Z Score Formula
The z score formula is z = (X − μ) / σ: take the value, subtract the mean, divide by the standard deviation. The result is the value's distance from the mean measured in standard deviations — the one number every normal-distribution table, chart and calculator on this site runs on.
The formula in words, and why each step earns its place
Subtract the mean first. X − μ centres the value: it asks "how far from average is this, in the original units?" A score 15 marks above a mean of 70 gives +15; a temperature 4 degrees below its average gives −4. The sign at this stage is the sign your z score will carry, and it already means something before any division happens.
Then divide by the standard deviation. σ rescales that distance into standard units. Fifteen marks means little until you know the spread: in a class with σ = 10 it is a large gap (z = 1.5); in one with σ = 30 it is unremarkable (z = 0.5). Division is what makes z scores comparable across different tests, different units and different subjects — a z of 1.5 in maths and a z of 1.5 in height are the same statement about standing, which no raw score can claim.
Population version vs sample version
The two versions are the same idea with different symbols, and the choice is bookkeeping, not mathematics: μ and σ when the population parameters are known (standardised tests publish theirs; quality specifications state theirs), x̄ and s when you computed the centre and spread from the data in front of you. What you may not do is mix the pairs — a population mean with a sample standard deviation is a small act of fiction that markers notice.
Five worked examples
Example 1 — exam marks
X = 85, μ = 70, σ = 10. z = (85 − 70) / 10 = 15 / 10 = 1.50. One and a half standard deviations above the mean; the table converts it to the 93.32nd percentile.
Example 2 — a below-average value (negative z)
A reading score of X = 54 with μ = 60, σ = 8: z = (54 − 60) / 8 = −6 / 8 = −0.75. The formula produces the negative sign by itself — never add it, remove it, or apologise for it. Below average is a location, not a mistake.
Example 3 — exactly average
X = μ always gives z = 0, whatever σ is, because the numerator is 0. A z score of 0 is the most ordinary result possible: the 50th percentile, dead centre of the curve.
Example 4 — comparing across subjects
A student scores 78 in maths (μ = 65, σ = 8) and 82 in English (μ = 72, σ = 12). Raw marks say English was better. Z scores disagree: maths z = (78 − 65) / 8 = 1.625; English z = (82 − 72) / 12 = 0.83. Relative to each class, the maths result (93.31st percentile at z = 1.63, roughly) is far stronger. This comparison — impossible with raw marks — is the formula's entire reason to exist.
Example 5 — the sample version
A lab sample of components has x̄ = 24.6 mm and s = 0.4 mm. A component measuring 25.4 mm: z = (25.4 − 24.6) / 0.4 = 2.00. Two sample standard deviations above the sample mean — far enough out (top 2.28%) that a quality engineer would look at it twice.
The formula run backwards: finding X
Rearrange once and the formula answers the reverse question — what raw value sits at a given z? X = μ + zσ. With μ = 500 and σ = 100 (the old SAT scale), the 90th percentile (z = 1.28) is X = 500 + 1.28 × 100 = 628. With μ = 60, σ = 8, a z of −0.75 gives X = 60 − 6 = 54 — Example 2's score, recovered. Cutoff-setting in admissions, grading and screening is this rearrangement plus a reverse table lookup; the pair of skills together covers nearly every applied z problem set in courses. The step-by-step version, with a calculator, is on how to calculate z score.
What the formula assumes (and what it does not)
The formula itself assumes nothing about shape — you can standardise any dataset, skewed or not, and the arithmetic is legal. What changes is the interpretation: the trip from z to a table probability ("z = 1.5 means top 6.68%") only works when the underlying distribution is approximately normal. On a heavily skewed distribution, z = 1.5 is still 1.5 standard deviations above the mean, but the share beyond it is whatever the data says, not 6.68%. Check the shape before quoting the percentile; the formula will not warn you, because formulas don't.
Two properties follow purely from the algebra and are worth stating once: standardising never changes the order of values (the largest X keeps the largest z), and the z scores of any dataset always have mean 0 and standard deviation 1 — which is why the standard normal table, built for exactly that mean and spread, can serve every dataset after conversion.
The same formula, grown up: standard error and the z-test
Later in any statistics course the formula quietly returns in a second costume. When the "value" being standardised is a sample mean rather than a single observation, the spread in the denominator becomes the standard error, σ/√n, and the formula becomes the z-test statistic: z = (x̄ − μ) / (σ/√n). Nothing about the logic changed — it is still "distance divided by spread" — but the spread now measures how much sample means wobble, which is why larger samples shrink the denominator and make the same gap more significant. Recognising the family resemblance early is worth real marks: students who meet the z-test as a stranger memorise it; students who meet it as the z score formula with a new denominator simply use it. The fences it is judged against live on the critical z values page.
Units: the formula's quiet superpower
Notice what happened to the units along the way: marks minus marks, divided by marks, leaves a pure number. A z score has no units — not marks, not centimetres, not seconds — and that is exactly why one table can serve every subject on earth. Physicists call this dimensional consistency; for our purposes it is simpler to remember that if your z score seems to carry a unit ("1.5 marks"), the division step was skipped, and the number on the page is a deviation, not a z score.
Frequently asked questions
What is the z score formula with mean and standard deviation?
z = (X − μ) / σ for a population: subtract the mean from the value, then divide by the standard deviation. For a sample, the same structure with the sample statistics: z = (x − x̄) / s. Both give the distance from the centre in standard deviation units.
Why do we divide by the standard deviation?
Division converts a raw distance (marks, centimetres, seconds) into a unit-free count of standard deviations. That is what makes results from different distributions comparable and lets one standard table serve every normal problem.
Can the z score formula give a negative answer?
Yes — whenever X is below the mean, the numerator is negative and so is z. About half of any dataset produces negative z scores. The negative table handles their probabilities.
What if the standard deviation is 0?
Then every value equals the mean, there is no spread to measure distance in, and z is undefined (division by zero). It can only happen when all values are identical — a dataset with nothing to analyse.
Is z the same as standardising in machine learning?
Yes — "z-score standardisation" (or normalisation) in data science is this exact formula applied feature by feature, usually with sample statistics. It puts variables on a shared scale so distance-based algorithms treat them fairly — the comparison trick from Example 4, industrialised.
Next: apply the formula with guidance on how to calculate z score, and look up your result on the z score table.