How this calculator works
For an individual observation, divide the deviation from the population mean by the population standard deviation. For a mean of independent observations, the sampling standard deviation is σ/√n. The percentile is the standard normal cumulative probability at z. A normal population or an adequate central limit approximation is required to interpret the probability; standardising alone does not establish normality.
Formula
Individual: z = (x - μ)/σ. Sample mean: z = (x̄ - μ)/(σ/√n). Percentile = Φ(z).
Worked example
A value of 130 with mean 100 and standard deviation 15 has z = 2 and normal percentile about 97.725%. A value equal to its mean has z = 0 and percentile 50%.
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Frequently asked questions
What does a negative z-score mean?
The value is below the mean. Its magnitude is the distance in standardising-scale units.
Should I use sample or population standard deviation?
These normal probabilities assume the population standard deviation. Tests using an estimated sample standard deviation generally use a t reference.
Why is a sample mean more extreme?
Independent observations reduce the variability of the mean to σ/√n, so the same raw deviation can be more unusual.