Z-Score Calculator
Find the z-score, percentile and probability for a value in a normal distribution.
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Reference values worked out by this calculator. Your z-score is highlighted.
How z-scores work
A z-score is one of the most useful numbers in statistics, because it puts any value on a common scale. This calculator works it out for you in seconds. It is free, with no sign-up, and it runs entirely in your browser. It is made for students checking a test score, teachers comparing classes, and researchers who want a quick standardised result.
What is a z-score?
A z-score, also called a standard score, tells you how many standard deviations a value is from the mean. A z-score of 0 means the value is exactly the mean, a z-score of 1 means it is one standard deviation above the mean, and a z-score of minus 1 means it is one standard deviation below it. The sign tells you the direction, and the size tells you how far away it is.
What does a z-score of 1.5 mean?
A z-score of 1.5 means the value is 1.5 standard deviations above the mean. In a normal distribution, about 93.32 percent of values fall below a z-score of 1.5, so the value sits at the 93rd percentile. Put simply, it is higher than about 93 out of every 100 values.
How do you calculate a z-score?
Subtract the mean from your value, then divide the answer by the standard deviation. The formula is z equals x minus μ, divided by σ, where x is your value, μ is the mean, and σ is the standard deviation. For example, a value of 85, a mean of 70 and a standard deviation of 10 gives 85 minus 70, which is 15, divided by 10, which is 1.5.
What is a percentile, and how does it relate to a z-score?
A percentile is the percentage of values in a normal distribution that fall below a given value. The percentile is found from the area under the normal curve to the left of the z-score. A z-score of 0 is the 50th percentile, because half the values fall below the mean, while a z-score of 1 is about the 84th percentile and a z-score of 2 is about the 98th percentile.
What is a good z-score?
There is no good or bad z-score on its own: it depends on what you are measuring. In exams, a positive z-score usually means a better than average result, while in medical tests a high z-score can mean a reading that needs attention. As a rule of thumb, most values fall between minus 2 and plus 2, so anything outside that range is unusual.
Worked example
A student scores 85 on a test where the class mean is 70 and the standard deviation is 10. The z-score is 85 minus 70, divided by 10, which is 1.5, so the student is 1.5 standard deviations above the mean. The percentile is 93.32 percent, so the score beats about 93 percent of the class, and the chance of a random student scoring higher is about 6.68 percent. Try it in the calculator above.
Why do exam boards and researchers use z-scores?
Z-scores let you compare results that were measured on different scales. A score of 85 on one test and 40 on another cannot be compared directly, but their z-scores can. The same idea is used to standardise test scores, compare lab results, and spot outliers in a data set. The OpenStax statistics textbook uses z-scores in its chapter on the normal distribution for exactly this reason.
Frequently asked questions
Can a z-score be negative?
Yes. A negative z-score simply means the value is below the mean. For example, a z-score of minus 1 means the value is one standard deviation below the mean, at about the 16th percentile.
What z-score is unusual?
As a rule of thumb, z-scores between minus 2 and plus 2 cover about 95 percent of values, so scores outside that range are uncommon. Z-scores beyond minus 3 or plus 3 cover 99.7 percent of values, so they are very rare and worth a second look.
What is the difference between a z-score and a percentile?
A z-score is measured in standard deviations from the mean, while a percentile is measured as a percentage of values below your value. They describe the same position in two ways: a z-score of 1.5 is the same as the 93.32nd percentile.
Does a z-score tell me the probability?
Yes, in a normal distribution. The percentile is the probability that a random value falls below yours, and 1 minus the percentile is the probability that it falls above yours. This calculator shows both, so you can see the chance in either direction.
What if my data is not normally distributed?
The z-score itself still works as a standardised distance from the mean, but the percentile and probabilities only hold if the data follows a roughly normal shape. For skewed data, the percentile can be misleading, and a different method may be better.
Can I use a z-score for a single test score?
Yes. That is one of its most common uses: a single score, a class mean and a standard deviation are all you need. You do not need the whole data set, just the summary numbers.
Does this calculator store my data?
No. The calculator runs entirely in your browser. Nothing you type is uploaded, stored or shared, and closing the page clears it.
More tools you may need
Need the spread of your data set first? Try our Standard Deviation Calculator. Testing whether a result is significant? Try our P-Value Calculator.
