# PDFNORM

## Purpose:

Returns the probability density function for a normal distribution.

## Syntax:

PDFNORM(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 value of the normal distribution function for
the given mean and standard deviation.

## Example:

pdfnorm(0)

returns 1, the value of the normal distribution with a mean of 0.0 and
a standard deviation of 1.0.

## Example:

pdfnorm(-8..0.01..8)

displays the normal distribution with a mean of 0.0 and a standard deviation

of 1.0 over the range -8 to 8.

## Example:

pdfnorm(-8..0.01..8, 0, 2.0)

Same as above except the standard deviation of the distribution is set
to 2.0.

## Example:

W1: hist(gnorm(10000, 1, 10, 3), 200, "pdf");lines

W2: pdfnorm(xvals(W1), 10, 3);overp(W1, lred)

Compares the calculated normal distribution of random values with mean
10 and standard deviation 3 in W1 with the analytic distribution in W2.
The overplotted histogram is normalized to provide an visual accurate
comparison.

## 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.

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 PROBN to calculate the normal cumulative
distribution function.

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

## See Also:

A2STD

CNF2STD

CONFX

ERF

GNORMAL

HISTOGRAM

INVPROBN

PROBN

RANDN

XCONF