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Normal Distribution Interval Probability Calculator

Compute the probability P(a≤X≤b) that a normal random variable falls in an interval, given its mean and standard deviation. Also shows lower and upper tail probabilities and z scores, with the distribution curve and the interval shaded on a graph.

Input

Compute the probability that a normal random variable with the given mean and standard deviation falls in the interval from a to b.

The center of the distribution

Spread of the distribution. Must be positive

Lower end of the interval

Upper end of the interval

Result

P(-1 ≤ X ≤ 1)

0.68268949

As a percentage 68.2689%

Lower probability P(X ≤ a)

0.15865525

Upper probability P(X ≥ b)

0.15865525

z score za

-1

z score zb

1

Phi(za)

0.15865525

Phi(zb)

0.84134475

Probability density and interval

How it works

  • The interval probability is computed as P(a≤X≤b)=Phi((b−mu)/sigma)−Phi((a−mu)/sigma), where Phi is the standard normal cumulative distribution function.
  • Phi(z) is expressed through the error function erf as Phi(z)=(1+erf(z÷√2))÷2, with erf evaluated by a series and a continued fraction for high accuracy.
  • A z score is the standardized value z=(x−mu)÷sigma. Here za standardizes a and zb standardizes b.
  • If a is greater than b, the smaller value is automatically used as the lower bound and the larger as the upper bound.
  • The standard deviation sigma must be a positive number. Values of zero or below produce an error.
  • The graph draws the probability density curve over a range of plus or minus four sigma around the mean and shades the interval from a to b.

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