Z-score:
where a value sits relative to its group

A raw number needs a context

An 85 on a test might be a triumph or an ordinary result. It depends on what everyone else scored and how spread out those scores were. In a class where the average was 70 and most marks fell within 10 points of that, 85 is clearly strong. In a class that averaged 82 with a narrow spread, it is just above the middle.

The z-score expresses that context as one number. It restates the value in units of standard deviation, measured from the mean, so scores from different tests, scales and even different quantities can sit on the same ruler.

The idea is older than the name. Statisticians call the result a standard score because it is the original value measured on the standard scale of the group. Other fields use their own versions: IQ scores are rescaled so that the mean is 100 and the SD is 15, and some exam boards report standardised marks with a fixed mean and spread, so the same logic sits behind many published scales.

The formula

The sign tells you the direction, the magnitude tells you the distance. A z of −1 is one deviation below the mean, a score of 60 in the example above.

Comparing values from different distributions

The most practical use of a z-score is a fair comparison. Say a student scored 85 in mathematics, where the class mean was 70 with SD 10, and 78 in science, where the mean was 60 and SD 9.

The raw marks suggest mathematics was the stronger result. The z-scores say the opposite: 78 in science is two deviations above the class, which is better relative performance than 1.5 in mathematics. This is the whole point of standardising.

Turning z into a percentile

If the data follow a roughly normal, bell-shaped distribution, a z-score maps to a percentile through the standard normal curve. The mapping is not linear, because most observations bunch near the centre.

For the 85 in the example, z = 1.5 corresponds to about the 93rd percentile, provided the marks are close to normal. A z-score is not a percentile itself, and reading z = 1.5 as 1.5 per cent or the 1.5th percentile is a common slip.

Software and spreadsheets will give the exact cumulative probability for any z through a normal distribution function, so you do not need to interpolate by hand. The table above is meant for building intuition and spot checks, particularly the pairs near 1 and 2 that appear most often in practice.

Computing z-scores from a small data set

When you only have a list of values, compute the mean and standard deviation first, then standardise each value. Take five test marks: 62, 70, 74, 80 and 84.

The example highlights a real choice. If your five marks are the entire group of interest, the population standard deviation applies. If they are a sample from a larger group, the sample SD with n − 1 is more appropriate. The two diverge noticeably in small data sets and converge as the count grows.

After standardising, the five z-scores always have a mean of 0 and an SD of 1, whatever the units of the original data. This property is what makes them comparable between different tests, and it is also a handy check on your arithmetic.

Rules of thumb and flagging outliers

Under a normal model, about 68% of values fall within one standard deviation of the mean, about 95% within two and about 99.7% within three. That gives a quick way to judge any z-score without a table.

• |z| below 1: ordinary, within the central two thirds of values.
• |z| between 1 and 2: noticeable but common, covering roughly 27% of values in total.
• |z| between 2 and 3: unusual, about 4.3% of values.
• |z| above 3: rare in a normal model, and a common trigger for checking the record for an error.

Treat a high z as a prompt to look closer, not as proof of a mistake. In skewed data, such as incomes or waiting times, large z-scores are routine, and the percentile mapping above no longer holds.

Working backwards, and the caveats

You can reverse the formula to find a raw value from a z-score: x = μ + z × σ. A z of 2 in mathematics (mean 70, SD 10) is 70 + 2 × 10 = 90. This is how cut-off marks for the top few per cent are found.

• Use the standard deviation of the right group: a class, a population or a reference sample, not another group's.
• For a sample, state whether you used the sample SD with n − 1.
• Small groups give shaky means and deviations, so z-scores computed from them are rough.
• Check the shape of the data before reading a percentile from the normal table.

Finally, remember that a z-score does not carry the units of the original data. A z of 1.5 can be a mark, a height or a delivery time. The units live in the mean and SD, so quote them alongside the z when you report a result.