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Normal Probability Calculator

The probability density function for a normally distributed random variable

$X$ with mean

$\mu$ and standard deviation

$\sigma$ is given by:

$f_X(x,\mu,\sigma) = \dfrac{1}{\sigma \sqrt{2 \pi}} e^{ - \dfrac{(x-\mu^2)}{2 \sigma^2}} \quad , \quad x \in \mathbb{R}$ The probabilities that the random variable

$X$ is between, below or above certain values are given by the areas:

$P( x_0 \lt X \lt x_1 ) = \displaystyle \int_{x_0}^{x_1} \dfrac{1}{\sigma \sqrt{2 \pi}} e^{ - \dfrac{(x-\mu^2)}{2 \sigma^2}} dx$
normal distribution probability between two x values

$P( X \lt x_0 ) = \displaystyle \int_{-\infty}^{x_0} \dfrac{1}{\sigma \sqrt{2 \pi}} e^{ - \dfrac{(x-\mu^2)}{2 \sigma^2}} dx$
normal distribution probability less than x value

$P( X \gt x_0 ) = \displaystyle \int_{x_0}^{\infty} \dfrac{1}{\sigma \sqrt{2 \pi}} e^{ - \dfrac{(x-\mu^2)}{2 \sigma^2}} dx$
normal distribution probability greater than x value

There are no closed form solutions to the above integrals and they are therefore computed numerically.
We present three calculators that compute the three probabilities given above.
Enter the mean and standard deviation as real numbers; the standard deviation must be positive.

Mean , Standard Deviation =

Decimal Places =
P (

$\lt X \lt$ ) ,

P (

$X \lt$ ) ,

P (

$X \gt$ ) ,

Normal Probability Calculator

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