Z-Score — How to Calculate

A z-score answers one question: how many standard deviations away from the mean a value sits, with the sign telling you which side. That single number lets you compare a value to its group instead of reading the raw number alone.

Here $x$ is the value, $\mu$ is the mean, and $\sigma$ is the standard deviation.

How To Read The Z-Score At A Glance

The sign and size of $z$ map directly onto position in the data:

The numerator $x - \mu$ is the raw distance from the center; the denominator $\sigma$ rescales that distance by the typical spread. So a z-score does not just say "this score is 14 points above average," it says whether 14 points is a small or large gap for that particular data set. The reading is always relative to the mean and standard deviation you used. Change those, and the z-score changes.

Population Versus Sample: Which Numbers To Divide By

The version you use depends on what your numbers describe.

The calculation step is identical, subtract the center and divide by the spread. Only the interpretation shifts with whether the values describe a whole population or a sample.

Worked Example

A test score is $84$, the class mean is $70$, and the standard deviation is $7$. Substitute:

Subtract first:

Then divide:

The z-score is $2$, so the score sits $2$ standard deviations above the mean.

Where Z-Scores Help And Where Students Trip

Z-scores are the right tool when you want to compare values from different scales, spot unusually high or low observations, or connect raw data to a normal-distribution model. That last use carries a condition: converting a z-score into a probability is most meaningful when a normal model is appropriate, or when the problem tells you to use one.

The most common confusion points:

• Dividing by the variance instead of the standard deviation. A z-score uses standard deviation.
• Ignoring the sign. A z-score of $-2$ is below the mean; $2$ is above it. They are not the same.
• Mixing data from different groups. The same value can have one z-score in one class and a different z-score in another if the mean or standard deviation differs.

Run It Yourself

Pick any value, mean, and standard deviation, compute the z-score, and put the result into words. For a sharper test, choose numbers that make the z-score come out negative and confirm your interpretation still matches the sign.