A z-score (or standard score) measures how many standard deviations a data point is from the mean. It converts raw scores into a standardised scale that enables comparison across different datasets.
The Z-Score Formula
z = (x − μ) ÷ σ
Where:
- x = individual data point
- μ (mu) = population mean
- σ (sigma) = population standard deviation
For a sample, replace μ with x̄ (sample mean) and σ with s (sample SD).
Worked Example
A student scores 72 on an exam. The class mean is 65, and the standard deviation is 8.
z = (72 − 65) ÷ 8 = 7 ÷ 8 = 0.875
This student scored 0.875 standard deviations above the mean.
Interpreting Z-Scores
| Z-score | Interpretation | Percentile (approx.) |
|---|---|---|
| −3 | Extremely below average | 0.1% |
| −2 | Well below average | 2.3% |
| −1 | Below average | 15.9% |
| 0 | At the mean | 50.0% |
| +1 | Above average | 84.1% |
| +2 | Well above average | 97.7% |
| +3 | Extremely above average | 99.9% |
The 68-95-99.7 Rule
In a normal distribution:
- 68% of data falls within ±1 standard deviation
- 95% within ±2 standard deviations
- 99.7% within ±3 standard deviations
Converting Z-Score to Percentile
Once you have a z-score, look up the standard normal table (Z-table) or use:
Percentile = Φ(z) × 100
Where Φ is the cumulative normal distribution function.
Example: z = 1.5 → Φ(1.5) = 0.9332 → 93.3rd percentile
Applications of Z-Scores
Finance:
- Altman Z-Score predicts bankruptcy risk
- Used in risk management to identify outliers
Healthcare:
- BMI for age z-scores for children
- Bone density (DXA) T-scores are a form of z-score
Quality control:
- Six Sigma uses z-scores to measure process capability
- A "6-sigma" process has a z-score of 6 (3.4 defects per million)
Standardising test scores:
- IQ scores: mean 100, SD 15 (a z-score of +2 → IQ 130)
- SAT scores: mean 1000, SD 200 (scaled from z-scores)
Comparing Scores Across Different Tests
Example: Alice scored 80 on Test A (mean 70, SD 10). Bob scored 55 on Test B (mean 40, SD 8).
Alice's z = (80 − 70) ÷ 10 = 1.0
Bob's z = (55 − 40) ÷ 8 = 1.875
Despite the lower raw score, Bob performed better relative to his peers.