Z-Score Calculator — standard score from value, mean and SD

How it works

A z-score puts a single value on a common scale so you can see how unusual it is, regardless of the original units. It measures the distance from the mean in units of standard deviations:

z = (x − mean) ÷ standard deviation

The top, x − mean, is how far the value sits above or below average; dividing by the standard deviation rescales that gap into "how many SDs". The sign tells you the direction — positive above the mean, negative below — and the size tells you how far out: a z of 1 is one SD above, a z of −2 is two SDs below, and z = 0 is exactly average.

Because every distribution's z-scores are on the same scale, you can compare values from completely different data sets — a test score and a height, say — by how extreme each is relative to its own group.

Worked example

A student scores 85 on a test where the mean is 70 and the standard deviation is 10:

z = (85 − 70) ÷ 10 = 15 ÷ 10 = 1.5

So the score is 1.5 standard deviations above the mean. A score of 60 in the same class gives z = (60 − 70) ÷ 10 = −1 (one SD below), and a score of exactly 70 gives z = 0. If the spread were wider — say SD 20 — the same 85 would be only z = 0.75, because the score is less unusual when scores vary more.

Frequently asked questions

How do I calculate a z-score?

Subtract the mean from your value, then divide by the standard deviation. For a value of 120 with mean 100 and SD 15: (120 − 100) ÷ 15 = 1.33. You need all three numbers — the value, the mean and the standard deviation of the group it belongs to.

What does a z-score tell me?

How far a value is from average, measured in standard deviations. A z of +1.5 means "1.5 SDs above the mean"; −0.5 means "half an SD below". It standardises values so extremes are easy to spot and so you can compare across different scales, since a z-score has no units.

What is a good or unusual z-score?

For roughly bell-shaped data, about 68% of values fall between z = −1 and +1, and about 95% between −2 and +2. So a z beyond ±2 is fairly unusual and beyond ±3 quite rare. "Good" depends on context — a high z is desirable for a test score but not for, say, blood pressure.

Can a z-score be negative or zero?

Yes. A **negative** z-score just means the value is below the mean, and its size still says how far (z = −2 is two SDs below). A z-score of **zero** means the value equals the mean exactly. Only the standard deviation must be positive.

How do I turn a z-score into a percentile or probability?

Look it up in a standard normal table or use a normal-distribution function: the table gives the proportion of a bell curve below that z. For instance z = 1.5 corresponds to about the 93rd percentile. That conversion assumes the data are approximately normally distributed.

What if the standard deviation is zero?

Then there's no z-score — every value equals the mean, so there's no spread to measure against and the formula would divide by zero. A z-score only makes sense when the data actually vary.