Descriptive statistics

Z-Score Calculator

How many standard deviations a value sits from the mean — and what share of the distribution falls below it.

Also called z stat calculator, standard score calculator, z-value calculator.

Updated October 2026Runs in your browser — nothing is uploadedVerified against R

How to read this result

A z-score puts any value on a common ruler. Subtract the mean, divide by the standard deviation, and a test score, a height and a share price all become comparable: z = 1.5 means "one and a half standard deviations above average" whatever the original units were.

The percentile assumes normality; the z-score does not. Standardising is arithmetic and always valid. Turning z into "the 93rd percentile" uses the normal distribution, and if your data are skewed that translation is wrong even though the z-score is right.

Rules of thumb. |z| beyond 2 covers about 5% of a normal distribution; beyond 3, about 0.3%. Those are the thresholds behind the common outlier rules — see outlier detection for why they misfire on small samples.

With n supplied, you are standardising a mean, not a value. The denominator becomes σ/√n, and z gets much larger for the same gap. That is the central limit theorem at work: means vary far less than individual observations, so the same absolute difference is far more surprising.

The formula

z = \frac{x - μ}{σ} or,forasamplemean z = \frac{\bar{x} - μ}{σ / \sqrt{n}}

$x$: the value being standardised
$μ, σ$: the mean and standard deviation of the distribution
$n$: the sample size, when standardising a mean

Worked example

One student against the class

A test has a mean of 70 and a standard deviation of 10. A student scores 85. How unusual is that?

Subtract the mean.
85 − 70 = 15
Divide by the standard deviation.
15 / 10 = 1.5
Look 1.5 up in the standard normal distribution.
Φ(1.5) = 0.9332
So the score sits above 93.3% of the distribution, and 6.7% score higher.
percentile = 93.3

z 1.50percentile 93.3%P(X > 85) 6.68%two-tailed p 0.1336

If instead this were the mean of 25 students, the standard error would be 10/√25 = 2 and z would be 7.5 — a class average of 85 is far more remarkable than one student scoring 85.

Checked against the standard normal distribution — pnorm(1.5) in R.

The calculator above is loaded with these numbers by the Load the worked example button.

Assumptions, and when to use something else

Standardising requires a mean and a standard deviation, and it requires them to be the right ones: the population values if you have them, the sample estimates if you do not. With a sample estimate and a small n, the resulting score is more properly a t statistic than a z, and the percentile should come from the t distribution.

The percentile output additionally assumes the distribution is normal. It is the one part of this page that fails on skewed data.

You want the probability rather than the scoreNormal distribution calculatorAreas under the curve for any mean and standard deviation.
Your sample is small and σ is estimatedt distribution calculatorThe heavier-tailed distribution that applies when σ comes from the data.
You are looking for outliersOutlier detectionThe z-score rule plus two more robust alternatives.
You want a rank rather than a normal percentilePercentile calculatorThe empirical percentile from the data itself, with no distributional assumption.

Questions people ask

What is a good z-score?

It depends on what you want. Above 0 is above average; above 1.65 is the top 5%; above 2.33 the top 1%. For quality control, |z| beyond 3 is the classic alarm threshold. There is nothing intrinsically good about any particular value.

Can a z-score be negative?

Yes — it simply means the value is below the mean. The sign carries the direction and the magnitude carries the distance.

What is the difference between a z-score and a t-score?

A z-score uses the known population standard deviation; a t-score uses one estimated from a sample. With small samples the estimate is uncertain, which makes the t distribution heavier-tailed than the normal — so the same score corresponds to a larger p-value.

How do I convert a z-score back to a raw value?

Multiply by the standard deviation and add the mean: x = μ + zσ. The normal distribution calculator does it directly in its inverse mode.