Z-score calculator
The z-score counts how many standard deviations your value sits above or below the mean. Positive is above, negative is below.
Show the step-by-step calculation
Z-Score Calculator
A z-score tells you how many standard deviations a value sits from the mean. Enter your value, mean, and standard deviation to get it instantly.
The z-score, also called the standard score, standardizes any value by re-expressing it in standard deviation units. The formula is z = (x - mean) / SD: subtract the mean to find how far the value is from center, then divide by the standard deviation to express that distance in units of spread. A z-score of 1.5 means the value is one and a half standard deviations above the mean. A z-score of -2 means two standard deviations below it. Because z-scores are unit free, they let you compare values from completely different scales, such as a test score against a height measurement.
This calculator computes the z-score live as you type, states the direction and distance in plain language, and expands the two arithmetic steps below the result. To build an interval around a mean using z-values, use the confidence interval calculator. If you need the standard deviation itself from raw data, the standard deviation solver computes it with full working.
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Z values for common percentiles
The table below lists standard normal quantiles: the z-score at which a given percentage of a normal distribution falls below it. These are the values behind the most common percentile and confidence statements.
| Percentile (standard normal) | z value |
|---|---|
| 90th percentile | 1.2816 |
| 95th percentile | 1.6449 |
| 97.5th percentile | 1.9600 |
| 99th percentile | 2.3263 |
The 97.5th percentile value of 1.9600 is the familiar two-sided 95 percent cutoff: 2.5 percent of a normal distribution lies above it and 2.5 percent below its negative, leaving 95 percent in between. The same z-values power the confidence levels in the confidence interval calculator.
Worked example
A student scores 85 on a test where the class mean is 70 and the standard deviation is 10. How many standard deviations above average is that score?
Step 1, subtract the mean from the value: 85 - 70 = 15. The score is 15 points above the class average.
Step 2, divide by the standard deviation: z = 15 / 10 = 1.5. The score of 85 sits 1.5 standard deviations above the mean. Under an approximately normal distribution of scores, that corresponds to roughly the 93rd percentile: the student outscored about 93 percent of the class. Enter 85, 70, and 10 in the calculator above and you will see exactly this result with the same steps.
Frequently asked questions
What is a z-score?
A z-score tells you how many standard deviations a value sits away from the mean of its distribution. The formula is z = (x - mean) / SD. A z-score of 0 means the value equals the mean, a z-score of 2 means it is two standard deviations above the mean, and a z-score of -1.5 means it is one and a half standard deviations below the mean.
What does a negative z-score mean?
A negative z-score simply means the value is below the mean. It says nothing about the value being bad or unusual on its own. A z-score of -0.4 is a perfectly typical value that happens to sit a little under the average, while a z-score of -3 is a rare value far below the mean.
What counts as an unusual z-score?
A common rule of thumb: absolute z-scores under 1 are very typical, between 1 and 2 are still common, between 2 and 3 are uncommon, and above 3 are rare. Under a normal distribution, only about 5 percent of values fall beyond 1.96 standard deviations from the mean, and under 1 percent fall beyond 2.5758.
How is a z-score related to percentiles?
If the underlying distribution is approximately normal, a z-score maps directly to a percentile. A z-score of 1.2816 is the 90th percentile, 1.6449 is the 95th, 1.9600 is the 97.5th, and 2.3263 is the 99th. The table on this page lists these standard normal quantiles. Without approximate normality, the z-score still measures distance from the mean, but the percentile mapping no longer holds exactly.
Can I compute a z-score for any data?
The arithmetic always works as long as the standard deviation is greater than zero. What changes is interpretation. For roughly symmetric, bell-shaped data the percentile interpretation is accurate. For skewed data, a z-score still measures distance in standard deviation units, but phrases like top 5 percent should not be read off the normal table.
What if the standard deviation is zero?
A standard deviation of zero means every value in the data set is identical, so there is no spread to measure against. Dividing by zero is undefined, and this calculator asks for a standard deviation greater than zero instead of returning a meaningless number.