Normal Distribution Calculator
Find the probability that a normally distributed value falls below or above a point — with its z-score and probability density.
How it works
z = (x − μ) ÷ σ · f(x) = 1 ÷ (σ√(2π)) · e^(−z²/2) · P(X < x) = Φ(z) = ½(1 + erf(z ÷ √2))
The normal distribution is the familiar symmetric bell curve defined by its mean μ and standard deviation σ. Converting a value to a z-score says how many standard deviations it sits from the mean, and the cumulative distribution Φ(z) gives the area under the curve to its left — the probability of a value falling below it. This calculator evaluates Φ with the Abramowitz & Stegun error-function approximation, accurate to about seven decimal places, and also reports the probability density f(x) at the point itself.
Worked example
For a distribution with mean 100 and standard deviation 15, a value of 120 gives z = (120 − 100) ÷ 15 ≈ 1.333. The area to the left is Φ(1.333) ≈ 0.9088, so P(X < 120) ≈ 90.88% and P(X > 120) ≈ 9.12%.
Sources: Normal distribution — Wikipedia
Frequently asked questions
What is the difference between the density and the probability?
The probability density f(x) is the height of the bell curve at a point, not a probability itself — for a continuous distribution the chance of any exact value is zero. Probabilities come from areas under the curve, such as P(X < x), which is why this tool reports both the height and the cumulative area.
How is P(X > x) found?
Because the total area under the curve is 1, the probability above a point is simply 1 minus the probability below it: P(X > x) = 1 − Φ(z). The two always sum to 100%.
What does the empirical rule say?
For a normal distribution about 68% of values fall within one standard deviation of the mean, 95% within two, and 99.7% within three. It is a quick sanity check: a z-score beyond ±3 marks a value that is rare under a normal model.
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