How this calculator works
- Step 1Distancex − μHow far the value is from the mean.
- Step 2Z-score÷ σMeasure that distance in standard deviations.
- Step 3ProbabilityΦ(z)The share of a normal distribution below z.
A z-score says how many standard deviations a value lies from the mean. It lets you compare values measured on different scales.
If the values follow a normal distribution, the z-score also gives the share of values below and above, worked out without lookup tables.
Worked example: a score of 85
- Z-score
z = (x − μ) ÷ σ
- xValue 85
- μMean 70
- σStandard deviation 10
- Distance from the mean85 − 7015
- Z-score15 ÷ 101.5
- Share belowΦ(1.5)93.31928%
So 85 is 1.5 standard deviations above the mean. A score of 55 in the same test has z = −1.5, the mirror image.
The 68–95–99.7 rule
Read the result
z = 1.96 leaves 2.49979% in the upper tail, which is why 1.96 appears in 95% confidence intervals.
P(X < x) is the percentile as a fraction. P(X > x) is the upper tail, worked out directly so tiny tails stay accurate.
Assumptions and limits
- Normal shapeProbabilities assume a bell-shaped distribution.
- σ above 0A zero spread has no z-score.
- Same unitValue, mean and σ must share a unit.
- DisplaySix decimals, or scientific notation for tiny tails.
Related statistics tools
A z-score places one value on the normal curve. Probabilities and tests build on it.
For the chance between two values, use the normal distribution calculator. The p-value calculator turns a test statistic into a p-value, and the standard deviation calculator finds the spread from raw data.
Common mistakes
Mistake: Dividing by the variance.
Instead: Divide by the standard deviation, its square root.
Mistake: Reading Φ(z) as the upper tail.
Instead: Φ(z) is the share below; the upper tail is 1 − Φ(z).
Mistake: Using normal probabilities for skewed data.
Instead: Check the shape first; z itself still works.
Checking the method
Every golden case is re-worked with exact fractions for z and the C library's erfc for probabilities.
The calculator uses a power series near the mean and a continued fraction in the tails. An independent script checks both to 12 significant digits.
Read how we test our calculators, or browse more maths calculators.