# INVPROBN

## Purpose:

Returns z value of the probability of X <= z for a normal distribution.

## Syntax:

INVPROBN(p, mean, std)

 p - A real or series. The probability value. mean - Optional. A real, the mean of the distribution. Defaults to 0.0. std - Optional. A real, the standard deviation of the distribution, where std > 0. Defaults to 1.0.

## Returns:

A real or series, the z value of the inverse normal cumulative distribution function with the given mean and standard deviation.

For the input probability p, returns the z value where P(X <= z) = p.

## Example:

invprobn(.5)

returns 0.0, the z value where 50% of the values are less than or equal to for a normal distribution with a mean of 0.0 and a standard deviation of 1.0.

In probabilistic terms, given the normal distribution N(0, 1), i.e. mean of 0, variance of 1:

P(X <= 0.0) = 0.5

## Example:

invprobn(.2, 10, 2)

returns 8.31675753, the z value where 20 percent of the values are less than or equal to for a normal distribution with a mean of 10.0 and a standard deviation of 2.0.

In probabilistic terms, given the normal distribution N(10, 4), i.e. mean of 10, variance of 4:

P(X <= 8.31675753) = 0.2

## Example:

probn(invprobn(.35))

returns 0.35 indicating that PROBN and INVPROBN are inverse functions.

## Example:

invprobn(0.01..0.01..0.99)

displays the inverse normal cumulative distribution function over the range 0.01 to 0.99.

## Remarks:

INVPROBN uses a minimax approximation by rational functions and the result has a relative error less than 1.15e-9. A last refinement by Halley's rational method is applied to achieve full machine precision. The algorithm was originally developed by Peter J. Acklam.

The result is NaN where std <= 0.

INVPROBN is much faster and more accurate than IVSNORMPB.

For 0.5 <= z < 1.0, invprobn(z) == ivsnormpb(z – 0.5) approximately.

See PROBN to return p for a given z such that P(X <= z) = p. INVPROBN is the inverse of PROBN.

See PDFNORM to generate the normal density function.

A2STD

CNF2STD

CONFX

ERF

ERFCINV

IVSNORMPB

PDFNORM

PROBN

XCONF