Z-Score Calculator
Convert any value into a standard z-score given the mean and standard deviation of its distribution. The calculator also reports the corresponding percentile and the probability of a value falling below or above the input.
Enter value, mean and SD
Enter a value, mean and standard deviation, then press Calculate.
The z-score formula
z = (x − μ) / σ
A z-score tells you how many standard deviations a value sits above or below the mean. Positive means the value is above the mean; negative means below.
| Symbol | Meaning |
|---|---|
x | The value you are standardising |
μ | Mean of the distribution |
σ | Standard deviation |
Interpretation
- z = 0 — the value equals the mean.
- z = 1 — one standard deviation above the mean (≈ 84th percentile).
- z = −1 — one standard deviation below the mean (≈ 16th percentile).
- |z| > 2 — unusual (about 5% of values).
- |z| > 3 — very rare (about 0.3% of values).
Worked examples
Example 1 — Test score
Score = 85, mean = 75, SD = 10.
z = (85 − 75) / 10 = 1.0 — one SD above the mean.
z = 1.0 (≈ 84th percentile)
Example 2 — Below the mean
Score = 62, mean = 75, SD = 10.
z = (62 − 75) / 10 = −1.3.
z = −1.3 (≈ 9.7th percentile)
Example 3 — Standard normal value
Value = 100, mean = 100, SD = 15.
z = 0 — the value is exactly at the mean.
z = 0 (50th percentile)
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Frequently asked questions
A z-score (standard score) measures how far a value is from the mean in units of standard deviations. It allows comparison of values from different distributions.
Under the standard normal distribution, the cumulative probability up to a z-score is Φ(z). This calculator computes Φ(z) using the error function and shows the corresponding percentile. For example, z = 1.0 corresponds to ≈ 84.13%.
Yes. A negative z-score means the value is below the mean. The magnitude tells you how many standard deviations away.