Normal Distribution Calculator
Find probabilities and z-scores for any normal distribution.
What the Normal Distribution Calculator does
The normal distribution is the symmetric bell curve defined entirely by its mean and standard deviation. Probabilities come from the area under the curve, which this calculator evaluates via the cumulative distribution function.
Formula
z = (x − μ) ÷ σP(X ≤ x) = Φ(z), the standard normal CDFP(a ≤ X ≤ b) = Φ(z_b) − Φ(z_a)
Inputs explained
| Input | Unit | Required | Notes |
|---|---|---|---|
| Mean (μ) | number | Yes | — |
| Standard deviation (σ) | number | Yes | Accepts more than 0. |
| Calculate | one of 4 options | Yes | — |
| x value | number | In some modes | Shown in 3 of the 4 modes. |
| Upper x value | number | In some modes | Shown Calculate is P(x₁ ≤ X ≤ x₂). |
| Probability | % | In some modes | Shown Calculate is Find x from a probability. |
How to use it
- Choose Calculate.
- Enter Mean (μ) and Standard deviation (σ).
- Fill in the remaining inputs the form shows for your choice.
- Select Calculate.
Worked example
IQ scores with a mean of 100 and standard deviation of 15 — what share exceed 130?
- Mean
- 100
- SD
- 15
- Mode
- P(X ≥ x)
- x
- 130
z = 2, so about 2.28% of people score above 130.
Frequently asked questions
What is the empirical rule?
For a normal distribution, roughly 68% of values lie within one standard deviation of the mean, 95% within two, and 99.7% within three.
Is my data normally distributed?
Many natural measurements approximate it, but income, wait times and financial returns are notably skewed. Plot a histogram before assuming normality.