Normal Distribution Calculator
For any bell curve, finds the share of values in a range, or the value at a chosen percentile.
P(lower < X < upper)
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P(X < lower) —
P(X > upper) —
Density at the lower value —
Value at the percentile —
z at the percentile —
Reading this —
The formula
z = (x − μ) ÷ σ P(X < x) = Φ(z)
Φ is the cumulative distribution function of the standard normal.
What to watch out for
Real data is rarely exactly normal, particularly in the far tails, so treat very small probabilities with care.
Frequently Asked Questions
What is the 68-95-99.7 rule?
About 68.27% of values lie within one standard deviation of the mean, 95.45% within two and 99.73% within three.
What z gives the 97.5th percentile?
1.95996, usually rounded to 1.96.