Z-Score Calculator
Choose whether to find z from a value or the value from a z-score. You also get the percentile, the area above, the two-tailed probability and the area between -z and +z.
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Formula
- z-score from a value:
z = (x - μ) / σ - Value from a z-score:
x = μ + z × σ - Percentile:
Φ(z), the standard normal cumulative probability, times 100 - Right tail:
1 - Φ(z); two-tailed:2 × (1 - Φ(|z|)); between -|z| and +|z|:1 - two-tailed
The tail areas assume the data follow a normal distribution. The z-score itself does not.
Worked example
x = 85, mean μ = 70, standard deviation σ = 10:
- z = (85 - 70) / 10 = 1.5
- Φ(1.5) = 0.9332, so the percentile is 93.32
- Right tail = 1 - 0.9332 = 0.0668
- Two-tailed = 2 × 0.0668 = 0.1336
- Between -1.5 and +1.5 = 1 - 0.1336 = 0.8664
Frequently asked questions
How do you calculate a z-score?
Subtract the mean from the value and divide by the standard deviation: z = (x - μ) / σ. For x = 85, μ = 70 and σ = 10 that is (85 - 70) / 10 = 1.5, so the value is 1.5 standard deviations above the mean.
What percentile does a z-score correspond to?
The percentile is the standard normal cumulative probability Φ(z), assuming the data are roughly normal. z = 1.5 gives Φ(1.5) = 0.9332, the 93.32nd percentile. z = 2 gives 97.73% and z = 0 gives exactly 50%.
What is the difference between one-tailed and two-tailed probability?
The one-tailed (right) probability is the area beyond z on one side only: 0.0668 for z = 1.5. The two-tailed probability counts both sides, beyond +1.5 and below -1.5, and is double that, 0.1336.
How do I find the original value from a z-score?
Use x = μ + z × σ. With mean 100, standard deviation 15 and z = -2, the value is 100 + (-2 × 15) = 70, which sits at the 2.28th percentile.