Z-Score Calculator
Turn a raw value into standard deviations from the mean so you can compare scores across different scales.
Z-Score Calculator
Calculate z-scores to standardize values and determine how many standard deviations from the mean
Example Scenarios
What a Z-score is
A Z-score (standard score) says how many standard deviations a value sits from the mean. Z = 0 is exactly average. Z = 1 is one SD above; Z = −1 is one SD below.
That makes cross-scale comparisons fair. An SAT score and an ACT score live on different number lines; both become comparable once you standardize them.
Formula
Z = (X − μ) / σ
X = raw value, μ = mean, σ = standard deviation
Reading the number
Positive
Above the mean. +1, +2, +3 = one, two, three SDs up. Beyond ±3 is rare under a normal curve.
Negative
Below the mean. Same scale in the other direction.
68-95-99.7 rule
In a normal distribution, about 68% of values fall within ±1 SD, about 95% within ±2, and about 99.7% within ±3.
Z-scores and percentiles
Under normality, Z maps to a percentile (share of the population below that value):
Z = +2 ≈ 97.7th percentile.
Worked examples
SAT: X=1250, μ=1060, σ=195
Z = (1250−1060)/195 ≈ 0.97 → roughly 83rd percentile
Height: X=74 in, μ=69, σ=3
Z = 5/3 ≈ 1.67
Significance cutoffs
In two-sided tests, |Z| > 1.96 ≈ 5% level; |Z| > 2.58 ≈ 1%; |Z| > 3.29 ≈ 0.1%. Statistical significance is not the same as practical importance—especially with huge n.
Assumptions matter: roughly normal data, known or well-estimated μ and σ, independent observations. Skewed data still give a computable Z; the percentile story does not hold.
Frequently Asked Questions
Z-score vs T-score?
Use Z when population σ is known (or n is large). Use t when σ is estimated from the sample, especially with smaller n—t has heavier tails.
Can |Z| be greater than 3?
Yes. Beyond ±3 is uncommon under normality; beyond ±4 often means an outlier or a data error.
What does Z = 0 mean?
The value equals the mean—50th percentile under a symmetric normal curve.
Data not normal?
You can still compute Z; do not trust normal-table percentiles. Prefer ranks, or transform first if that fits the analysis.
Sample instead of population?
Using sample mean and SD gives a standardized score. Fine for large n; for small samples, lean on t-based methods.
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