# 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.

## See Also:

A2STD

CNF2STD

CONFX

ERF

ERFCINV

IVSNORMPB

PDFNORM

PROBN

XCONF