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Z-Score Calculator

Turn a raw value into a z-score — how many standard deviations it sits from the mean — and convert any z into a percentile rank and one/two-tailed probability.

z-score
1.4444
Percentile rank
92.57%
Two-sided p
0.1486
z = (85 − 72) ÷ 9 = 1.4444 · 42.6% of the distribution lies on the far side of x.

The Empirical Rule (Normal Distribution)

RangeCoverage
μ ± 1σ68.27%
μ ± 2σ95.45%
μ ± 3σ99.73%
μ ± 1.96σ95.00% (the CI constant)

The Formula Behind the Result

  1. Enter the raw value, the mean, and the standard deviation of the distribution.
  2. The tool computes z = (x − μ)/σ and its percentile rank under the normal curve.
  3. Reverse mode: enter a z value to get the percentile, the one-tailed and two-tailed p-values, and the raw value for any μ/σ.
  4. The empirical-rule reference shows what ±1σ, ±2σ, and ±3σ cover so results have context.
Formula & Technical Standard
z = (x − μ)/σ. Percentile = Φ(z), computed with a high-accuracy rational approximation of the normal CDF (abrams & Stigler style, |ε| < 1e-7).

Worked Example

Test score context
Input: x = 85, μ = 72, σ = 9
Output: z = 1.44 → 92.5th percentile
The score is 1.44 standard deviations above average, beating about 92.5% of the distribution.
Quality control
Input: z = 3
Output: 99.87th percentile, two-tailed p ≈ 0.0027
The "six sigma" tradition starts here: observations beyond ±3σ happen under 0.3% of the time in a normal process.

Common Questions

What does a negative z-score mean?

The value sits below the mean. A z of −1.5 is one and a half standard deviations below average — the percentile formula mirrors it (about 6.7%).

Can I compare z-scores from different distributions?

Yes — that is the point of standardization. A z of 2 on a physics test and a z of 2 on a language test represent the same relative standing, which is why z-scores power standardized comparisons.

How accurate is the percentile computation?

The calculator uses a double-precision rational approximation of Φ(z) accurate to roughly 7 decimal places across the practical range (|z| < 8). For |z| beyond that, percentiles are effectively 0% or 100% in double precision anyway.

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P-Value Calculator

Convert a test statistic (z, t, or χ²) into its p-value — one- and two-sided — with plain-language guidance on what the number does and does not mean.

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Confidence Interval Calculator (Mean)

Compute a confidence interval for a population mean from raw data or summary statistics, using the t-distribution — correct for the small samples the normal formula gets wrong.

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Average, Median & Mode Calculator

Paste a list of numbers and get mean, median, mode, range, count, sum, and standard deviation with the formulas shown.