Z-Score Formula
Calculate the z-score with z = (x - μ) / σ.
Find how many standard deviations a value is above or below the mean.
The Formula
z = (x - μ) / σ
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