Correlation | Linear Regression | Variance and Standard Deviation | Normal Distribution

Normal Distribution Calculator

Calculations using the formula for the normal distribution and the standard normal distribution. Please enter the mean and standard deviation, as well as an input value to determine the function value of the normal distribution at this point. Standard gives the values ​​for the standard normal distribution, i.e. mean 0 and standard deviation 1.
If, however, you want to have the normal distribution represented as a graph, then refer to the function graph plotter with the function norm(0#1#x) as an example for the standard normal distribution.

Round to   0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15   decimal places.

Normal distribution

Mean value μ:
Standard deviation σ:
Input value x:
Function value f(x):

The formula for the normal distribution is:
f(x) = 1 / ( σ * √ 2π ) * e^{-1/2 * [(x-μ)/σ]²}

This image shows two graphs of the normal distribution, created with the function grapher linked above. The blue graph shows the standard normal distribution, the red graph a normal distribution with a mean of 2 and a standard deviation of 0.5. The red graph is much steeper and has a smaller deviation of the values ​​from each other. This gives the impression that it has its maximum at around 0.8, the above calculator for μ=2, σ=0.5 and x=2 gives f(x)=0.79788456, i.e. almost 0.8. The maximum value for the standard normal distribution at the center μ=0 is half as large, since the standard deviation is twice as large here. Deriving this connection algebraically from the above formula is rather difficult, you cannot see it immediately from the formula.
The curve of the normal distribution is also known as the Gaussian bell curve. Bell curve, of course, because of its shape. Like so much in mathematics, it goes back to Carl Friedrich Gauss. The normal distribution is the most important and most commonly used probability distribution, but not the only one. In addition to other centrally centered distributions, there are also curves that are steep to the left and right. The Gaussian curve is often only an approximation; for example, if you take lengths or heights, there is a natural lower limit of 0, but no such upper limit. The distribution is therefore expected to be steep to the left, but for example body heights are nevertheless fairly well normally distributed.

Last updated on 06/24/2025. Author: Jürgen Kummer

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