Z Score Calculator

Calculate Z-scores (standard scores) and percentiles. Determine how far a value is from the mean in standard deviations.

The data point to analyze

Population or sample mean

Population or sample standard deviation (cannot be zero)

Z-Score Formula:\nZ = (X - μ) / rho�\n\nWhere:\nX = individual data value\nμ = mean (average)\nrho� = standard deviation\n\nPercentile from Z:\nP = Φ(Z) * 100\nΦ = cumulative distribution function
Example 1:\nTest Score (X): 85\nMean (μ): 70\nStandard Deviation (rho�): 10\n\nZ = (85 - 70) / 10 = 1.5\nPercentile: ~93.3%\nInterpretation: Score is 1.5 SD above mean\n\nExample 2:\nHeight (X): 165 cm\nMean: 170 cm\nSD: 8 cm\n\nZ = (165 - 170) / 8 = -0.625\nPercentile: ~26.6%\nInterpretation: 0.625 SD below mean

What is a Z-score?

Z-score measures how many standard deviations a value is from the mean. Formula: Z = (X - μ) / rho�. Example: Test score 85, mean 70, SD 10 → Z = (85-70)/10 = 1.5. This score is 1.5 standard deviations above average. Positive = above mean, negative = below.

How do I interpret a Z-score?

Z=0: exactly average. Z=1: 1 SD above mean (~84th percentile). Z=2: 2 SD above (~98th percentile). Z=-1: 1 SD below (~16th percentile). Z>3 or Z<-3: very rare (outlier). About 68% of data falls between Z=-1 and Z=1 in normal distribution.

What is the difference between Z-score and percentile?

Z-score: distance from mean in standard deviations. Percentile: percentage of data below a value. They're related but different. Z=0 → 50th percentile. Z=1 → 84th percentile. Z=1.96 → 97.5th percentile. Z=2.58 → 99th percentile. Z-scores work for any distribution, percentiles for ranking.

When do I use Z-scores?

Comparing data from different scales (SAT vs ACT), identifying outliers (|Z|>3), standardizing data for analysis, calculating probabilities in normal distribution, quality control (Six Sigma), medical reference ranges, standardized test scoring.

What is a good Z-score?

Depends on context. For positive traits (IQ, test scores): Z>0 is above average, Z>2 is excellent. For negative traits (error rates, defects): Z<0 is good. In finance, Altman Z-score >2.99 indicates healthy company. Always consider what's being measured.