Free z-score calculator
Calculate the z-score for a data point — enter a value, mean, and standard deviation to see how many standard deviations from the mean it falls, updated live, as you type.
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z = (x − μ) ÷ σ. How many standard deviations x is from the mean.
Results are estimates. Consult a professional.
How the z-score calculator works
A z-score (also called a standard score) expresses how many standard deviations a value sits above or below the mean of its distribution. It converts raw scores to a common scale, making it possible to compare values from different data sets or look up probabilities in a standard normal table.
Worked example: test score 78, mean 70, σ = 8
A student scores 78 on an exam. The class mean is μ = 70 and the standard deviation is σ = 8. How many standard deviations above the mean is this score?
Common z-scores and percentiles
For a standard normal distribution, each z-score maps to a cumulative percentile — the percentage of values at or below that point.
| z-score | Cumulative percentile | Interpretation |
|---|---|---|
| −3.0 | 0.13% | Far below average (bottom 0.13%) |
| −2.0 | 2.28% | Well below average (bottom 2.3%) |
| −1.0 | 15.87% | Below average (bottom 16%) |
| 0.0 | 50.00% | Exactly average |
| +1.0 | 84.13% | Above average (top 16%) |
| +2.0 | 97.72% | Well above average (top 2.3%) |
| +3.0 | 99.87% | Far above average (top 0.13%) |
Source: standard normal distribution table (Z-table). Values assume a perfectly normal distribution.
Tips for using z-scores
Z-scores are most useful when comparing across different scales or looking up probabilities.
- Use population parameters — the z-score formula uses the population mean μ and population std dev σ. If you only have a sample, use the t-score formula instead (which accounts for extra uncertainty).
- Interpret the sign — a positive z-score means the value is above the mean; a negative z-score means it is below. A z-score of 0 is exactly average.
- Use the z-table for probabilities — once you have a z-score, look up its cumulative probability to find what percentage of the distribution falls below that value.
- Compare unlike scales — z-scores let you compare a height in centimetres to a weight in kilograms by standardising both to the same dimensionless scale.
- Flag unusual values — z-scores beyond ±3 are rare in a normal distribution (< 0.3% of data) and often flag outliers worth investigating.
Accuracy and limitations
Z-score calculations are arithmetically straightforward and exact up to floating-point precision. The key limitation is distributional: z-score to percentile conversions assume the underlying data follows a normal distribution. For skewed, bimodal, or heavy-tailed data, the standard normal table will give misleading percentile estimates. Additionally, using sample statistics (x̄ and s) instead of population parameters (μ and σ) introduces extra uncertainty that the z-score formula does not capture — use a t-score in those cases.
Key terms
About this calculator
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