Normal distribution calculator
Example: if student heights average 170 cm with an SD of 8 cm, what share are taller than 180 cm?
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How to use it
- Enter mean (μ), standard deviation (σ), probability of, limit a and limit b (for "between").
- The result appears straight away and updates as you edit.
- Copy the result, or copy a link that keeps the same values to share it.
How it is calculated
z = (x - μ) ÷ σ. P(X < a) = Φ(zₐ), P(X > a) = 1 - Φ(zₐ), P(a < X < b) = Φ(z_b) - Φ(zₐ).
Example
Inputs: Mean (μ) = 100, Standard deviation (σ) = 15, Probability of = Below a, Limit a = 130, Limit b (for "between") = 145. Result: Probability: 0.97725; Percentage: 97.725%; z for a: 2.
Limits and caveats
- Results hold only if the data really follows a normal distribution.
- For whole-number counts you may need a continuity correction of ± 0.5.
Questions about this tool
What is the 68-95-99.7 rule?
In a normal distribution about 68% of values lie within one SD of the mean, 95% within two and 99.7% within three.
Is "below" the same as "at or below"?
Yes for a continuous distribution, since any single exact value has probability zero.
How do I get a value from a probability?
That is the inverse problem; use NORM.INV in Excel or work back from a z table.