How to use this normal distribution calculator
Enter the mean and standard deviation of a normal distribution, plus the value x you are interested in. The calculator standardises x to a z-score and returns P(X < x) and P(X > x), the probabilities on each side of your value.
From a value to a probability
Every normal distribution question is secretly the same question: what proportion of the curve lies beyond this point? The recipe never changes: convert x to a z-score with z = (x − μ)/σ, then read the cumulative probability Φ(z). This calculator performs both steps and shows them.
Heights of adult women average 162 cm with SD 6.5 cm. What proportion are shorter than 170 cm? z = (170 − 162) ÷ 6.5 ≈ 1.23, and Φ(1.23) ≈ 0.891, about 89% are shorter than 170 cm, and about 11% taller. Enter μ = 162, σ = 6.5, x = 170 above and the calculator reproduces it exactly.
Between two values
For P(a < X < b), calculate P(X < b) and subtract P(X < a). Run this calculator twice. The subtraction works because the cumulative probabilities nest: the area below b contains the area below a, and the difference is the slice between them. It is the single most-used move in applied probability.
Frequently asked questions
How do you find probability in a normal distribution?
Standardise the value to a z-score, z = (x − μ)/σ, then find the cumulative probability Φ(z), the proportion of the distribution below x.
What does P(X < x) mean?
The probability that a randomly drawn value from the distribution is less than x, equivalently, the proportion of the population below x, or the area under the curve to its left.
How do I find the probability between two values?
Compute P(X < upper) and subtract P(X < lower). The difference is the probability of falling between them.