Z Score Calculator
Find the z-score of any value from a mean and standard deviation, or straight from your own data. See the percentile and a clear step-by-step solution.
Z Score Calculator
Choose how you want to calculate, enter your numbers and get your result instantly.
Step-by-Step Solution
What Is a Z Score?
A z-score tells you how many standard deviations a value is above or below the mean of its data set. It lets you compare a single value to the rest of the data in a way that does not depend on the units used.
A z-score of 0 means the value is exactly equal to the mean. A positive z-score means the value is above the mean, and a negative z-score means it is below the mean.
Here x is your value, μ is the mean and σ is the standard deviation. When you work with a sample, the sample mean and the sample standard deviation are used in the same way.
How to Calculate a Z Score
You can find a z-score by hand in three steps.
1. Find the mean and standard deviation
If they are not given, calculate them from your data. Our Average Calculator and Standard Deviation Calculator can help.
2. Subtract the mean from your value
Calculate x − μ. This is how far your value is from the mean.
3. Divide by the standard deviation
Divide that distance by σ. The result is the z-score.
Z Score Calculator Examples
Example 1: a value above the mean
A student scores 85 on a test. The class mean is 70 and the standard deviation is 10.
z = (85 − 70) ÷ 10 = 1.5. The score is 1.5 standard deviations above the mean, which is about the 93.32nd percentile if the scores follow a normal distribution.
Example 2: a value below the mean
A machine part measures 40 mm. The mean is 50 mm and the standard deviation is 5 mm.
z = (40 − 50) ÷ 5 = −2. The part is 2 standard deviations below the mean, which is about the 2.28th percentile.
Example 3: from a data set
Find the z-score of 9 in this population data:
The mean is 5 and the population standard deviation is 2, so z = (9 − 5) ÷ 2 = 2.
How to Interpret a Z Score
The further a z-score is from zero, the more unusual the value is compared with the rest of the data. For data that follows a normal distribution, the following guide applies:
| Z Score Range | Share of Values in That Range |
|---|---|
| Between −1 and 1 | About 68% |
| Between −2 and 2 | About 95% |
| Between −3 and 3 | About 99.7% |
Many people treat a z-score beyond 2 or below −2 as unusual, and beyond 3 or below −3 as very unusual. These are only rules of thumb, not strict limits.
Z Score and Percentile
If your data is roughly normally distributed, a z-score can be turned into a percentile. The percentile is the share of values that fall below yours. Our calculator shows this as the area to the left of the z-score under the standard normal curve, and also the area to the right.
For example, a z-score of 0 is the 50th percentile, because half of the values lie below the mean. Keep in mind that the percentile is only reliable when the data is close to normal.
Sample or Population?
When you calculate from a data set, you need to choose between a population and a sample. Choose population when your data includes every member of the group you care about. Choose sample when your data is only a part of a larger group.
The only difference is the standard deviation: a population divides the squared distances by n, while a sample divides by n − 1. The sample choice gives a slightly larger standard deviation, so the z-score is slightly smaller. To see the details, try our Variance Calculator.
More Z Score Examples
These extra worked examples show how to compare scores from different tests, how to work backwards from a z-score to a value, how the sample and population choices differ, and how to tell whether a value is unusual. Remember that z = (x − μ) ÷ σ, and that a percentile from a z-score is only reliable when the data is roughly normal.
Example 4: Comparing two different tests
A student scores 78 in Maths, where the class mean is 70 and the standard deviation is 5. The same student scores 85 in English, where the class mean is 75 and the standard deviation is 10. Which result is better compared with the class?
Answer: Maths is better compared with the class, even though the English mark is higher.
Maths: z = (78 − 70) ÷ 5 = 8 ÷ 5 = 1.6. English: z = (85 − 75) ÷ 10 = 10 ÷ 10 = 1. A z-score of 1.6 is about the 94.52nd percentile, and a z-score of 1 is about the 84.13rd percentile (if both sets of scores are roughly normal). The raw marks cannot be compared directly because the tests have different means and spreads, but z-scores put them on the same scale.
Example 5: Finding the value from a z-score
A test has a mean of 100 and a standard deviation of 15. What score has a z-score of −1.2, and what score has a z-score of 2?
Answer: a z-score of −1.2 is a score of 82, and a z-score of 2 is a score of 130.
Rearrange the formula to x = μ + z × σ. For z = −1.2: x = 100 + (−1.2) × 15 = 100 − 18 = 82. For z = 2: x = 100 + 2 × 15 = 130. Check: (82 − 100) ÷ 15 = −1.2 and (130 − 100) ÷ 15 = 2.
Example 6: Sample or population from a data set
Find the z-score of 8 in the data 4, 8, 6, 5, 7, first as a population and then as a sample.
Answer: the population z-score is about 1.4142, and the sample z-score is about 1.2649.
The mean is (4 + 8 + 6 + 5 + 7) ÷ 5 = 6. The deviations are −2, 2, 0, −1 and 1, and the squared deviations are 4, 4, 0, 1 and 1, which add up to 10. Population: the variance is 10 ÷ 5 = 2 and the standard deviation is √2 ≈ 1.4142, so z = (8 − 6) ÷ 1.4142 ≈ 1.4142. Sample: the variance is 10 ÷ 4 = 2.5 and the standard deviation is √2.5 ≈ 1.5811, so z = (8 − 6) ÷ 1.5811 ≈ 1.2649. The sample standard deviation is larger, so the z-score is a little smaller.
Example 7: Is this value unusual?
Adult heights in a group have a mean of 170 cm and a standard deviation of 6 cm. How unusual is a height of 188 cm, and how unusual is a height of 176 cm?
Answer: 188 cm has a z-score of 3, which is very unusual. 176 cm has a z-score of 1, which is quite ordinary.
For 188 cm: z = (188 − 170) ÷ 6 = 18 ÷ 6 = 3. That is about the 99.87th percentile, so only about 0.13% of heights are taller (for roughly normal data). For 176 cm: z = (176 − 170) ÷ 6 = 6 ÷ 6 = 1, which is about the 84.13th percentile. Many people treat a z-score beyond 2 or below −2 as unusual, and beyond 3 or below −3 as very unusual, but these are rules of thumb and not strict limits.
Common mistakes when finding a z-score
A common mistake is subtracting in the wrong order, which flips the sign of the z-score. Another is using the wrong standard deviation, such as the sample value when the data is a whole population. Also remember that a percentile from a z-score is only reliable when the data is close to normal. If you need the standard deviation first, use our Standard Deviation Calculator, and to find the rank of a value directly in your own data, try our Percentile Calculator.
Created by Sajid Ali, SmartStatCalc · Last updated: 7 November 2026
Found a mistake or have a suggestion? Contact us.
How to Use Our Z Score Calculator
- Choose a calculation method: use the mean and standard deviation, or your own data set.
- Enter your numbers in the boxes.
- For a data set, choose population or sample, and enter a value x if you want one specific z-score.
- Click the Calculate Z Score button.
- Read the z-score, the percentile and the step-by-step solution.
Click Reset at any time to clear everything and start again.
Why Use SmartStatCalc?
SmartStatCalc is built to make everyday statistics quick and easy to understand. Instead of working through the formula by hand, you can enter your numbers and see a clear result right away.
Who Can Use a Z Score Calculator?
Students use z-scores in statistics and science classes, and teachers use them to compare test results. Researchers, analysts and quality teams use them to find unusual values and to compare data measured on different scales.
Frequently Asked Questions
What is a z-score in simple words?
A z-score shows how many standard deviations a value is away from the mean.
What is the z-score formula?
The formula is z = (x − μ) ÷ σ, where x is the value, μ is the mean and σ is the standard deviation.
Can a z-score be negative?
Yes. A negative z-score means the value is below the mean.
What does a z-score of 0 mean?
It means the value is exactly equal to the mean.
What is a good z-score?
It depends on the situation. A higher z-score means a value is further above the mean, which is good for a test score but not for something like a defect rate.
Can the standard deviation be zero?
If the standard deviation is zero, all values are the same and a z-score cannot be calculated, because the formula would divide by zero. Our calculator shows a message in this case.
How do I find a z-score from a data set?
First find the mean and standard deviation of the data, then use the z-score formula. Our calculator does both steps for you when you choose the data set method.
Is this Z Score Calculator free?
Yes. SmartStatCalc's Z Score Calculator is free to use online.
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If you use this calculator in your work, you can cite it like this:
SmartStatCalc. (2026). Z Score Calculator. https://smartstatcalc.com/z-score-calculator/