CalcKitHome›Math & Stats›Z-score
Z-Score Calculator
Your numbers
The mean and standard deviation describe the population or group your value comes from. The standard deviation must be greater than zero.
A positive z-score means the value is above the mean; a negative z-score means it is below the mean.
85 is 2.00 standard deviations above the mean of 75. About 97.7% of values fall below 85, so it is around the 97.7th percentile.
Example: x = 85, mean = 75, standard deviation = 5 — z = 2.00, about the 97.7th percentile.
Behind the numbers
How the z-score is calculated
A z-score standardizes a value so you can compare it with any other value on the same scale, no matter what units the original numbers use. Subtract the mean, then divide by the standard deviation: the result is the number of standard deviations the value sits from the mean. The percentile is the share of a normal distribution that falls below that z-score, found from the standard normal cumulative distribution function.
where x is the value, μ is the mean and σ is the standard deviation. Worked example: x = 85, μ = 75, σ = 5 gives z = (85 − 75) ÷ 5 = 2.00. The percentile is the area under the normal curve to the left of 2.00, about 97.7%, so 85 is around the 97.7th percentile. To work backwards, use x = μ + zσ: with μ = 100, σ = 15 and z = 1.50, x = 100 + 1.50 × 15 = 122.5.
Worked examples
Example z-scores
| Value (x) | Mean (μ) | SD (σ) | Z-score | Percentile |
|---|---|---|---|---|
| 85 | 75 | 5 | 2.00 | 97.7th |
| 60 | 70 | 10 | −1.00 | 15.9th |
| 100 | 80 | 8 | 2.50 | 99.4th |
| 72 | 75 | 5 | −0.60 | 27.4th |
| 90 | 75 | 5 | 3.00 | 99.9th |
| 122.5 | 100 | 15 | 1.50 | 93.3rd |
A z-score of 0 means the value equals the mean, right at the 50th percentile. In a normal distribution about 68% of values fall between z = −1 and z = 1, about 95% fall between z = −2 and z = 2, and about 99.7% fall between z = −3 and z = 3.
Using z-scores the right way
The sign tells you the direction: a positive z-score is above the mean, a negative z-score is below it. A z-score of −1.00 for a test score of 60 with a mean of 70 means the score is 1 standard deviation below the class average.
The percentile only means what it says if the data is roughly normal. For skewed data, the z-score is still a fair way to compare values, but the percentile from the normal curve will not match the real share of values below it.
Use the reverse mode when you know where a value should sit: a z-score of 1.50 with a mean of 100 and a standard deviation of 15 means a value of 122.5, which is around the 93.3rd percentile.
Do not mix up a z-score with a percentile. A z-score of 2.00 is not the 200th percentile — it is the 97.7th percentile, because percentiles run from 0 to 100 while z-scores have no upper limit.
Good to know
Your z-score questions, answered.
What is a z-score?
A z-score tells you how many standard deviations a value is from the mean of its distribution. It is calculated as z = (x − mean) / standard deviation. A z-score of 2.00 means the value sits 2 standard deviations above the mean, and a z-score of −1.00 means it sits 1 standard deviation below the mean.
What does a z-score of 2.00 mean?
A z-score of 2.00 means the value is 2 standard deviations above the mean. About 97.7% of values in a normal distribution fall below it, so it is around the 97.7th percentile and only about 2.3% of values are higher.
How do I find the percentile from a z-score?
The percentile is the area under the normal curve to the left of the z-score, which is the value of the standard normal cumulative distribution function at that z-score. For example, a z-score of 1.50 gives a percentile of about 93.3%, and a z-score of −1.00 gives about 15.9%.
Can I work backwards from a z-score to a value?
Yes. Rearrange the formula to x = mean + z times the standard deviation. For example, with a mean of 100 and a standard deviation of 15, a z-score of 1.50 gives x = 100 + 1.50 times 15 = 122.5. Choose the “Find x from z, mean and standard deviation” mode on this calculator to do it.