Normal Distribution Calculator
Find normal-distribution probabilities: below or above a value, within or outside a range, for any mean and standard deviation.
Change the values and press Calculate to work out your own figures.
Probability for a value
Enter the mean, standard deviation and a value.
Show the working
- z = (x − μ) ÷ σ = (1.96 − 0) ÷ 1 = 1.96
- P(X ≤ x) = Φ(z) = 0.9750021049
Probability in a range
Enter the mean, standard deviation and the two ends of a range.
Show the working
- z₁ = -1, z₂ = 1
- P = Φ(z₂) − Φ(z₁) = 0.6826894921
How to use it
Enter the mean, standard deviation and a value. Enter the mean, standard deviation and the two ends of a range.
Probabilities can be typed as decimals, fractions or percentages. Show the working gives the formula with your numbers in it.
Key facts
z = (x − μ) ÷ σ; P(X ≤ x) = Φ(z)
Tails far from the mean
The calculator works the upper tail directly, so tiny probabilities such as P(Z > 8) ≈ 6.2 × 10⁻¹⁶ keep their accuracy.
Questions
What percentage of values are within 1 standard deviation?
About 68.27%.
What is P(Z ≤ 1.96)?
0.9750021049.
Formulas
z = (x − μ) ÷ σ; P(X ≤ x) = Φ(z)
Sources
Limitations
- Probabilities can be typed as decimals (0.25), fractions (1/4) or percentages (25%).