Z-Score — How to Calculate

A z-score tells you how far a value is from the mean in units of standard deviation. That makes it useful when you want to compare a score to the rest of a group, not just read the raw number by itself.

The basic formula is

$$z = \frac{x - \mu}{\sigma}$$

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

If $z$ is positive, the value is above the mean. If $z$ is negative, the value is below the mean. If $z = 0$, the value is exactly at the mean.

How To Calculate A Z-Score

Use these steps:

1. Start with the value $x$.
2. Subtract the mean.
3. Divide by the standard deviation.

That is all the calculation is doing: turning a raw distance from the mean into a standardized distance.

What The Formula Means Intuitively

The numerator $x - \mu$ tells you how far the value is from the center. The denominator $\sigma$ rescales that distance by the typical spread of the data.

So a z-score does not just say "this score is 14 points above average." It says whether 14 points is a small gap or a large one for that particular data set.

Worked Example

Suppose a test score is $84$, the class mean is $70$, and the standard deviation is $7$.

Plug those numbers into the formula:

$$z = \frac{84 - 70}{7}$$

First subtract:

$$84 - 70 = 14$$

Then divide:

$$z = \frac{14}{7} = 2$$

The z-score is $2$. That means the score is $2$ standard deviations above the mean.

A Quick Way To Read The Answer

• $z = 1$ means one standard deviation above the mean.
• $z = -1.5$ means one and a half standard deviations below the mean.
• A larger absolute value such as $|z| = 3$ means the value is relatively far from the mean.

This interpretation works only relative to the mean and standard deviation you used. Change those, and the z-score changes too.

Common Mistakes

One common mistake is dividing by the variance instead of the standard deviation. A z-score uses standard deviation, not variance.

Another mistake is ignoring the sign. A z-score of $-2$ is not the same as $2$. The first is below the mean, and the second is above it.

It is also easy to mix data from different groups. A score can have one z-score in one class and a different z-score in another class if the mean or standard deviation changes.

When People Use Z-Scores

Z-scores are useful 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 needs a condition: converting a z-score into a probability is most meaningful when a normal model is appropriate, or when the problem explicitly tells you to use one.

Population Vs. Sample Numbers

In many statistics formulas, the z-score is written with population symbols:

$$z = \frac{x - \mu}{\sigma}$$

If you only have a sample mean $\bar{x}$ and sample standard deviation $s$, people often standardize with

$$\frac{x - \bar{x}}{s}$$

The calculation step is the same, but the interpretation depends on whether those values describe a full population or only a sample.

Try Your Own Version

Pick any value, mean, and standard deviation, then compute the z-score and explain the result in words. If you want a useful next case, solve a similar problem where the z-score comes out negative and check that your interpretation still matches the sign.