Z-score

Before you rely on it

What to check

It is only interpretable as a probability if the distribution is roughly normal.

The common error

Where people go wrong with z-score

Reading probabilities off a z-score from skewed data. The score computes, the interpretation does not hold, and nothing warns you.

Worked through

Z-score in plain numbers

A test has a mean of 70 and a standard deviation of 8. A student scores 82. The z-score is (82 − 70) ÷ 8, which is 1.5. She sits one and a half standard deviations above the mean. If scores follow a bell curve, about 93% of students score below her, and a student with 58 has a z-score of −1.5, near the 7th percentile. Compare an 82 on this test with a 90 on another test whose mean is 80 and standard deviation is 10: the second student has a z-score of 1.0, so the 82 was the stronger result. Z-scores put different scales on the same footing.

Easily confused

Z-score and its neighbours

A z-score is not a percentile, a t-score or a raw score; a percentile gives the share of the group below a value. The z-score gives distance from the mean in standard deviation units, and it maps to a percentile only if the distribution is roughly normal; a t-score uses a different mean and scale, and the t-statistic in hypothesis testing is a different idea again. Standardizing means converting to z-scores. The Altman Z-score in finance is an unrelated formula for bankruptcy risk that shares only the name.

In real decisions

Where z-score turns up

Teachers and researchers use z-scores to compare results across tests and groups. Statisticians use them in hypothesis tests and confidence intervals. Quality control uses them to flag unusual measurements, such as those beyond 3. Pediatricians use them to describe growth measures against reference populations; analysts use them to standardize variables before comparing or combining them; a calculator converts a value, mean and standard deviation to a z-score and vice versa.

Know the edges

Where z-score stops being reliable

Interpretation as a percentile assumes a normal distribution, and many real datasets are skewed. Outliers distort the mean and standard deviation on which z depends. Small samples give unstable estimates. A large z-score marks an unusual value and does not say why. Look at the data and the context before drawing conclusions.

Quick answers

What people ask about z-score

How do I convert a z-score to a percentile? Use a standard normal table or a calculator. A z of 1.0 is about the 84th percentile, and a z of 2.0 about the 98th, if the data are roughly normal.

What z-score is unusual? Values beyond 2 or below −2 occur about five percent of the time in a normal distribution. Beyond 3 is rare, about three in a thousand.

Can I use z-scores on skewed data? You can compute them, but the percentile interpretation will be off. Consider a transformation or a non-parametric method.