Normal distribution calculator
Probabilities and z-scores with the area under the curve shaded as you type.
p = 0.841345
(x − μ) ÷ σ
Work backwards from a probability
The x with this much area below it
The empirical rule
| Range | Values | Contains |
|---|---|---|
| μ ± 1σ | 85 to 115 | 68.27% |
| μ ± 2σ | 70 to 130 | 95.45% |
| μ ± 3σ | 55 to 145 | 99.73% |
How it works
The normal distribution is the bell curve: symmetric about its mean, with the spread set by the standard deviation. It appears constantly not because nature prefers it, but because the central limit theorem drives sums and averages toward it regardless of what they started as.
z = (x − μ) / σ
f(x) = (1 / σ√(2π)) · e^(−z²/2) The z-score is the whole trick
A z-score says how many standard deviations a value sits from the mean. Converting to z lets one table serve every normal distribution — which is why z-tables exist at all, and why 1.96 (95%) and 2.576 (99%) are worth memorising.
68–95–99.7
About 68% of values fall within one standard deviation of the mean, 95% within two, and 99.7% within three. For IQ scores (μ = 100, σ = 15) that puts about 2.3% of people above 130 — which matches the usual figure for the “gifted” threshold.
When not to use it
Normality is an assumption, not a guarantee. Income, city sizes, file sizes and insurance claims are all strongly right-skewed, and assuming a bell curve badly understates the tail. A histogram of your actual data is the cheapest check available.
Sources
The cumulative distribution uses Graeme West's implementation of Hart's algorithm (accurate to about 1 × 10⁻¹⁵); the inverse uses Acklam's approximation with a Halley refinement. Both are verified against published z-table values in the test suite.