z-scores — Core Maths Compendium

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Understanding z-scores

Click each step below to reveal it — work through them in order the first time round.

Step 1 · Why we standardise

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Different normal distributions have different means and standard deviations, which makes them hard to compare directly. Standardising converts any normal value into a z-score, which tells you how many standard deviations above or below the mean that value is — on a universal scale that works for any normal distribution.

Step 2 · The z-score formula

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$z = \frac{x-\mu}{\sigma}$, where $x$ is the value, $\mu$ is the mean, and $\sigma$ is the standard deviation. A positive z-score means the value is above the mean; a negative z-score means it's below the mean.

Step 3 · Interpreting z-scores

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A z-score of 0 means the value equals the mean exactly. A z-score of 2 means the value is 2 standard deviations above the mean (an unusually high value). Z-scores let you compare values from completely different normal distributions on the same scale.

Step 4 · Using z-scores to find probabilities

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Once you've found a z-score, you can look it up on the standard normal distribution table (or use your calculator's normal distribution function) to find the probability of a value being less than, greater than, or between certain values.

Worked example

A student scores 68 in a test with mean 60 and standard deviation 8. Calculate their z-score, and interpret it.

$z=\frac{68-60}{8}=\frac{8}{8}=1$
The student's score is exactly 1 standard deviation above the mean — a fairly good, but not extreme, result.

Test yourself

Past-paper style question

In Maths, students' scores are normally distributed with mean 55 and standard deviation 10. In English, scores are normally distributed with mean 62 and standard deviation 6. Freya scores 70 in Maths and 71 in English.

(a) Calculate Freya's z-score for each subject.
(b) Using your answers, determine in which subject Freya performed relatively better compared to her peers, explaining your reasoning. [5 marks]

Show the answer

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(a) Maths: $z=\frac{70-55}{10}=\frac{15}{10}=1.5$. English: $z=\frac{71-62}{6}=\frac{9}{6}=1.5$.

(b) Both z-scores are exactly equal (1.5), meaning Freya performed equally well relative to her peers in both subjects — she is 1.5 standard deviations above the mean in each subject, even though her raw scores (70 vs 71) and each subject's mean/spread are different.

Practice

z-scores worksheet

Five short questions on z-scores. Work through them, then reveal the mark scheme to check.

A value of 45 comes from a normal distribution with mean 40 and standard deviation 5. Calculate its z-score.
A z-score of −2 comes from a distribution with mean 100 and standard deviation 15. Find the original value, x.
Explain what a z-score of 0 tells you about a value.
Two students' heights are compared: Student A is 180 cm tall in a population with mean 175 cm, sd 5 cm. Student B is 170 cm tall in a population with mean 162 cm, sd 4 cm. Use z-scores to determine who is relatively taller compared to their own population.
Explain why standardising values into z-scores is useful when comparing data from two different normal distributions.

Mark scheme

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$z=\frac{45-40}{5}=$ 1.
$-2=\frac{x-100}{15} \Rightarrow x-100=-30 \Rightarrow x=$ 70.
A z-score of 0 tells you the value is exactly equal to the mean of its distribution.
Student A: $z=\frac{180-175}{5}=1$. Student B: $z=\frac{170-162}{4}=2$. Student B has the higher z-score (2 vs 1), so Student B is relatively taller compared to their own population, even though their raw height (170cm) is lower than Student A's (180cm).
Because different distributions can have different means and standard deviations, raw values aren't directly comparable; standardising converts any value onto the same universal scale (number of standard deviations from its own mean), allowing fair comparison regardless of the original distribution's parameters.

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