Z-Score Formula Guide

Anatomy of the Z-Score Formula

The formula z = (x − μ) / σ has three components. The numerator (x − μ) is the deviation: how far the raw value x sits from the mean μ. If x is above the mean, the deviation is positive; if below, negative; if equal, zero.

The denominator σ is the standard deviation of the distribution. Dividing by σ scales the deviation into "standard deviation units" — making the result unit-free. A z-score of +1 always means "one standard deviation above the mean," regardless of whether the original units were centimetres, dollars, or milliseconds.

The result z tells you: (a) the direction — positive z means above average, negative means below average; (b) the magnitude — z = 2.0 means the value is farther from average than z = 0.5; (c) under a normal distribution — the exact percentile position the value occupies.

Step-by-step worked example: A student scores 82 on a maths exam. The class mean is μ = 70 and standard deviation is σ = 10. Step 1 — Deviation: x − μ = 82 − 70 = 12. Step 2 — Standardise: z = 12 / 10 = 1.2. Interpretation: the student scored 1.2 standard deviations above the class mean. Under a normal distribution, this corresponds to approximately the 88th percentile — scoring higher than about 88% of the class.

Reading a Z-Table — Converting Z to Probability

A z-table (also called a standard normal table) gives the cumulative probability P(Z ≤ z) — the proportion of the normal distribution that falls at or below a given z-score. This is the area under the normal curve to the left of z.

Most z-tables give values for z between −3.49 and +3.49. To use one: look up the row for the first two digits of z, then the column for the second decimal place. For z = 1.2: row 1.2, column 0.00 → P(Z ≤ 1.20) = 0.8849, meaning 88.49% of values fall below z = 1.2.

From one table lookup you can derive three probabilities: (1) P(X ≤ x) = Φ(z) — the value from the table directly. (2) P(X ≥ x) = 1 − Φ(z) — the right tail. (3) P(−|z| ≤ Z ≤ |z|) = 2Φ(|z|) − 1 — the two-tailed probability, used in hypothesis testing.

For the student example (z = 1.2): P(X ≤ 82) = 0.8849 (88.49th percentile). P(X ≥ 82) = 1 − 0.8849 = 0.1151 (11.51% scored higher). P(60 ≤ X ≤ 80) at z = −1.0 and z = 1.0: Φ(1.0) − Φ(−1.0) = 0.8413 − 0.1587 = 0.6827 ≈ 68.3% (the empirical rule for ±1σ).

The Empirical Rule (68–95–99.7 Rule)

For any approximately normal distribution, the empirical rule gives three key probability bounds based on z-scores:

|z| ≤ 1 (within one standard deviation of mean): 68.27% of values. Roughly two-thirds of any normal dataset falls within ±1σ of the mean.

|z| ≤ 2 (within two standard deviations): 95.45% of values. Only about 1 in 22 observations falls outside this range.

|z| ≤ 3 (within three standard deviations): 99.73% of values. Values with |z| > 3 are genuinely rare — occurring less than 3 times per 1,000 observations. In quality control, "six sigma" targets defect rates of 3.4 per million by demanding |z| ≤ 6.

The empirical rule is useful as a quick sanity check: if you calculate a z-score of 4.5 for a value that is supposed to come from a normal distribution, either the data point is an extraordinary outlier or your mean and standard deviation estimates are wrong.

Z-Score vs T-Score — When to Use Which

Both z and t standardise a value relative to a distribution. The critical difference is what you know about the population standard deviation σ.

Use a z-score when: (a) the population standard deviation σ is known (from historical data, specification sheets, or prior research); or (b) the sample is large (n ≥ 30), in which case the sample standard deviation s is a reliable enough estimate of σ that the distinction is negligible.

Use a t-score when: σ is unknown and must be estimated from the sample (the usual situation in research), AND the sample size is small (n < 30). The t-distribution has heavier tails than the normal distribution, accounting for the additional uncertainty introduced by estimating σ from limited data. As n increases, the t-distribution converges to the standard normal — for n = 30, t and z critical values differ by only about 0.03 at the 95% level.

Practical rule: in real data analysis, σ is almost never known. Use t unless you have a specific reason to know σ (e.g., you are a manufacturer with a decades-long process record). Calculators and software can handle either; the choice matters most for small samples.

Real-World Applications of Z-Scores

Standardised testing (SAT, GRE, IQ): All are reported with reference to a normalised scale. An IQ of 130 corresponds to z = (130−100)/15 = +2.0 — the 97.7th percentile. A GRE Verbal score is converted to a percentile by computing z = (score − population mean) / σ and looking up Φ(z).

Quality control (Six Sigma): Manufacturing processes are evaluated by their z-score — how many standard deviations the process mean is from the nearest specification limit. A "six sigma" process has z = 6, meaning only 3.4 defects per million opportunities. Z-scores quantify process capability.

Finance — identifying unusual returns: If a stock fund has a mean monthly return of 1.2% and standard deviation of 3.5%, a month with −6% return has z = (−6 − 1.2)/3.5 = −2.06 — an unusually bad month at the 2nd percentile. Z-scores help distinguish genuine shocks from routine volatility.

Medical reference ranges: Lab test reference ranges are usually set at the 2.5th to 97.5th percentiles (±1.96 standard deviations). A result outside this range has |z| > 1.96 and is flagged as abnormal — not because it cannot occur naturally, but because it is rare enough to warrant investigation.