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Statistics

Normal Distribution Calculator

For a normally distributed variable X ~ N(μ, σ), this calculator finds the probability of landing below or above a value x. It converts x to a z-score and evaluates the cumulative distribution function Φ(z).

Normal Distribution Calculator

Probabilities for a normal distribution N(μ, σ).

Try:
AnswerP(X ≤ 1.2) = 0.88493, P(X ≥ 1.2) = 0.11507
  1. DistributionX ~ N(μ = 0, σ = 1)
  2. Z-scorez = (x − μ) / σ = 1.2
  3. P(X ≤ x)Φ(z) = 0.88493
  4. P(X ≥ x)1 − Φ(z) = 0.11507
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Worked examples

P(X ≤ 115) for IQ with μ = 100, σ = 15 Find the cumulative probability for a normal distribution modelling IQ scores. P(Z ≤ 1.96) for the standard normal The classic critical value that gives roughly 97.5% of the standard normal mass.

Key terms

Standard deviation A measure of how spread out the values in a data set are around the mean. Z-score The number of standard deviations a value lies above or below the mean. Normal distribution The bell-shaped probability distribution defined by a mean and a standard deviation.

Frequently asked questions

What is the cumulative probability?

P(X ≤ x) is the area under the normal curve to the left of x — the chance of a value at or below x.

How accurate is the result?

The cumulative probability uses a high-accuracy approximation of the error function, correct to several decimal places.

What is the standard normal distribution?

It is the normal distribution with mean 0 and standard deviation 1; any normal variable can be standardized to it.

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