Calculate a z-score from a value, mean, and standard deviation. See how many standard deviations the value is from the mean and estimate its percentile in a normal distribution.
Calculate a standard score from a value, mean, and standard deviation or convert a z-score to a percentile.
Enter a value, mean, and standard deviation to calculate its z-score.
A z-score shows how far a value is from the mean of a distribution, measured in standard deviations.
A positive z-score means the value is above the mean, a negative z-score means it is below the mean, and a z-score of 0 means the value is exactly equal to the mean.
In this formula, X is the value being measured, μ is the mean, and σ is the standard deviation.
Suppose a score is 85, the mean is 70, and the standard deviation is 10.
| Value | Amount |
|---|---|
| Score (X) | 85 |
| Mean (μ) | 70 |
| Standard Deviation (σ) | 10 |
The score is 1.50 standard deviations above the mean. In a standard normal distribution, this corresponds to approximately the 93.32nd percentile.
| Z-Score | Interpretation |
|---|---|
| 0 | The value is equal to the mean |
| +1 | One standard deviation above the mean |
| −1 | One standard deviation below the mean |
| +2 | Two standard deviations above the mean |
| −2 | Two standard deviations below the mean |
The farther the z-score is from 0, the farther the value is from the mean relative to the spread of the distribution.
A z-score can also be converted to a percentile using the standard normal distribution. The percentile represents the approximate percentage of values that fall below the selected z-score.
| Z-Score | Approximate Percentile |
|---|---|
| −2.00 | 2.28% |
| −1.00 | 15.87% |
| 0.00 | 50.00% |
| 1.00 | 84.13% |
| 1.50 | 93.32% |
| 2.00 | 97.72% |
Z-scores and percentiles both describe relative position, but they do it in different ways.
| Measure | What It Shows | Example |
|---|---|---|
| Z-Score | How many standard deviations a value is from the mean | Z = 1.50 means 1.5 standard deviations above the mean |
| Percentile | The approximate percentage of values below a score | 93.32nd percentile means about 93.32% fall below it |
Use a z-score when you know the mean and standard deviation. Use the Percentile Calculator when you have a raw data set and want to find percentile values or percentile ranks directly.
Z-scores are commonly used in statistics, education, research, and data analysis to compare values measured on different scales.