Z-Score Calculator
A z-score says how many standard deviations a value sits from the mean: z = (x − μ) ÷ σ. A z of 1.4 means the value is 1.4 standard deviations above average, which under a normal curve puts it at roughly the 92nd percentile — higher than about 92% of the group.
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Z-score
1.4000
Area to the right
8.08%
Details
Updates as you typeThe single observation you want to place in the distribution.
The average of the group your value is being compared against.
How widely the data is spread. Must be greater than zero.
Summary
Above averageZ-score
1.4000
- Percentile (area to the left)
- 91.92%
- Area to the right
- 8.08%
- Two-tailed p-value
- 0.1615
- Distance from the mean
- 14.0000
- Standard deviations away
- 1.4000
- 85 is 1.4 standard deviations above the mean. About 91.9% of a normal distribution falls below it, which is the 92nd percentile.
- The percentile, the tail areas, and the p-value all assume the underlying data is normally distributed. The z-score itself does not — it is just a distance, measured in standard deviations.
How this is calculated
- Your value (x)
- 85.0000
- Mean (μ)
- 71.0000
- Deviation (x − μ)
- 14.0000
- Standard deviation (σ)
- 10.0000
- z = (x − μ) ÷ σ
- 1.4000
- Φ(z), area to the left
- 91.92%
- 1 − Φ(z), area to the right
- 8.08%
- Two-tailed p = 2 × (1 − Φ(|z|))
- 0.1615
Compare scenarios
See how one change moves the result
- CurrentYour inputs as they stand1.4000Current
- Your value (x)106.53.5500
- Mean of the distribution89-0.4000
- Standard deviation12.51.1200
Frequently asked questions
What does a z-score of 1.4 actually mean?
It means the value sits 1.4 standard deviations above the mean. Under a normal distribution that is the 92nd percentile: about 92% of the group scores lower and about 8% scores higher. A negative z-score is the mirror image — z = −1.4 beats only about 8% of the group.
How is the percentile worked out?
The percentile is the area under the standard normal curve to the left of your z-score, written Φ(z). That integral has no exact formula in ordinary functions, so this page uses the Abramowitz and Stegun approximation 26.2.17, which the authors bound at an error below 7.5 in 100 million. The tests check it against published z-tables and against numerical integration of the bell curve.
What is a good or bad z-score?
Neither. A z-score only measures distance, so whether far from the mean is good depends entirely on what you measured. As a rough guide to rarity: about 68% of a normal distribution falls within one standard deviation of the mean, 95% within two, and 99.7% within three, so anything past z = 3 is genuinely rare.
Why is the two-tailed p-value different from the tail area?
The area to the right of z answers a one-sided question — how likely is a value at least this high. The two-tailed p-value doubles it to cover both directions, answering how likely a value is at least this far from the mean either way. At z = 1.96 the right tail is about 2.5% and the two-tailed p-value is about 0.05, which is where that familiar threshold comes from.
Why does the standard deviation have to be more than zero?
A z-score divides the distance from the mean by the standard deviation. If the spread were zero, every observation would equal the mean, and there would be no unit left to measure distance in — the division is undefined rather than infinite. The calculator refuses the input instead of returning a meaningless number.