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Z-Score Calculator

A Z-Score (standard score) measures exactly how many standard deviations a raw observation lies above or below the mean of a distribution. This free Z

Mathematics
Formula & Methodology
Z = (X - μ) ÷ σ where X is the raw score, μ is the mean, and σ is the standard deviation

To compute a Z-score: 1. Subtract the distribution mean (μ) from the raw score (X) to determine the distance from center. 2. Divide that difference by the standard deviation (σ). A positive Z-score indicates the observation is above the mean; a negative Z-score indicates it is below the mean; a Z-score of 0 is exactly on the mean.

Z-Score (+1)

1

Overview

A Z-Score (standard score) measures exactly how many standard deviations a raw observation lies above or below the mean of a distribution. This free Z-Score Calculator computes standard scores, normal cumulative percentiles, p-values, and statistical interpretations.

Z-scores standardize disparate testing scales onto a single universal metric. This allows direct comparisons between SAT and ACT exams, heights of men versus women, asset returns against benchmark indexes, and clinical laboratory biomarker deviations.

Worked Examples

Example 1: Test Score of 85 (Mean = 75, SD = 10)

Inputs: Score X = 85, Mean μ = 75, SD σ = 10

  1. Difference from mean: 85 - 75 = 10.
  2. Divide by SD: 10 ÷ 10 = +1.00.
  3. Percentile: Z = +1.00 corresponds to the 84.13th percentile.
Final Answer: Z = +1.0000 (84.13th Percentile)

Example 2: Score Below Mean (62, Mean = 70, SD = 4)

Inputs: X = 62, μ = 70, σ = 4

  1. Difference: 62 - 70 = -8.
  2. Z-Score: -8 ÷ 4 = -2.00.
  3. Percentile: Z = -2.00 corresponds to the 2.28th percentile.
Final Answer: Z = -2.0000 (2.28th Percentile)

Example 3: Comparing SAT and ACT Scores

SAT 1300 (μ=1050, σ=200) vs ACT 29 (μ=21, σ=5)

  1. SAT Z-Score: (1300 - 1050) ÷ 200 = 250 ÷ 200 = +1.25.
  2. ACT Z-Score: (29 - 21) ÷ 5 = 8 ÷ 5 = +1.60.
  3. The ACT score of 29 represents a higher relative percentile (+1.60 > +1.25).
Final Answer: ACT is higher relative performance (+1.60 vs +1.25)

How to Use This Calculator

  1. Enter your raw observed score (X) in the first field.
  2. Enter the population or sample mean in the second field.
  3. Enter the standard deviation in the third field.
  4. Click "Calculate" to determine the Z-score and percentile ranking.
  5. Review the p-values and distribution interpretation below.

Reference Table

Z-ScoreStandard DeviationsCumulative PercentileArea to Right (P-value)
-3.003 SD below mean0.13%0.9987
-2.002 SD below mean2.28%0.9772
-1.001 SD below mean15.87%0.8413
0.00Exactly on mean50.00%0.5000
+1.001 SD above mean84.13%0.1587
+2.002 SD above mean97.72%0.0228
+3.003 SD above mean99.87%0.0013

Frequently Asked Questions

What does a Z-score of 0 indicate?
A Z-score of 0 means the raw score is exactly equal to the distribution mean, sitting precisely at the 50th percentile in a normal distribution.
What is considered an outlier using Z-scores?
In standard statistical practice, observations with an absolute Z-score |Z| > 2.0 or |Z| > 3.0 are considered unusual outliers, representing less than 5% or 0.3% of a normal population.
Can Z-scores be calculated for non-normal distributions?
Yes. The formula Z = (X - μ) / σ works for any distribution with a defined mean and standard deviation. However, percentiles derived from the standard normal curve assume normality.
How do you convert a Z-score back into a raw score?
Rearrange the formula: X = μ + (Z × σ). Multiply the Z-score by the standard deviation and add the mean.
Can the standard deviation be negative?
No. Standard deviation is always positive, so the divisor must strictly be greater than zero.

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Reviewed & Authored by: Dr. Elizabeth Vance, PhD Applied Mathematics · Last Updated: 2026-10-06
Computed via standard Gaussian z-transformation and Hart-Abramowitz error function approximations.