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 varibale \( X \) 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 less 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 ( \( X \gt \)
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More References and links

  1. Inverse Normal Probability Calculator.
  2. Normal Distribution Problems with Solutions
  3. Elementary Statistics and Probability Tutorials and Problems
  4. Statistics Calculators, Solvers and Graphers