*PROC CAPABILITY and General Statements* |

## Descriptive Statistics

This section provides computational details for the
descriptive statistics
which are computed with the
PROC CAPABILITY statement.
These statistics can also be
saved in the OUT= data set by specifying the keywords
listed in Table 7.1 in the OUTPUT statement.
Standard algorithms (Fisher 1973) are used to compute
the moment statistics. The computational methods
used by the CAPABILITY procedure are consistent with
those used by other SAS procedures for calculating
descriptive statistics. For details on statistics
also calculated by base SAS software, see *SAS Language
Reference: Dictionary*.

The following sections give specific
details on several statistics calculated by the
CAPABILITY procedure.

*Mean*

The sample mean is calculated as

where *n* is the number of nonmissing values for a variable,
*x*_{i} is the *i*^{ th} value of the variable,
and *w*_{i} is the weight associated with the
*i*^{ th} value of the variable.
If there is no WEIGHT= variable, the formula reduces to
.

*Sum*

The sum is calculated as , where *n*
is the number of nonmissing values for a variable, *x*_{i}
is the *i*^{ th} value of the variable,
and *w*_{i} is the
weight associated with the
*i*^{ th} value of the variable.
If there is no WEIGHT= variable, the formula reduces
to .

*Sum of the Weights*

The sum of the weights is calculated as ,where *n* is the number of nonmissing values for a variable
and *w*_{i} is the weight associated with the *i*^{ th} value
of the variable. If there is no WEIGHT= variable, the
sum of the weights is *n*.

*Variance*

The variance is calculated as
`
`

where *n* is the number of nonmissing values for a variable,
*x*_{i} is the *i*^{ th} value of the variable,
is the weighted mean, *w*_{i} is the weight
associated with the *i*^{ th} value of
the variable, and *d* is the divisor controlled by the
VARDEF= option in the PROC CAPABILITY statement.
If there is no WEIGHT= variable, the formula reduces to
`
`

The standard deviation is calculated as
`
`

where *n* is the number of nonmissing values for a variable,
*x*_{i} is the *i*^{ th}
value of the variable, is the weighted mean, *w*_{i} is the weight associated with
the *i*^{ th}
value of the variable, and *d* is the divisor
controlled by the VARDEF= option in the PROC CAPABILITY
statement. If there is no WEIGHT= variable, the formula
reduces to
`
`

*Skewness*

The sample skewness is calculated as

where *n* is the number of nonmissing values for a variable
and must be greater than 2,
*x*_{i} is the *i*^{ th} value of
the variable, is the sample average, and *s* is
the sample standard deviation.
The sample skewness can be positive or negative; it
measures the asymmetry of the data distribution and
estimates the theoretical skewness
,where and are the second and third
central moments. Observations that are normally
distributed should have a skewness near zero.

*Kurtosis*

The sample kurtosis is calculated as

where *n* > 3. The sample kurtosis measures the
heaviness of the tails of the data distribution. It
estimates the adjusted theoretical kurtosis denoted
as , where ,and is the fourth central moment. Observations
that are normally distributed should have a kurtosis
near zero.

*Coefficient of Variation (CV)*

The coefficient of variation is calculated as

Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.