Summary: This z score calculator computes percentile and reverse z with live steps, formulas and a chart. It accepts labeled numeric inputs, works offline through file:// and includes source-backed explanations for students, analysts and researchers.
Z score calculator for z=(x-mu)/sigma, percentile from normal CDF, reverse x from z, formula and bell curve. The calculator works offline, updates instantly and includes a worked example, plain-text formula, MathML, references and structured data.
Default values are loaded. Click any field and edit it; results and chart update automatically.
| Input or setting | Result or interpretation | Use this when |
|---|---|---|
| z=0 | 50th percentile | standardized distance |
| z=1.96 | about 97.5th percentile | standardized distance |
| z=-1 | about 15.9th percentile | standardized distance |
z = (x - mu) / sigma; x = mu + z*sigma; percentile = Phi(z)A z-score tells how many standard deviations a value is above or below the mean.
x=85, mu=70, sigma=10
z = 1.5; percentile about 93.3%
This page uses the shared statistics core for distribution functions, quantiles and exact integer counting where needed. The formula is shown in plain text so screen readers and search engines can parse it reliably.
Use Load example in the calculator to reproduce this reference result.
{
"tool": "Z Score Calculator",
"input": "x=85, mu=70, sigma=10",
"output": "z = 1.5; percentile about 93.3%",
"formula": "z = (x - mu) / sigma; x = mu + z*sigma; percentile = Phi(z)"
}| Calculator | Example input | Expected output |
|---|---|---|
| Z Score Calculator | x=85, mu=70, sigma=10 | z = 1.5; percentile about 93.3% |
What does p = 0.03 mean? If the null hypothesis and model assumptions were true, a result at least this extreme would occur about 3% of the time. The American Statistical Association cautions that a p-value alone does not measure effect size, practical importance or the probability that Hâ‚€ is true.3
For most classroom and professional reports, pair the calculator result with the question you are answering. A mean or median summarizes location, but spread explains consistency. A confidence interval estimates plausible values, while a hypothesis test evaluates compatibility with a null model. Regression and correlation describe association, so they should be reported with a chart and residual or outlier review. When a result is statistically significant, still ask whether the effect is large enough to matter in the real setting.
| Statistic | Small | Medium | Large | Use |
|---|---|---|---|---|
| Cohen's d | 0.2 | 0.5 | 0.8 | t-test effect size |
| Cramér's V | 0.1 | 0.3 | 0.5 | chi-square association |
| |r| | 0.10 | 0.30 | 0.50 | correlation strength |
| R² | 0.01 | 0.09 | 0.25 | variance explained |
Use z = (x − μ)/σ, where x is your data point, μ is the mean, and σ is the standard deviation. The z-score tells you how many standard deviations a value sits above or below the mean. Suppose a student scores 85 on an exam where μ = 75 and σ = 5: z = (85 − 75)/5 = 2. That student scored two SDs above average. Negative z means below the mean; positive z means above. Z-scores let you compare values from different distributions on a common scale.
Look up the cumulative probability — the area to the left of your z under the standard normal curve. That value is your percentile. Example: z = 1.0 corresponds to a cumulative area of about 0.8413, so the 84th percentile. In Excel, =NORM.S.DIST(1, TRUE) returns 0.8413. A z of 0 sits at the 50th percentile (the mean), z = −1 at the 16th, z = 2 at the 97.7th. Standard normal tables give these values directly. Multiply the cumulative probability by 100 to express it as a percentile.
Use the inverse standard normal function. For the 90th percentile, find the z value with 0.90 of the area to its left, which is z ≈ 1.2816. In Excel: =NORM.S.INV(0.90) returns 1.2816. For the 25th percentile, =NORM.S.INV(0.25) gives roughly −0.6745. Once you have the z-score, you can convert back to a raw score on your specific distribution using x = μ + zσ. Standard tables can also be read in reverse — locate the desired probability inside the table and trace back to the z value.
For the 90th percentile (the value below which 90% of the distribution lies), z ≈ 1.2816. That's from =NORM.S.INV(0.90) in Excel. Be careful with terminology: 1.645 is the z for the 95th percentile, often used as the 5% one-tailed critical value. People sometimes confuse "90th percentile" with "90% confidence" — they're different. A 90% two-tailed CI uses z = 1.645 (which has 5% in each tail). The 90th percentile, in contrast, has 90% of the area to its left, so z = 1.2816.
Plug into z = (x − μ)/σ. Suppose a student's height is 180 cm in a population with μ = 170 cm and σ = 8 cm: z = (180 − 170)/8 = 1.25. That student is 1.25 SDs above the mean — taller than roughly 89% of the population. Try another: a temperature reading of 65°F where μ = 70 and σ = 5 gives z = −1, meaning one SD below average. The formula works in any context: heights, exam scores, lab measurements — anywhere you have x, μ, and σ.
A negative z-score means the value sits below the mean. The size tells you how far below: z = −1 is one SD below average, z = −2 is two SDs below, and so on. Example: an IQ score of 85 in a distribution with mean 100 and SD 15 gives z = (85 − 100)/15 = −1, so that score is one SD below the average IQ. Negative z values correspond to percentiles below 50%. The further negative, the rarer the value — z = −3 puts you below roughly 0.13% of the distribution.
Rearrange the z formula: x = μ + zσ. So if a student has a z of 1.5 on an exam where μ = 70 and σ = 8, the raw score is 70 + 1.5 × 8 = 82. For a negative z = −1.2 with μ = 100 and σ = 15: x = 100 + (−1.2) × 15 = 82. This conversion is essential for translating standardised scores (like IQ percentiles or test results) back into the original measurement scale, where they're easier to interpret in context.
Use the standard normal CDF. For a left-tailed test, p = P(Z < z) — the cumulative area to the left. For a right-tailed test, p = P(Z > z) = 1 − P(Z < z). For a two-tailed test, p = 2 × P(Z > |z|). Example: z = 2.0 in a two-tailed test gives p ≈ 2 × 0.0228 = 0.0456. In Excel, =NORM.S.DIST(2, TRUE) returns the cumulative area, and =1-NORM.S.DIST(2, TRUE) gives the right-tail probability. Always pick the tail that matches your alternative hypothesis.