Z-Score Calculator

Standardize ยท normal CDF ยท percentile ยท 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 โˆ’ ฮผ) / ฯƒ
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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 = โˆ’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% โˆ’2ฯƒ โˆ’1ฯƒ ฮผ +1ฯƒ +2ฯƒ
RangeZ-Score% of DataExample
ฮผ ยฑ 1ฯƒ|Z| โ‰ค 168.27%Most values (typical)
ฮผ ยฑ 2ฯƒ|Z| โ‰ค 295.45%Nearly all values
ฮผ ยฑ 3ฯƒ|Z| โ‰ค 399.73%Virtually all values

Common Z-Scores & Percentiles

Z-ScorePercentileInterpretation
โˆ’3.00.13%Extremely low
โˆ’2.02.28%Significantly low
โˆ’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 = ยฑ6) 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 โˆ’ ฮผ) / ฯƒ. A positive Z means above the mean, negative means below, and zero means exactly at the mean.
Use the standard normal CDF: Percentile = ฮฆ(z) ร— 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 ยฑ1ฯƒ, 95% within ยฑ2ฯƒ, and 99.7% within ยฑ3ฯƒ. These correspond to Z-score ranges of โˆ’1 to 1, โˆ’2 to 2, and โˆ’3 to 3.
Left tail P(Z โ‰ค z) is the area under the curve to the left of z. Right tail P(Z โ‰ฅ z) is the area to the right. They always sum to 1. โ€œBetween ยฑzโ€ gives the central area, and โ€œOutside ยฑzโ€ gives the combined two-tail area.
Use Z-scores when the population standard deviation is known or n > 30. Use t-scores when ฯƒ 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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