Z-Score Calculator

Convert a raw score to a z-score and percentile, go from z back to a raw score, or standardise a whole data set with z-scores and percentiles for every value.

The Z-Score Calculator computes z = (x − μ)/σ, the number of standard deviations a value lies from the mean, and converts it to a percentile and tail probabilities under the standard normal distribution. It also works in reverse (z → raw score) and standardises an entire data set.

Free, runs in your browser. Nothing you enter is uploaded.

How to use it

Pick a mode.
Enter the value, mean and SD (or paste a data set).
Read z, percentile and probabilities.

What you enter

Value, mean, standard deviation

What you get

z-score
Percentile and tail probabilities
Raw score from z

Formula

z = (x − μ) / σ   |   Percentile = Φ(z) × 100

z expresses a value in standard-deviation units so scores from different scales can be compared. Φ is the standard normal cumulative distribution function.

Worked example

Input: x = 85, μ = 70, σ = 10

Result: z = 1.5 · percentile 93.32 % · P(X > 85) = 0.0668

(85 − 70)/10 = 1.5; about 93 % of a normal population scores below 85.

Questions

What does a z-score of 2 mean?

The value is two standard deviations above the mean, roughly the 97.7th percentile of a normal distribution.

Can a z-score be negative?

Yes; negative z-scores are below the mean, positive above it, and zero is exactly the mean.

Limitations

Percentiles assume the population is normally distributed.

References

Abramowitz M, Stegun IA (1972). Handbook of Mathematical Functions, §26.2.

More Statistics

All free tools · By Dr. Zubair Khalid, DVM, MS, PhD.