Z-Scores Explained: Comparing Values on Different Scales

The formula and how to read it

z = (x − mean) ÷ standard deviationx is the value; the mean and standard deviation describe the group it belongs to.

• z = 0 means the value equals the mean.
• z = +1 means one standard deviation above the mean; z = −1 means one below.
• The sign shows the direction, and the size shows how unusual the value is within its group.
• Many published score scales are simply rescaled z-scores. A scale with a mean of 100 and a standard deviation of 15 turns any z into 100 + 15 × z.

A z-score has no units. Whether the original data was in points, centimetres or dollars, the result is just a count of standard deviations, which is exactly what makes comparison possible.

Worked example: which result is stronger?

Test A has a mean of 70 and a standard deviation of 5. Test B has a mean of 500 and a standard deviation of 100. A student scores 78 on A and 640 on B.

Test A's result is 1.6 standard deviations above its mean and Test B's is 1.4, so the 78 on Test A is the relatively stronger performance, even though 640 is a much bigger number. The z-score calculator does the same arithmetic for any inputs.

Turning a z-score into a percentile

If the data follows a normal (bell-shaped) distribution, each z-score matches a fixed percentile. A z of 1.6 corresponds to about the 94.5th percentile, meaning around 94.5% of the group scored lower. A z of 1.4 corresponds to about the 91.9th percentile. A percentile is a rank, not a percentage of marks: the 94.5th percentile means beating about 94.5% of the group, not scoring 94.5%.

A few anchor points are worth remembering:

Because the curve is symmetric, about 68.3% of values lie within ±1, 95.4% within ±2 and 99.7% within ±3 of the mean. For any value in between, the normal distribution calculator gives the exact area, and the z-score to p-value calculator gives the tail probabilities used in hypothesis tests.

Working backwards: from a z-score to a value

Rearranging the formula gives the value that matches a chosen z-score:

x = mean + z × standard deviation

Worked example: adult heights in a group have a mean of 170 cm and a standard deviation of 8 cm. A height of 186 cm has z = (186 − 170) ÷ 8 = 2, while 158 cm has z = (158 − 170) ÷ 8 = −1.5. Going the other way, a z of 1.96 on Test B corresponds to 500 + 1.96 × 100 = 696, the cut-off above which about 2.5% of a normal distribution lies.

Using z-scores to spot unusual values

A common rule of thumb flags values with a z-score beyond ±2 as unusual and beyond ±3 as very unusual. In a normal distribution, only about 4.6% of values fall outside ±2 and about 0.3% outside ±3.

Treat this as a prompt to look closer, not as proof of an error. In a data set of a thousand values, you would expect a few beyond ±3 by chance alone. And the outlier itself inflates the mean and standard deviation used to compute the z-score, which can make a single extreme value look less extreme than it is.

Where z-scores can mislead

• Non-normal data. The percentile table only holds for roughly bell-shaped data. For skewed data, a z of 2 may not mean the 97.7th percentile.
• Wrong reference group. A z-score is only meaningful relative to the group whose mean and standard deviation you used.
• Small samples. With few values, the sample mean and standard deviation are themselves uncertain, so the z-score is rough.
• Sample versus population. If you use the population mean and standard deviation, the result is a z-score. When you estimate them from a small sample, tests normally use a t statistic instead.