Statistics calculator

Z Score Calculator

ValueMeanStandard deviation

Z score: -

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Calculation explained

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The formula, your substituted values, and the evaluated result are shown below.

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How the result changes when the first input changes

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This calculator applies the z score relationship. Enter values in the stated units and review the result. Check assumptions and use appropriate professional guidance for safety-critical decisions.

About this calculator

Calculate z score using clearly labelled inputs.

Calculator guide

How the z-score calculator works

A z-score tells you how many standard deviations a value sits above or below the mean:

z = (value − mean) ÷ standard deviation

A positive z is above the mean, a negative z is below it, and 0 is exactly average.

Worked example: an exam result

Your mark: 78
Class mean: 65
Standard deviation: 8
z = (78 − 65) ÷ 8 = 1.625

If the marks are roughly normally distributed, a z of 1.625 means you scored higher than about 95% of the class (the 94.8th percentile).

Reading z-scores

z-score	Approx. percentile (normal distribution)
−2.0	2.3%
−1.0	15.9%
0	50%
+1.0	84.1%
+1.645	95%
+2.0	97.7%
+3.0	99.9%

About 68% of values fall within ±1 standard deviation, 95% within ±1.96, and 99.7% within ±3.

What z-scores are used for

Comparing results on different scales: a z of 1.2 in maths and 0.8 in English shows the stronger result relative to each class.
Spotting outliers: values beyond ±3 are unusual and worth checking.
Hypothesis tests and confidence intervals: z = 1.96 marks the edges of a 95% interval.
Quality control: manufacturing tolerances are often set in standard deviations.

Common mistakes

Using the variance instead of the standard deviation (the SD is the square root of the variance).
Reading percentiles off a normal table when the data is heavily skewed.
Mixing up population and sample standard deviations on small samples.

What does a z-score of 2 mean?

The value is 2 standard deviations above the mean, higher than about 97.7% of a normal distribution.

Can a z-score be negative?

Yes. A negative z-score means the value is below the mean.

How do I convert a z-score to a percentile?

Look it up in a standard normal table or use a statistics function. For example, z = 1 is about the 84th percentile.

Written by Sam Alex, civil engineer. About the author