Z-score calculator
Turn a raw value into a z-score and standard-normal probabilities: one-sided and two-sided p values, percentile, ±1σ/2σ/3σ interval coverage, or z-scores for a whole data set.
Runs in your browserEvery computation happens in your browser — your data never leaves this device.
Input
Standard-normal probabilities come from an in-house erf approximation (Abramowitz & Stegun 7.1.26, error < 1.5e-7). Nothing is sent anywhere.
Result
0.0668070.13361493.32%Within ±2σWhat this tool does
- Convert an exam score or a body-measurement figure into a standard score once you know the mean and the standard deviation of the group.
- Screen a data set for outliers: enter the raw values to get the mean, the standard deviation and the z-score of every point — |z| > 3 is the usual red flag.
- Translate a z-score into a p value and a percentile without opening a standard-normal table: z = 1.96 is a two-sided p of about 0.05 and the 97.5th percentile.
- Make sense of the "how many sigmas" phrasing: see which of the ±1σ / ±2σ / ±3σ bands a value lands in, with the theoretical coverage of each band.
Example
Input
Raw value 85, mean 70, standard deviation 10 (single-value mode)
Output
Z-score 1.5000, one-sided p 0.066807, two-sided p 0.133614, percentile 93.32%, inside ±2σ; ±1σ interval 60.00 ~ 80.00 with 68.27% coverage, ±2σ 50.00 ~ 90.00 with 95.45%, ±3σ 40.00 ~ 100.00 with 99.73%
The one-sided p value is P(Z ≥ z); the two-sided value is 2 × P(Z ≥ |z|). Switch to data-set mode and enter 2, 4, 4, 4, 5, 5, 7, 9 to get a sample size of 8, a mean of 5.0000, a sample standard deviation of 2.1381 and the z-score of each point.
Frequently asked questions
How accurate are the standard-normal probabilities?
The error function uses the rational approximation from Abramowitz & Stegun, Handbook of Mathematical Functions, formula 7.1.26, with an absolute error below 1.5e-7, and Φ(z) = 0.5 × (1 + erf(z / √2)) turns it into p values and percentiles. Φ(0) is exactly 0.5 and Φ(1.96) ≈ 0.975002, matching printed tables to six decimals.
Should I read the one-sided or the two-sided p value?
It depends on the direction of your hypothesis: use one-sided when you only care whether the value is higher, and two-sided when you care about any difference. For the same z the two-sided p is roughly twice the one-sided one, so always state which one you report.
Which standard deviation does the data-set mode use?
The sample standard deviation (divided by n − 1) by default, which is the right choice when the data is a sample of a larger population. If your data is the entire population, call the logic with { sample: false } to get the population standard deviation (divided by n); the result always tells you which one was used.
What happens if the standard deviation is 0?
The tool refuses with "the standard deviation must be greater than 0", because σ is the denominator of z = (x − μ) / σ and z is undefined at σ = 0. When every value in a data set is identical, the per-point z-scores are shown as 0 instead of raising an error.
Is this percentile the same as a percentile rank?
The number shown here is Φ(z) × 100, the probability of falling below the value under a normal distribution. It assumes normality; for a skewed sample the empirical percentile rank computed from the data itself will differ.
Keywords：z-scorez score calculatorstandard scorestandard normalp valuepercentilez 分数标准分数正态分布p 值百分位标准差