A z score calculator shows how many standard deviations a value lies above or below the mean. A score of 85 on a test with mean 70 and standard deviation 10 has z = 1.5, about the 93rd percentile.
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How to use the z score calculator
- Enter the value x.
- Enter the mean.
- Enter the standard deviation.
- Read the z-score.
How the z score calculator works
The z score calculator measures how far a value lies from the mean in units of standard deviation. z = (x − μ) ÷ σ, where x is your value, μ is the mean and σ is the standard deviation. A positive z means the value is above the mean, a negative z means below, and zero means it equals the mean. The standard deviation must not be zero. The result is shown to four decimal places.
Z score calculator worked example
A test has a mean of 70 and a standard deviation of 10. A score of 85 gives z = (85 − 70) ÷ 10 = 1.5, so it is one and a half standard deviations above average. If scores follow a normal distribution, about 93.32% of people score below 1.5 standard deviations above the mean. The z score calculator gives the z value; the table adds the matching percentile from a standard normal table.
| x | Mean | SD | z | Percentile |
|---|---|---|---|---|
| 85 | 70 | 10 | 1.5 | 93.32% |
| 60 | 70 | 10 | −1.0 | 15.87% |
| 100 | 70 | 10 | 3.0 | 99.87% |
| 72 | 70 | 10 | 0.2 | 57.93% |
| 50 | 70 | 10 | −2.0 | 2.28% |
From z to percentile
For normally distributed data, the empirical rule says about 68% of values lie within one standard deviation of the mean, about 95% within two and about 99.7% within three. So z = −1 is near the 16th percentile, z = 0 is the 50th, and z = 2 is near the 98th. For exact percentiles, look up the z value in a standard normal table or use a p-value tool. These percentiles hold only if the data are close to normal.
Comparing different scales
Z-scores let you compare values measured on different scales. A student who scores 85 on a test with mean 70 and SD 10 (z = 1.5) did relatively better than one who scores 90 on a test with mean 80 and SD 10 (z = 1.0), even though 90 is the higher raw score. Researchers use the same idea to combine measures with different units, and clinicians use z-scores for growth charts and bone density reports.
Spotting unusual values
A common rule of thumb flags values with |z| above 3 as possible outliers, because they are rare in normal data. Use the rule with care: in small samples the mean and SD are themselves pulled by extreme values, and skewed data can produce many large z-values that are not errors. Look at the data before deleting anything.
Population or sample statistics
If you know the true population mean and standard deviation, use them. More often you estimate them from a sample, and the z score calculator then describes the value relative to that sample. For inference about a sample mean, divide by the standard error, σ ÷ √n, instead of σ.
Working backward from z
You can rearrange the formula to find the raw value for a target z: x = μ + z × σ. With mean 70 and SD 10, the score at z = 2 is 70 + 2 × 10 = 90, and the score at z = −0.5 is 65. This is useful for setting cut-offs, such as the score needed to be in the top 2.5%, which is about z = 1.96. Use the z score calculator to check your answer.
Limits of z-scores
A z-score tells you position, not importance. In heavily skewed data, such as incomes, a z-score of 2 may not be rare at all. Always look at a histogram before converting z-scores into percentiles, and report the mean and standard deviation you used so others can check the result.
Tips
- Check the SD is a standard deviation, not a variance.
- Keep the sign: negative means below the mean.
- Use percentiles only for roughly normal data.
- Round z to two decimal places when using tables.
Related converters and calculators
Related calculators:
- Standard deviation calculator: SD
- P-value calculator: tail areas
- Statistics calculator: summary stats
- Confidence interval calculator: intervals
Sources
OpenStax Introductory Statistics 2e, 6.1: The standard normal distribution; NIST/SEMATECH e-Handbook: Normal distribution table.
Formula and method
z = (x - mean) / sd. Requires non-zero sd.
Standard score for one observation relative to a stated mean and SD. Not a full hypothesis test.
Questions about this calculator
What is the z-score formula?
z = (x − mean) ÷ standard deviation.
What does a negative z mean?
The value is below the mean.
What percentile is z = 1.5?
About 93.32% for normal data.
What counts as an outlier?
Often |z| above 3, used with care.
Can the SD be zero?
No.
Check the assumptions and units before relying on a result. Financial, tax and health estimates do not replace advice for your individual circumstances.
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