OneTaskBy Tim King · Updated
Z-Score Calculator
Z Score to Percentile, Tail Areas and Two-Tailed P-Value, With Working
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How the Z-Score Calculator works
This z-score calculator tells you how many standard deviations a value sits above or below the mean. Enter the value, the mean and the standard deviation and you get the z-score, the percentile it falls at, the area in each tail of the normal curve and the two-tailed p-value, with the working written out line by line.
A z-score, also called a standard score, puts any measurement on a common scale. A score of 0 is exactly average, +1 is one standard deviation above, and −2 is two below. Because the scale is the same for every data set, you can compare an exam mark with a reaction time or a birth weight and see which one is more unusual.
How the z-score calculator works
The formula is z = (x − μ) ÷ σ, where x is your value, μ is the mean and σ is the standard deviation. The calculator then looks up the standard normal cumulative distribution Φ(z), which is the share of a normal population below your value. One minus that is the area to the right, and twice the smaller tail is the two-tailed p-value.
First example. A test has a mean of 70 and a standard deviation of 10. A mark of 85 gives z = (85 − 70) ÷ 10 = 1.5, which is the 93rd percentile.
Second example. A baby weighs 2.6 kg where the mean is 3.4 kg and σ is 0.5 kg. The z-score is −1.6, so only about 5.5% of babies are lighter.
How many standard deviations is "unusual"? Values beyond ±2 happen about 5% of the time and values beyond ±3 only 0.27% of the time, so most fields treat |z| > 2 as worth a second look.
Common z-scores with the share of a normal distribution below and above them
| Z-score | Below (percentile) | Above | Two-tailed p |
|---|---|---|---|
| -3 | 0.135% | 99.87% | 0.0027 |
| -2.576 | 0.5% | 99.5% | 0.01 |
| -2.326 | 1.001% | 99% | 0.02002 |
| -2 | 2.275% | 97.72% | 0.0455 |
| -1.96 | 2.5% | 97.5% | 0.05 |
| -1.645 | 4.998% | 95% | 0.09997 |
| -1.5 | 6.681% | 93.32% | 0.13361 |
| -1.282 | 9.992% | 90.01% | 0.19984 |
| -1 | 15.87% | 84.13% | 0.31731 |
| -0.674 | 25.02% | 74.98% | 0.50031 |
| -0.5 | 30.85% | 69.15% | 0.61708 |
| 0 | 50% | 50% | 1 |
| 0.5 | 69.15% | 30.85% | 0.61708 |
| 0.674 | 74.98% | 25.02% | 0.50031 |
| 1 | 84.13% | 15.87% | 0.31731 |
| 1.282 | 90.01% | 9.992% | 0.19984 |
| 1.5 | 93.32% | 6.681% | 0.13361 |
| 1.645 | 95% | 4.998% | 0.09997 |
| 1.96 | 97.5% | 2.5% | 0.05 |
| 2 | 97.72% | 2.275% | 0.0455 |
| 2.326 | 99% | 1.001% | 0.02002 |
| 2.576 | 99.5% | 0.5% | 0.01 |
| 3 | 99.87% | 0.135% | 0.0027 |
| 3.5 | 99.98% | 0.023% | 0.00047 |
Tips
- Population or sample? Use the population SD when you know it; with a sample, a t-score is more honest for small n.
- Negative is fine. A negative z-score just means the value is below the mean.
- Normal data only. Percentiles from z assume a roughly bell-shaped distribution; skewed data needs the real percentile.
FAQ
What does a z-score of 2 mean?
It means the value is two standard deviations above the mean. In a normal distribution about 97.7% of values fall below it, so it sits in the top 2.3%. Combined with the matching lower tail, only about 4.6% of values are at least that far from the mean in either direction.
How do you convert a z-score to a percentile?
Look up the standard normal cumulative probability Φ(z) and multiply by 100. A z-score of 1 gives Φ(1) ≈ 0.8413, the 84th percentile. This calculator does the lookup for you, so you never need a printed z-table, and it works for any z-score, not just the rows a table happens to print.
Can a z-score be greater than 3?
Yes. Nothing limits the size of a z-score; it is just a distance measured in standard deviations. Values beyond 3 are rare in normal data, about 1 in 370 on either side combined, so a very large z-score often points to an outlier, a data entry mistake, or a distribution that is not normal.
What is the difference between a z-score and a t-score?
A z-score uses the known population standard deviation. A t-score uses the standard deviation estimated from a sample, and its distribution has heavier tails to allow for that extra uncertainty. With samples above roughly 30 the two give almost the same answers.
See also: all Statistics tools, Arithmetic, Number Theory, Algebra and Geometry, Trigonometry and Calculus.
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