z-Scores

Is a height of 190\,\text{cm} "tall"? Is a test score of 85 "good"? You genuinely cannot say — not from the number alone. 190\,\text{cm} is towering in a room of jockeys and unremarkable in a line-up of basketball centres; 85 is brilliant if the class average was 60 and mediocre if it was 90. To judge a value you need to know two things about the crowd it came from: where the middle is, and how spread out the crowd is.

The z-score bottles exactly that judgement into a single number. It measures a value in units of standard deviations from the mean:

z = \frac{x - \mu}{\sigma}.

It answers one universal question: how many standard deviations is x away from the mean? Whatever the value measured — centimetres, marks, seconds, dollars — the z-score comes out as a pure "how unusual is this" number that any two situations can be compared on.

Reading the sign

So z = -2 means "two standard deviations below the mean", and z = 1.5 means "one and a half standard deviations above it". The size of |z| tells you how unusual the value is — by the empirical rule, a z beyond \pm 2 is already quite rare, and one beyond \pm 3 is a genuine outlier.

Worked example 1 — compute and interpret

A cohort's exam scores are normal with mean \mu = 72 and standard deviation \sigma = 8. Priya scored 88. What is her z-score, and what does it mean?

Plug straight into the formula:

z = \frac{x - \mu}{\sigma} = \frac{88 - 72}{8} = \frac{16}{8} = 2.

Her z = +2: she is two standard deviations above average. That is a strong result — by the empirical rule only about 2.5\% of the cohort sits above z = 2, so she is roughly in the top 1 in 40. Notice the units cancelled: the 16 marks in the numerator and the 8 marks in the denominator leave a bare 2.

Worked example 2 — compare across different tests

Here is where z-scores earn their keep. Sam scores 90 on a hard test (mean 70, SD 10). Lee scores 95 on an easy test (mean 90, SD 2). Lee's raw mark is higher — but who actually did better relative to their test?

Standardise both:

z_{\text{Sam}} = \frac{90 - 70}{10} = 2.0, \qquad z_{\text{Lee}} = \frac{95 - 90}{2} = 2.5.

Lee's z = 2.5 beats Sam's z = 2.0, so Lee's result is the more exceptional — even though on the hard test 90 "feels" more impressive. The raw numbers are not comparable; the z-scores are. Whenever a spread differs, only 2 points above a tight mean (\sigma = 2) can outrank 20 points above a loose one (\sigma = 10).

Worked example 3 — turn a z-score into a percentile

Because the empirical rule tells you the area under a bell curve, a z-score can be read as a rough percentile — what fraction of the crowd you are above. Take Priya's z = +2 from example 1.

The rule says 95\% lies within 2\sigma, so 5\% lies outside, split as 2.5\% in each tail. The fraction below z = 2 is therefore everything except the upper tail:

100\% - 2.5\% = 97.5\%.

Priya is at about the 97.5th percentile — higher than roughly 97.5\% of the cohort. The same reasoning gives quick percentiles at the landmark z-scores: z = 0 \to 50\text{th}, z = 1 \to 84\text{th} (since 100\% - 16\% = 84\%), z = -1 \to 16\text{th}.

Slide the z-marker

Drag the slider to move the marker to x = z on the standard normal curve. Read off where it lands: near the centre for small |z|, far out in a thin tail as |z| grows toward 3.

A tempting mistake: "I computed z-scores, so now my data is normal and I can use the 68–95–99.7 rule." Not so. Subtracting the mean and dividing by \sigma only shifts and rescales the values so they have mean 0 and SD 1. It slides and stretches the picture — it does not change its shape.

If the original data was skewed, the z-scores are exactly as skewed. The arithmetic is perfectly valid — every value still has a well-defined z — but you may not then claim "95\% lie within z = \pm 2", because that percentage is a property of the bell shape, which the raw data never had. z-scores standardise position, never shape. Turning a value into a percentile via the empirical rule only works when the distribution was already (approximately) normal to begin with.

z-scores are the machinery behind the scoreboards you already know. Standardised tests like the SAT and IQ scales don't report raw marks at all — they convert each person's raw score to a z-score, then map that onto a fixed, friendly scale. IQ, for instance, is built to have mean 100 and SD 15, so an IQ of 130 is simply z = +2 dressed up in nicer numbers. "Grading on a curve" is the same trick: a teacher standardises the class's raw marks and assigns letter grades by z-score bands, so the shape of the class decides the grades, not a fixed pass mark.

Sports analysts lean on it hardest of all. How do you compare a hitter from the 1920s with one from today, when the whole league's averages have drifted? You standardise each player against their own era's mean and spread. A "+3 z-score season" is dominant whether it happened in 1927 or 2024 — the z-score quietly cancels the era, the league, and the units, and leaves only "how far above everyone else they stood".

See it explained