Z Score Calculator

What the z-score calculator does

A z-score says how many standard deviations a value sits from the mean. This calculator works it out from three numbers: the raw score, the mean, and the standard deviation. The result places your value on a common scale, so you can see at a glance whether it is ordinary or unusual.

Standardising a value like this is the first step behind percentiles, probabilities from the normal curve, and comparing scores measured on different scales. Below is how it works and how to read the answer.

How to use it

1. Enter the raw score, the value you want to place.
2. Enter the mean and standard deviation of the distribution it came from.
3. Press Calculate for the z-score, or Reset to clear it.

How the z-score is worked out

It is one subtraction and one division. Take how far the value is from the mean, then measure that distance in standard deviations:

z = (raw score minus the mean) ÷ the standard deviation

Subtracting the mean recentres the value so the mean becomes zero, and dividing by the standard deviation rescales it so one whole unit is one standard deviation. Whatever the original units were, the z-score comes out as a plain number.

Reading a z-score

The sign tells you which side of the mean the value is on. A positive z-score is above the mean, a negative one is below, and a z-score of zero is exactly on it. The size tells you how far. A z-score of 1 is one standard deviation out, a 2 is two, and so on.

That gives you an instant sense of how unusual a value is. On a bell-shaped distribution, most values fall between about minus 2 and 2, so a z-score beyond that is genuinely uncommon, and one beyond 3 is rare. A z-score of 1.5 is high but unremarkable; a z-score of 3.5 is the kind of value that makes you look twice.

Why standardising helps

The real power of a z-score is that it lets you compare things measured on different scales. A score of 85 on one test and 90 on another cannot be compared directly, because the tests had different means and spreads. Turn each into a z-score and they land on the same scale, and now you can say which result was actually stronger relative to its own group.

It is also the doorway to probability. Once a value is a z-score, the standard normal curve turns it into a percentile or a p-value, which is why standardising is the step that so many other calculations start from.

A worked example

Suppose a test has a mean of 70 and a standard deviation of 10, and you scored 85.

Your distance from the mean is 85 minus 70 = 15, and dividing by the standard deviation gives 15 ÷ 10 = 1.5. So your score sits one and a half standard deviations above the mean, comfortably above average but not extraordinary.

Entering your values

Enter the raw score, the mean, and the standard deviation. The mean and standard deviation should describe the distribution the raw score belongs to. The z-score is shown to four decimal places with trailing zeros trimmed. A negative result simply means the value is below the mean, which is perfectly normal.

Questions people ask

What is a z-score?

A measure of how many standard deviations a value is from the mean. It is found by subtracting the mean from the value and dividing by the standard deviation.

What does a negative z-score mean?

That the value is below the mean. The sign shows the side of the mean, and the size shows how many standard deviations away it is.

What counts as a high z-score?

On a bell-shaped distribution, most values fall between about minus 2 and 2. Beyond that is uncommon, and beyond 3 is rare, so those are the values worth a closer look.

Why turn a score into a z-score?

To compare values from different scales on equal terms, and to unlock probabilities and percentiles from the normal curve, which start from the z-score.

References

A quick note on where the methods here come from. The z-score, or standard score, is set out in the NIST/SEMATECH e-Handbook of Statistical Methods, the US government's public statistics reference. OpenStax Introductory Statistics is a free, widely used textbook that covers z-scores and the standard normal distribution.

1. NIST/SEMATECH e-Handbook of Statistical Methods (the normal distribution and standard scores). https://www.itl.nist.gov/div898/handbook/
2. OpenStax, Introductory Statistics (z-scores and the standard normal distribution). https://openstax.org/details/books/introductory-statistics-2e