Formula
z = (x − μ) ÷ σP(X < x) = Φ(z)P(X > x) = 1 − Φ(z)P(a < X < b) = Φ(z_b) − Φ(z_a) How it works
The normal distribution is the symmetric bell-shaped curve that describes heights, test scores, measurement errors and many other quantities. It is fully described by two numbers: the mean μ, which sets its centre, and the standard deviation σ, which sets its width.
Probabilities correspond to areas under the curve. To find one, convert each x value into a z-score, which measures distance from the mean in standard deviations, and then look up the cumulative probability Φ(z) — the area to the left of z under the standard normal curve. This calculator computes Φ directly, so you don’t need a printed z-table.
The default example uses IQ scores, which are scaled to a mean of 100 and a standard deviation of 15. The probability of a score between 85 and 115 — within one standard deviation — is about 68.3%, matching the empirical 68-95-99.7 rule.
Frequently asked questions
Does it matter whether I use < or ≤?
Not for a continuous distribution like the normal: the probability of hitting any exact value is zero, so P(X < x) and P(X ≤ x) are equal.
What if my data aren’t normal?
These probabilities will then be approximations at best. Check a histogram first; skewed data, such as incomes, are often better modelled after a log transformation.