Z-Score Calculator

A z-score says how many standard deviations a value lies above or below the mean: z = (x − μ) / σ. A positive z is above the mean, a negative z below it. For an IQ score of 130 on a scale with mean 100 and standard deviation 15, z = (130 − 100) / 15 = 2, so the score is two standard deviations above average.

Use the standard normal cumulative distribution Φ(z). The area to the left of z is Φ(z), the area to the right is 1 − Φ(z), the area between two z-scores is the difference of their Φ values, and the two-tailed area outside ±z is 2 × (1 − Φ(|z|)). For z = 1.96 the left area is 0.9750, and 95% of the curve lies between −1.96 and 1.96.

Use the inverse normal function, which the Percentile tab computes directly. Common values: the 90th percentile is z ≈ 1.2816, the 95th is 1.6449, the 97.5th is 1.9600 and the 99th is 2.3263. The 50th percentile is always z = 0, and percentiles below 50 give negative z-scores — the 10th percentile is −1.2816.

Use the population version (divide by n) when your numbers are the entire group you care about, such as every score in one class. Use the sample version (divide by n − 1) when the data is a sample standing in for a larger group. The sample standard deviation is slightly larger, so z-scores computed with it are slightly closer to zero; the From data tab lets you switch between the two.

More precise. Φ(z) is computed with Hart’s double-precision rational approximation, which our tests check against numerical integration of the normal curve to within 1e-12 across z = −8 to 8. Printed tables round z to two decimals and the area to four, so small differences in the fourth decimal come from the table’s rounding, not the calculator.

In a normal distribution about 95.45% of values fall within ±2 standard deviations and 99.73% within ±3, so a |z| above 2 happens roughly 4.6% of the time and above 3 only about 0.27% of the time. Many courses call |z| > 2 unusual and |z| > 3 a possible outlier. These percentages only hold if the data is approximately normal.