Z-Score Calculator

Formula and what each part means

z = (x − μ) ÷ σ

• x is the value you are standardizing
• μ (mu) is the mean of the population or dataset
• σ (sigma) is its standard deviation, which must be positive

A positive z is above the mean, a negative z below it, and z = 0 is exactly at the mean. Because z is measured in standard deviations, it has no units: inches, points and dollars all become comparable.

If you are working with a sample and have the sample mean (x̄) and sample standard deviation (s), the same formula applies with those values. The calculator does not compute a mean or standard deviation from raw data; enter them from your dataset or textbook.

From z-score to percentile

The percentile is the area under the standard normal curve to the left of z, written Φ(z). NIST's Engineering Statistics Handbook notes that this integral has no simple closed form and is computed numerically. This calculator uses the classic Abramowitz–Stegun polynomial approximation, which matches printed z-tables to the two decimals shown.

Z-table of common values

The 1.0, 2.0 and 3.0 rows are the source of the "68–95–99.7 rule": about 68%, 95% and 99.7% of a normal distribution lies within one, two and three standard deviations of the mean.

Left tail, right tail, or both?

The calculator's percentile is the left-tail area, P(Z < z). The other two areas follow from it:

• Right tail, P(Z > z) = 100% − percentile. For z = 1.75: 100 − 95.99 = 4.01%.
• Two-tailed, P(|Z| > |z|) = 2 × the smaller tail. For z = 1.75: 2 × 4.01 = 8.02%.

This matters for hypothesis tests. A two-tailed test at the 5% level rejects when |z| > 1.96; a one-tailed test at 5% uses 1.645. Note that many printed tables, including the NIST table, give the area from 0 to z instead of from −∞ to z; add 0.5 to those values to get the percentile shown here.

Percentile from the other direction

To find the value at a given percentile, run the formula backwards: x = μ + z × σ. For the 90th percentile, z ≈ 1.282; with μ = 72 and σ = 8, x = 72 + 1.282 × 8 = 82.3. CDC's growth chart documentation lists the z-scores that correspond exactly to common percentiles: −1.881 (3rd), −1.645 (5th), −1.282 (10th), −0.674 (25th), 0 (50th), 0.674 (75th), 1.036 (85th), 1.282 (90th), 1.645 (95th) and 1.881 (97th).

When the normal percentile misleads

• Skewed data. Income, home prices and waiting times have long right tails. A z of +2 there may correspond to a much lower percentile than 97.7%.
• Small samples. With a handful of observations, the sample mean and standard deviation are uncertain, and a t-distribution is more appropriate for inference.
• Growth and body measurements. CDC's growth charts handle skewed measurements such as weight with the LMS method, which adjusts for skewness before converting to a z-score. A plain (x − μ) ÷ σ will not reproduce a pediatric weight-for-age percentile.
• Outliers. A |z| above 3 is rare in a true normal distribution (about 0.27% of values). In real data it is often a measurement error or a sign that the distribution is not normal, rather than a genuine extreme.

Comparing scores from different tests

The main practical use of z-scores is putting different scales on one footing. Suppose a student scores 78 on a chemistry exam (class mean 70, SD 5) and 85 on a history exam (mean 80, SD 10). The raw history score is higher, but:

• Chemistry: z = (78 − 70) ÷ 5 = 1.6, about the 94.5th percentile
• History: z = (85 − 80) ÷ 10 = 0.5, about the 69.1st percentile

Relative to classmates, the chemistry result is far stronger. Standardized test reports, scaled scores and many grading curves rely on this idea.

Population vs sample standard deviation

The standard deviation you enter changes the answer, so make sure it is the right one. A population SD divides the sum of squared deviations by n; a sample SD divides by n − 1, which makes it slightly larger, especially for small samples. With 10 observations the sample SD is about 5% larger (√(10/9) ≈ 1.054), so every z-score shrinks by the same proportion. Statistics software and spreadsheets usually default to the sample version (STDEV.S in Excel and Google Sheets); textbook problems that give σ mean the population value. If you only have raw data, compute the mean and standard deviation first (a mean, median and mode or standard deviation calculator will do it), then enter them here together with the value you want to standardize.