Normal distribution takes special role in the probability theory. This is most common continues probability distribution, commonly used for random values representation of unknown distribution law.
Probability density function
Normal distribution probability density function is the Gauss function:
where μ — mean,
σ — standard deviation,
σ ² — variance,
Median and mode of Normal distribution equals to mean μ.
The calculator below gives probability density function value and cumulative distribution function value for the given x, mean and variance:
Cumulative distribution function
Normal distribution cumulative distribution function has the following formula:
where, erf(x) - error function, given as:
Normal distribution quantile function (inverse CDF) given as inverse error function:
p lays in the range [0,1]
Standard normal distribution quantile function (σ =1, μ=0) looks like this:
This function is called the probit function.
Calculator below gives quantile value by probability for specified by mean and variance normal distribution( set variance=1 and mean=0 for probit function).