Free Z-score Calculator (+ Guide)

What is a z-score calculator?

A z-score calculator turns a value into standard deviations from the mean, and finds the share of a normal distribution below and above it.

This one also works backward, from a percentile to its z-score and value.

How it works

The calculator works it out on your device, and your numbers stay there.

The formula

OpenStax's Introductory Statistics defines the z-score as z = (x − μ) ÷ σ. A positive z sits above the mean, and a negative one below it.

Its example: with a mean of 5 and a standard deviation of 2, a value of 11 has a z-score of 3.

The empirical rule

For a normal distribution, about 68% of values sit within 1 standard deviation of the mean, about 95% within 2, and about 99.7% within 3.

That's why a z-score past 2 or 3 is rare, and worth a second look.

A worked example

Here's the example the calculator loads, OpenStax's own: a value of 11, a mean of 5 and a standard deviation of 2.

The z-score is 3. About 99.87% of values fall below it and 0.13% above, and the two-tailed p-value is about 0.0027.

Mistakes to avoid

These are the slips that come up most.

• Using the variance instead of the standard deviation
• Reading a percentile from a z-score on data that isn't normal
• Mixing up one-tailed and two-tailed p-values
• Forgetting the sign: a negative z-score is below the mean

Who is the z-score calculator for?

It's for students, researchers and anyone comparing a number to the rest, and these tools help around it.

• Standard deviation calculator – for the spread of your numbers
• Statistics calculator – for every statistic at once
• Probability calculator – for the chance of events

Find the mean and standard deviation first, then let the curve show you where any value stands.