Z-Score Calculator

How to use the z-score calculator

1. Pick a mode: raw score → z, z → probability, or percentile → score.
2. Raw score mode: enter the score x, the mean μ and the standard deviation σ of the group you are comparing against.
3. z mode: enter a z-score, for example a test statistic, to read its probabilities.
4. Percentile mode: enter a percentile between 0 and 100 plus μ and σ to get the matching z and score.
5. Read the percentile, the one- and two-tailed probabilities and the share of values within ±z. The score result appears in the raw score and percentile modes.

Formulas used

• z-score: z = (x − μ) / σ, the number of standard deviations a value lies above (positive) or below (negative) the mean.
• Percentile: Φ(z) × 100, where Φ is the standard normal cumulative distribution, the share of values below z.
• Left tail P(Z < z) = Φ(z); right tail P(Z > z) = 1 − Φ(z); two-tailed p = 2 × P(Z > |z|).
• Within ±z: the share of values between −|z| and +|z|.
• Percentile → score: z = Φ⁻¹(p), then x = μ + z × σ.

As the NIST/SEMATECH e-Handbook notes, the normal cumulative distribution has no simple closed formula and is computed numerically. The calculator evaluates it through the error function, with tails computed directly rather than as 1 − Φ, and inverts it with Acklam's approximation refined by Halley steps. In tests the results agree with reference statistical software to better than one part in a trillion. They also match the NIST table, which lists the area from 0 to z: 0.47500 for z = 1.96, so P(Z < 1.96) = 0.97500.

Z-score examples

1. Score of 85 when the mean is 70 and σ is 10

z = (85 − 70) / 10 = 1.5, the 93.32nd percentile. The right-tail probability is 0.0668072, the two-tailed p-value 0.133614, and 86.64% of a normal distribution lies within ±1.5 σ.

2. A scale with mean 100 and σ 15

A score of 130 gives z = 2, the 97.72nd percentile. Going the other way, the 90th percentile is z = 1.281552, a score of about 119.22, and the 25th percentile is z = −0.67449, about 89.88. A score of 78 against a mean of 85 and σ of 6 is z = −1.1667, the 12.17th percentile.

3. Comparing two different tests (hypothetical numbers)

1,350 on a test with mean 1,050 and σ 210 is z = 1.4286 (92.34th percentile). 27 on a test with mean 21 and σ 5 is z = 1.2 (88.49th). Relative to each group, the first result is stronger.

Reading the result: normality, tails and t

• The percentile needs normal data. A z-score always exists, but converting it to a percentile with Φ is only right if the data are roughly bell-shaped. Take commute times, where most are short and a few are very long. The normal curve misplaces both ends: it gives some probability to times below zero, which can't happen, and it understates how common very long commutes are. The z-score is still a correct distance from the mean; the percentile read from Φ is what goes wrong. For skewed data, rank the actual values instead.
• One or two tails. The two-tailed p-value is the one used in a two-sided z-test, where a difference in either direction counts. Use a one-tailed value only when the hypothesis names the direction in advance.
• μ and σ estimated from a small sample? A z-score treats them as known. With few observations, a t-based method is more appropriate: see the confidence interval calculator. To compute the mean and standard deviation of your data first, use the standard deviation calculator.