# Inverse Normal distribution

This tool calculates the Inverse of Normal Cumulative Distribution Function. One of its uses is to calculate percentiles of a Normal distribution.

## Inverse Normal Distribution formulas

### Notations

X : a random variable with a normal distribution

`mu` : mean of variable X

`sigma` : its Standard deviation

The inverse of the cumulative distribution function is also called the 'quantile function'.

We denote Q the quantile function and F the cumulative distribution function of variable X. We have,

`F(x) = 1/(sigma*sqrt(2pi))*\int_-oo^xe^(-1/2*(t-mu)^2/sigma^2)\ dt`

`Q(x) = F^(-1)(x)`

For a probability p, quantile function Q gives a q value that verifies,

`q = Q(p) = F^(-1)(p)`

By definition of F, we have,

`P(X < q) = p`

`P(X < q)` is the pobability that X is less than q.

In addition to quantiles, this tool calculates, for a given probability p, the values q such that,

`P(X > q) = p`

`P( |X - mu| < q) = p`

`P( |X - mu| > q) = p`

### Example of use : quartiles calculator

To calculate the first quartile Q_{1}(or 25th percentile) of the standard Normal distribution, we input the following values in the calculator,

`mu = 0`

`sigma = 1`

`p=0.25`

We get, Q_{1} = -0.67448975

for p = 0.75, we get Q_{3} = 0.67448975

for p = 0.5, we get the mean as expected, Q_{2} = 0.

## See also

Normal distribution Probabilities

Statistics Calculators