# PROBN

## Purpose:

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

## Syntax:

PROBN(z, mean, std)

z |
- |
A real or series. The *z* 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 probability that a value is less than or equal
to the input z value for a normal distribution with
the given mean and standard deviation.

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

## Example:

probn(0)

returns 0.5, the probability that a value is less than or equal to 0.0
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:

probn(2, 1, 2) - probn(0, 1, 2)

returns 0.382925, the probability that a value is less than or equal
to 2 and greater than or equal to 0.0 for a normal distribution with a
mean of 1.0 and a standard deviation of 2.0

In probabilistic terms, given the normal distribution N(1,
4), i.e. mean of 1, variance of 4:

P(0.0 <=
X <= 2.0) = 0.382925

## Example:

1 – probn(.5)

returns 0.30853754, the probability that a value is greater than 0.5
for a normal distribution with a mean of 0.0 and a standard deviation
of 1.0

## Example:

probn(invprobn(.3))

returns 0.3, indicating that PROBN and INVPROBN
are inverse functions.

## Example:

probn(-3..0.01..3)

displays the normal cumulative distribution function from –3 to 3.

## Remarks:

The result is NaN where **std** ≤ 0.

The probability density function, *f*(*x*),
for normally distributed random values is:

where *μ*
is the mean, *σ* is the standard
deviation and *σ*^{2}
is the variance. The cumulative distribution function, Φ(*x*),
is:

where erf(*x*)
is the error function implemented by ERF.
PROBN uses ERF to evaluate the area
under the normal distribution curve. Note that probn(z)
returns the area from -∞
to z.

Approximately 68% of the values from a normal distribution are within
one standard deviation away from the mean. Approximately 95% of the values
lie within two standard deviations and approximately 99.7% are within
three standard deviations. This is known as the *3-sigma rule*.

See INVPROBN to calculate the
inverse normal cumulative distribution function. PROBN and INVPROBN
are inverse functions.

See PDFNORM to generate the normal density
function.

See GNORMAL to generate a series
of normally distributed random values.

## See Also:

A2STD

CNF2STD

CONFX

ERF

GNORMAL

INVPROBN

PDFNORM

XCONF