A Z-Score Calculator
A z-score restates a value as how many standard deviations it sits from the mean. That rescaling removes the original units, which is exactly what lets you compare things that were never commensurable.
Z-score
1
- Distance from the mean (original units)
- 10
- Distance ignoring direction (|z|)
- 1
Values update as you type. This calculator covers the single scenario its formula assumes — see Common Mistakes for what it leaves out.
The formula this calculator uses
z = (x - mean) / standard deviation
- x
- The value you are standardising
- mean
- Average of the distribution the value came from
- SD (sigma)
- Standard deviation of that same distribution
- z
- Signed distance from the mean, measured in SD units
How to check the result by hand
- 1
Fix the reference distribution first
Decide what population the value belongs to. The mean and standard deviation must both describe that same population, or the result describes nothing.
- 2
Subtract the mean
x - mean gives the signed deviation in original units. Here 85 - 75 = 10, so the value sits ten points above average. Keep the sign; it is the entire directional content.
- 3
Divide by the standard deviation
10 / 10 = 1.0. Read the result as "one standard deviation above the mean" rather than as a bare number, which keeps the interpretation attached to the figure.
- 4
Sanity-check the magnitude
Most ordinary observations land between -2 and 2. Anything beyond roughly 3 is genuinely rare in bell-shaped data, and beyond that you should suspect a mismatch of distribution before celebrating a finding.
- 5
State whether normality is being assumed
If you go on to quote a percentile, that step needs the bell curve. Say so explicitly, because the z-score by itself justifies no such statement.
For worked examples, common mistakes and the limits of this formula, read the full How To Calculate A Z-Score page.