worldofcalcsZ-Score Calculator
Z-score
1
z = (x − μ) ÷ σ = (85 − 75) ÷ 10 = 1
Below x (percentile)
84.13%
Above x
15.87%
Within ±z of the mean
68.27%
85 is 1 standard deviation above the mean, higher than about 84.13% of values.
What is a z-score?
A z-score says how many standard deviations a value is from the mean. A z-score of 1 means one standard deviation above average; −2 means two below. For data that follows a normal (bell-shaped) distribution, the z-score gives the percentile: the share of values below it. This calculator gives the z-score with its percentile and the probabilities above, below and within that range.
Z-score formula
z = (x − μ) ÷ σ · Percentile = Φ(z)
x is the value, μ the mean and σ the standard deviation. Φ(z) is the area under the standard normal curve to the left of z, which a z-table lists.
The 68–95–99.7 rule
- About 68% of normally distributed values lie within 1 standard deviation of the mean, 95% within 2 and 99.7% within 3.
- z = 1.96 marks the middle 95%, used for 95% confidence intervals.
- A z-score above 3 or below −3 is rare, and often treated as an outlier.
- Z-scores let you compare results from different tests: 80 in a test with mean 70 and σ 5 (z = 2) is better than 85 with mean 80 and σ 10 (z = 0.5).
- Percentiles from z-scores assume a normal distribution. For skewed data, rank the values instead.
Worked example: an exam score
- A student scores 85 in an exam with mean 75 and standard deviation 10
- z = (85 − 75) ÷ 10 = 1
- Φ(1) ≈ 0.8413, so the score is at about the 84th percentile
- About 15.87% of students scored higher
- About 68.27% of scores lie within 1 standard deviation of the mean (65 to 85)
Common z-scores
Percentile and the share of values within ±z of the mean.
| z | Below z (percentile) | Within ±z |
|---|---|---|
| -2 | 2.28% | 95.45% |
| -1 | 15.87% | 68.27% |
| 0 | 50.00% | 0.00% |
| 0.5 | 69.15% | 38.29% |
| 1 | 84.13% | 68.27% |
| 1.5 | 93.32% | 86.64% |
| 1.645 | 95.00% | 90.00% |
| 1.96 | 97.50% | 95.00% |
| 2 | 97.72% | 95.45% |
| 2.576 | 99.50% | 99.00% |
| 3 | 99.87% | 99.73% |
From the standard normal distribution.