Z-Score Formula
Reference for z-score z = (x − μ) / σ measuring standard deviations from the mean.
Covers z-table lookup, p-value interpretation, and test score comparison.
The Formula
The z-score tells you how many standard deviations a data point is from the mean. A positive z-score means the value is above the mean. A negative z-score means it is below the mean.
Variables
| Symbol | Meaning |
|---|---|
| z | The z-score (number of standard deviations from the mean) |
| x | The individual data value |
| μ | The population mean |
| σ | The population standard deviation |
Example 1
Exam scores have a mean of 75 and standard deviation of 10. A student scored 92.
x = 92, μ = 75, σ = 10
z = (92 - 75) / 10
z = 17 / 10
z = 1.7 — The student scored 1.7 standard deviations above the mean.
Example 2
Average height of adults is 170 cm with a standard deviation of 8 cm. Someone is 158 cm tall.
x = 158, μ = 170, σ = 8
z = (158 - 170) / 8
z = -12 / 8
z = -1.5 — This person is 1.5 standard deviations below the average height.
When to Use It
Use the z-score formula when:
- Comparing values from different data sets with different scales
- Determining if a value is unusual or an outlier (z > 2 or z < -2 is often considered unusual)
- Looking up probabilities in a standard normal distribution table
- Standardizing data for statistical analysis
Key Notes
- Formula: z = (x − μ) / σ: Measures how many standard deviations a data point x lies from the mean μ. A z-score of +2 means x is 2 standard deviations above average; −1.5 means 1.5 below. The sign conveys direction.
- Standard normal distribution: After z-transformation, data follows N(0, 1). A z-table (or calculator) gives the cumulative probability P(Z < z) — the fraction of the distribution below that z-score.
- The 68-95-99.7 rule: For any normal distribution, ~68% of values fall within ±1σ, ~95% within ±2σ, and ~99.7% within ±3σ. A z-score of ±3 is rare (3 per 1,000); ±4 is very rare (1 per 31,000).
- Z vs t-score: Use z when the population standard deviation σ is known and the sample size n ≥ 30. Use the t-distribution for small samples or when σ is estimated from the sample — t accounts for the extra uncertainty in estimating σ.
- Applications: Z-scores standardize test results (SAT, IQ), detect outliers (|z| > 3 flags unusual observations), support quality control (Six Sigma targets z = 6), and underpin hypothesis testing and confidence intervals.