 100 and a standard deviation of 15. An IQ of 130 has z = (130 − 100) ÷ 15 = 2, two standard deviations above the mean. Going the other way, a z-score of 2 on the same scale corresponds to 100 + 2 × 15 = 130.

From a data list: the twelve scores 70, 85, 90, 90, 95, 100, 100, 105, 110, 110, 115, 130 sum to 1,200, so their mean is 100. Their squared deviations sum to 2,700, so as a population the standard deviation is √(2,700 ÷ 12) = 15 and a score of 130 again has z = 2. Treated as a sample the standard deviation is √(2,700 ÷ 11) = 15.667 and the z-score is 1.9149.

Assumptions and limitations

• A z-score only says how far a value is from the mean in standard deviations. Turning it into a percentile or probability needs a distribution; for a normal distribution roughly 68% of values lie within 1 SD, 95% within 2 and 99.7% within 3.
• The mean and standard deviation you enter are taken as given. If they come from a sample, the z-score is relative to that sample.
• In the data-list mode, choose population when the values are the whole group of interest and sample when they are a sample from a larger one. Every entry in the list must be a number.

Frequently asked questions