How to use
- Enter the mean μ and standard deviation σ.
- Type the score to convert under "Score (x)".
- Read the z-score, percentile and the share above and below.
Options explained
- z-score
- z = (x − μ) / σ. z = 1.25 means 1.25 standard deviations above the mean; a negative z is below the mean.
- Percentile
- The estimated share of scores below x, assuming a normal distribution. It is only accurate if the scores are roughly normal.
- Two-tailed p
- The p-value when z is a test statistic, the probability of |Z| being at least this large. |z| > 1.96 gives p < 0.05.
Worked example
Mean 75, SD 8, and a score of 85:
| Result | Value |
|---|---|
| z | 1.25 |
| Percentile | 89.44% |
| Share above 85 | 10.56% |
| Two-tailed p | 0.2113 |
Assuming normally distributed scores, 85 beats about 89% of people. The calculator opens on the standard normal (μ = 0, σ = 1) with x = 1.5, giving P(Z < 1.5) = 0.9332.
Common z-values
| z | Share below | Meaning |
|---|---|---|
| −2 | 2.28% | Very low |
| −1 | 15.87% | Below average |
| 0 | 50% | Exactly average |
| 1 | 84.13% | Above average |
| 1.645 | 95% | One-sided 0.05 critical value |
| 1.96 | 97.5% | Two-sided 0.05 critical value |
| 2 | 97.72% | Very high |
| 2.576 | 99.5% | Two-sided 0.01 critical value |
Frequently asked questions
How do I compare scores from two different tests?
Convert both to z-scores. 85 in English (mean 75, SD 8) is z = 1.25; 90 in math (mean 82, SD 10) is z = 0.8, so the English result is relatively better.
How do I get the raw score from z?
x = μ + z × σ. Or set the mean to 0 and SD to 1 and use "From probability".
What is a T-score?
T = 50 + 10z, a z-score rescaled to mean 50 and SD 10 to avoid negatives and decimals. z = 1.25 is T = 62.5.
I only have a list of scores. Where do I get the mean and SD?
Paste the list into the statistics calculator to get the mean and standard deviation, then come back.
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