What Is a Z-Score? Standard Deviations Above the Mean

The formula and what each part means

The z-score subtracts the mean and then divides by the standard deviation: z = (x − μ) / σ. The subtraction re-centers the data so the mean sits at zero, and the division rescales it so one unit equals one standard deviation. What comes out is a pure number saying how many standard deviations, and in which direction, the value lies from the center.

Take an exam with mean μ = 70 and standard deviation σ = 8. A score of x = 86 gives z = (86 − 70) / 8 = 16 / 8 = 2, so it is two standard deviations above average. A score of 62 gives z = (62 − 70) / 8 = −1, one below. The sign carries the direction: positive is above the mean, negative below, and the size tells you how unusual the value is.

Reading the standard normal curve

When data follow a normal distribution, z-scores map onto the standard normal curve — a bell shape with mean 0 and standard deviation 1. The empirical rule summarizes it: about 68% of values lie within z = ±1, about 95% within ±2, and about 99.7% within ±3. A z-score beyond ±3 is genuinely rare, occurring for well under 1% of a normal population.

Because the curve is symmetric, negative and positive z-scores mirror each other. A z of +1 and a z of −1 sit the same distance from center on opposite sides. This symmetry is what lets a single z-table cover every case: look up the positive value, then use symmetry for the negative one.

Turning a z-score into a percentile

A percentile is the share of values that fall below yours. The area under the standard normal curve to the left of a z-score gives that share directly. A z of 0 sits at the 50th percentile — half the values are below it. A z of +1 corresponds to about the 84th percentile, and +2 to about the 97.7th, so a score two standard deviations above the mean beats roughly 98% of the group.

This is where z-scores earn their keep for comparison. Suppose you score 86 on a test with mean 70 and σ = 8 (z = 2), while a friend scores 130 on a different test with mean 100 and σ = 20 (z = 1.5). Even though 130 is the bigger raw number, your z of 2 places you higher on the curve — about the 98th percentile versus the 93rd — so relative to each group, you did better.

Frequently asked questions

What does a negative z-score mean?

A negative z-score simply means the value is below the mean. A z of −1.5, for instance, sits one and a half standard deviations below average. The sign shows direction, not quality.

Do I need normally distributed data to compute a z-score?

No. The formula z = (x − μ) / σ works for any data set. But turning a z-score into a percentile using the standard normal curve is only valid when the data are approximately normal.

What is the difference between σ and a z-score?

The standard deviation σ is the spread of the data in its own units. A z-score expresses one value's distance from the mean measured in those σ units, so it is unitless. σ describes the whole set; a z-score describes one point.

What counts as an unusually high z-score?

As a rule of thumb, |z| above 2 is fairly uncommon (outside about 95% of values) and |z| above 3 is rare (outside about 99.7%). Many analysts flag values beyond ±3 as potential outliers worth checking.