How to use it
- Choose What to find: a share below, above or between two values, or the value at a percentile.
- Type the Mean μ and the Standard deviation σ of the normal distribution.
- Type the Value x, the Second value x₂ for a range, or the Percentile for a value at a percentile.
- Read Share of the values or Value at the percentile, then How it is worked out.
How the z-score is calculated
The z-score says how many standard deviations a value lies above or below the mean: z = (x − μ) ÷ σ. A z-score of 0 is the mean, and a z-score of 1.5 lies one and a half standard deviations above it.
When the values follow a normal distribution, the share of them below a value is Φ(z), where Φ is the standard normal distribution function. Φ(1.96) = 0.9750, so 97.5% of the values lie below a value with z = 1.96.
z = (x − μ) ÷ σ
share below x = Φ(z)
share between x₁ and x₂ = Φ(z₂) − Φ(z₁)
value at percentile p = μ + Φ⁻¹(p) × σPercentiles and the value at a percentile
A percentile is the value below which a given share of the distribution lies. The 90th percentile, for example, has 90% of the values below it. The calculator finds its z-value with the inverse of Φ and turns that into a value with the mean and the standard deviation.
The results are exact for a normal distribution. Real data are rarely exactly normal, especially in the tails, so check that the data are roughly bell-shaped before you rely on a percentile.
What the calculator does not cover
- Skewed data, such as incomes or waiting times. For these, use the data's own percentiles rather than a normal curve.
- The uncertainty of a mean or standard deviation estimated from a sample. The confidence interval calculator covers that.
- Tests of whether data are normal, and distributions other than the normal.
Worked example: exam scores with mean 70 and SD 8
A score of 82 has z = (82 − 70) ÷ 8 = 1.5000. Φ(1.5) = 0.933193, so 93.32% of the scores lie below 82.
The score at the 90th percentile is 80.2524: the z-value for 90% is 1.28155, and 70 + 1.28155 × 8 = 80.2524.
Frequently asked questions
What is a z-score?
A z-score is the distance of a value from the mean, measured in standard deviations. A z-score of +2 means the value is two standard deviations above the mean, and a negative z-score means it lies below the mean.
What share of a normal distribution lies within one or two standard deviations of the mean?
About 68.27% of the values lie within one standard deviation of the mean and about 95.45% within two, from Φ(1) − Φ(−1) and Φ(2) − Φ(−2). These shares do not depend on the mean or the standard deviation.
Why is the 90th percentile at z = 1.28 and not at 0.9?
A z-score is a distance in standard deviations, not a probability. The 90th percentile is the z-value with 90% of the normal distribution below it, which is 1.2816. The 0.9 is the probability itself.
Should I use the population or the sample standard deviation?
Use the standard deviation of the distribution you are describing. For a sample, the sample standard deviation is the usual estimate, and the z-score then describes the sample as if it were the whole population.
Can I use this calculator for data that are not normal?
Only as a rough guide. The results are exact for a normal distribution. For skewed data, such as incomes, the share below a value can be quite different, and the data's own percentiles are more reliable.
Are the numbers I enter sent to a server?
No. The calculation runs in your browser, and nothing you type is sent to our servers.
Sources
- NIST e-Handbook of Statistical Methods, 1.3.6.6.1: Normal distribution
- NIST e-Handbook of Statistical Methods, 1.3.6.7.1: Cumulative distribution function of the standard normal
- Wikipedia: Standard score
Figures and text last checked on Tuesday, November 10, 2026.