What Is a Z-Score and How Do You Calculate It?

You scored 85 on a test. Is that impressive? It depends on how everyone else did. A z-score answers that by measuring how far your value sits from the average, counted in standard deviations. It turns one raw number into a fair, shared scale, so you can compare it against the whole group and against scores from other tests too.

What a Z-Score Is

A z-score describes a single value by comparing it to the whole group. It tells you how many standard deviations that value is from the mean. The mean is the average, and the standard deviation measures how spread out the data is.

Our sibling guide on standard deviation covers that spread in detail. A z-score, also called a standard score, uses it to turn any raw number into a common scale. That lets you compare values from different tests fairly. A positive z-score sits above the mean, a negative one sits below, and a z-score of zero lands right on it.

Think about comparing a math score and a reading score. The raw points cannot be lined up, because the two tests use different scales. Z-scores fix this. Once both scores become z-scores, they share one scale, so you can see which result was stronger relative to its own group.

The Z-Score Formula

The z-score formula is short and easy to use:

z = (x – mean) / SD

Here x is your raw value, mean is the group average, and SD is the standard deviation. You subtract the mean from your value, then divide by the standard deviation. The result is a single number that captures both direction and distance at once.

The top part, x minus mean, is the raw distance from average. Dividing by SD rescales that distance into standard-deviation units. That is why a z-score has no units of its own. The same formula works whether you use a population or a sample standard deviation. Just know which spread your data uses, since conventions vary by course and software.

You can also read the formula in reverse. If you know a z-score, you can find the raw value with x = mean + z times SD. This is handy when a problem gives you a z-score and asks for the original number behind it.

A Worked Example

Suppose a class takes a test. The mean score is 70 and the standard deviation is 10. Ana scored 85. How unusual is that?

Plug the numbers into the formula:

z = (85 – 70) / 10
z = 15 / 10
z = 1.5

Ana’s score is 1.5 standard deviations above the mean. That is clearly above average.

Now take Ben, who scored 60. His z-score is:

z = (60 – 70) / 10
z = -10 / 10
z = -1.0

Ben’s score sits 1 standard deviation below the mean. The negative sign tells you it is below average. With just the formula, you can place any score on the same scale and see how it compares.

Notice how the z-scores rank the two students without needing the raw points. Ana at 1.5 is above average, and Ben at -1.0 is below. The gap between their z-scores, which is 2.5, shows how far apart they landed in standard-deviation terms.

Reading Positive and Negative Z-Scores

The sign and the size of a z-score both carry meaning. The sign tells you the direction from the mean, and the size tells you the distance. Here is how to read common values:

• z = 0: the value equals the mean, right in the middle.
• Positive z: the value is above the mean. Larger positive numbers are further above.
• Negative z: the value is below the mean. More negative numbers are further below.
• z near 0, say between -1 and 1: fairly typical, close to average.
• z beyond +2 or below -2: unusual, far from the center.

A z-score of 1.5 and a z-score of -1.5 sit the same distance from the mean. They just fall on opposite sides of it.

When you compare two z-scores, the absolute value is what shows how extreme each one is. A z-score of -2.3 is more unusual than a z-score of 1.2, even though one is negative. Ignore the sign for a moment, and the bigger number is the more surprising result.

Z-Scores and the 68-95-99.7 Rule

Z-scores connect to the standard normal distribution. This bell-shaped curve has a mean of 0 and a standard deviation of 1. When data is roughly normal, z-scores follow a pattern called the empirical rule, or the 68-95-99.7 rule.

The rule describes how much data falls near the mean:

• About 68% of values fall within +/-1 standard deviation of the mean.
• About 95% of values fall within +/-2 standard deviations.
• About 99.7% of values fall within +/-3 standard deviations.

So a z-score between -1 and 1 covers most of the group. A z-score past +/-3 is rare. This rule only holds for roughly normal data, so treat it as a guide, not a guarantee.

Using Z-Scores to Spot Outliers

The empirical rule also helps you flag unusual values. Many analysts treat any value with a z-score beyond +/-3 as a possible outlier. It is a rough screen, not a firm rule. Still, it gives a quick, consistent place to start. To turn z-scores into exact ranks, see our guide on percentiles and quartiles.

Frequently Asked Questions About Z-Scores

Sources

This article is for general education only, not professional statistical advice. Methods and notation can vary by field, course, or software, so follow your textbook or instructor for exact conventions. Reviewed for accuracy by Prof. Dr. Khalil Mudassar, PhD. Last updated September 11, 2026.