How to use the z-score calculator
- Choose what to find: the z-score and percentile of a value, the probability between two values, or the value at a percentile.
- Enter the value (or two values, or a percentile).
- Enter the mean and standard deviation of the distribution. If you already have z-scores, enter a mean of 0 and a standard deviation of 1, and the "values" become z-scores.
- Read the result, the working, and the table of values at whole z-scores.
The z-score formula
x is the value, μ the mean and σ the standard deviation. A z-score of 0 is exactly average; +1 is one standard deviation above the mean, −2 is two below. Φ is the cumulative normal distribution. The calculator computes it from the complementary error function to about 15 significant digits, so it agrees with printed z-tables and with spreadsheet functions such as NORM.S.DIST, and it stays accurate far out in the tails. The reverse step uses an inverse-normal approximation polished with two Halley iterations.
Worked example
The calculator starts with an exam score of 85, where the class mean is 70 and the standard deviation is 10:
- z = (85 − 70) ÷ 10 = 15 ÷ 10 = 1.5
- Φ(1.5) = 0.933193, so the score is at about the 93rd percentile: 93.32% of scores are lower
- 6.68% of scores are higher, and 86.64% fall within 1.5 standard deviations of the mean (55 to 85)
For the probability between 60 and 85: z = −1 and z = 1.5, so P = Φ(1.5) − Φ(−1) = 0.933193 − 0.158655 = 0.7745, or 77.45%.
For the score at the 90th percentile: z = 1.28155, so x = 70 + 1.28155 × 10 = 82.8155.
Common z-scores
| z | Share below | Share above | Used for |
|---|---|---|---|
| 1.0000 | 84.13% | 15.87% | One standard deviation |
| 1.2816 | 90.00% | 10.00% | Top 10% |
| 1.6449 | 95.00% | 5.00% | 90% confidence, one-tailed 5% test |
| 1.9600 | 97.50% | 2.50% | 95% confidence |
| 2.3263 | 99.00% | 1.00% | Top 1% |
| 2.5758 | 99.50% | 0.50% | 99% confidence |
| 3.0000 | 99.87% | 0.13% | Three standard deviations |
Negative z-scores mirror these: z = −1.96 has 2.5% below it. The 68-95-99.7 rule comes from the same curve: about 68.27% of values lie within one standard deviation of the mean, 95.45% within two and 99.73% within three.
When percentiles from z-scores are valid
The percentile and probability results assume the data follows a normal distribution, the symmetric bell curve. Heights, measurement errors and many test scores come close. Incomes, house prices, waiting times and anything with a hard floor at zero usually don't: they have a long right tail, and z-score percentiles can be well off. The z-score itself is still a useful measure of how unusual a value is, whatever the shape. To check the shape of your data, look at the skewness in the statistics calculator.
Frequently asked questions
What is a good z-score?
It depends on what is being measured. For test scores, anything above 1 beats about 84% of people and above 2 beats about 98%. In quality control, a z-score beyond ±3 usually flags a value worth investigating. A z-score near 0 simply means the value is typical.
Can a z-score be negative?
Yes. A negative z-score means the value is below the mean. A score of 55 in a class with mean 70 and standard deviation 10 has z = −1.5, which is the 6.68th percentile.
How do I convert a z-score to a percentile?
Find the area under the standard normal curve to the left of the z-score and multiply by 100. z = 1.5 gives 0.9332, so the 93rd percentile. Set the mean to 0 and the standard deviation to 1 here to enter a z-score directly.
What is the difference between a z-score and a t-score?
A z-score uses the known standard deviation of the whole population. When you estimate the standard deviation from a small sample, the t-distribution is more appropriate, because it allows for the extra uncertainty. The p-value calculator handles both.
Related calculators
- Standard Deviation Calculator — Sample and population standard deviation, variance and mean of a list of numbers, with every step shown.
- P-Value Calculator — p-value from a z-score, t-score or chi-square statistic, one- or two-tailed, with significance at 0.05 and 0.01.
- Confidence Interval Calculator — Confidence interval and margin of error for a mean (t or z) or a proportion, at any confidence level.
- Statistics Calculator — Descriptive statistics for a list: mean, median, mode, quartiles, IQR, outliers, variance, SD and skewness.
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