Z-Score Calculator

Standardize · normal CDF · percentile · reverse lookup

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Convert scores to Z-scores, find probabilities, percentiles, and more.

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What Is a Z-Score?

A Z-score (standard score) measures how many standard deviations a data point is from the mean. It standardizes values from different distributions onto a common scale, making comparisons possible.

Z-Score Formula:  Z = (x − μ) / σ
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Positive Z

Value is above the mean. Z = 1.5 means 1.5 standard deviations above average.

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Negative Z

Value is below the mean. Z = −2.0 means 2 standard deviations below average.

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Zero Z

Value equals the mean exactly. The 50th percentile on a normal distribution.

The 68-95-99.7 Rule (Empirical Rule)

68% 95% −2σ −1σ μ +1σ +2σ
RangeZ-Score% of DataExample
μ ± 1σ|Z| ≤ 168.27%Most values (typical)
μ ± 2σ|Z| ≤ 295.45%Nearly all values
μ ± 3σ|Z| ≤ 399.73%Virtually all values

Common Z-Scores & Percentiles

Z-ScorePercentileInterpretation
−3.00.13%Extremely low
−2.02.28%Significantly low
−1.015.87%Below average
0.050.00%Mean / Average
+1.084.13%Above average
+1.64595.00%90% CI critical value
+1.96097.50%95% CI critical value
+2.097.72%Significantly high
+2.57699.50%99% CI critical value
+3.099.87%Extremely high

Real-World Applications

Standardized Testing

SAT, ACT, and IQ scores use Z-scores to rank test-takers against the population and set percentile-based cutoffs.

Quality Control

Manufacturing uses Six Sigma (Z = ±6) to maintain defect rates below 3.4 per million.

Finance & Risk

Stock returns are standardized for risk comparison. Value-at-Risk (VaR) models use Z-scores for tail risk estimation.

Medical Research

Growth charts, lab results, and BMI use Z-scores to compare patients against age/sex-matched reference populations.

Frequently Asked Questions

A Z-score measures how many standard deviations a value is from the mean. The formula is Z = (x − μ) / σ. A positive Z means above the mean, negative means below, and zero means exactly at the mean.
Use the standard normal CDF: Percentile = Φ(z) × 100%. For example, Z = 1.96 corresponds to the 97.5th percentile. This calculator performs the lookup automatically using jStat.
For normal distributions: 68% of data falls within ±1σ, 95% within ±2σ, and 99.7% within ±3σ. These correspond to Z-score ranges of −1 to 1, −2 to 2, and −3 to 3.
Left tail P(Z ≤ z) is the area under the curve to the left of z. Right tail P(Z ≥ z) is the area to the right. They always sum to 1. “Between ±z” gives the central area, and “Outside ±z” gives the combined two-tail area.
Use Z-scores when the population standard deviation is known or n > 30. Use t-scores when σ is unknown and the sample is small. The t-distribution has heavier tails but approaches the normal as n increases.
You can calculate Z-scores for any data, but the probability and percentile interpretations assume normality. For non-normal data, consider log or Box-Cox transformations first. The Central Limit Theorem helps when working with sample means of large samples.

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