What a z-score means
A z-score (standard score) measures how far a value sits from the mean, in units of standard deviation: z = (value − mean) ÷ standard deviation. A z of 0 is exactly the mean; +1.5 means 1.5 standard deviations above it. For a value of 85, mean 70, SD 10: (85 − 70) ÷ 10 = 1.5.
The subtraction shifts the mean to zero and the division rescales so one unit equals one standard deviation, giving a unit-free result. You need the spread of the data before you can standardise anything, which is what the standard deviation calculator provides.
How to use this calculator
Enter the value, the mean, and the standard deviation. A positive z-score is above the mean, negative is below. To turn a z-score into a probability, use the p-value calculator.
The mean and standard deviation must describe the same group the value came from, and all three must share the same units. A standard deviation of zero makes the z-score undefined, since every value in such a data set is identical.
Interpreting the size of a z-score
In a roughly normal distribution the empirical rule gives a quick sense of scale:
- About 68% of values lie within z = −1 to +1.
- About 95% lie within z = −2 to +2.
- About 99.7% lie within z = −3 to +3.
So a z-score of 1.5 is above average but not unusual, while anything beyond 3 is rare and often flagged as an outlier. These percentages assume the data is approximately normal — for a strongly skewed distribution the z-score still measures distance from the mean, but the probabilities attached to it no longer hold. To express the same position as a rank out of a hundred, use the percentile calculator.
Frequently asked questions
- How do I calculate a z-score?
- Subtract the mean from the value, then divide by the standard deviation. (85 − 70) ÷ 10 = 1.5.
- What does a z-score of 1.5 mean?
- The value is 1.5 standard deviations above the mean.
- What does a negative z-score mean?
- The value is below the mean by that many standard deviations.
- How do I get a probability from a z-score?
- Use the standard normal distribution — a p-value calculator converts a z-score into a tail probability.
- Should I use the sample or the population standard deviation?
- Use the population figure when you have data for the whole group, and the sample figure when your data stands in for a larger group. The sample version is slightly larger, so it produces slightly smaller z-scores — worth noting with small data sets.