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The summation represents a summation of .
Based on the variance definition, vardef, the variance divisor d is computed as
for vardef=DF, degrees of freedom | ||
for vardef=N, number of observations |
The skewness is a measure of the tendency of the deviations from the mean to be larger in one direction than in the other. The sample skewness is calculated as
for vardef=DF | ||
for vardef=N |
where c_{3n} = [n/((n-2))][1/((n-1))].
The kurtosis is primarily a measure of the heaviness of the tails of a distribution. The sample kurtosis is calculated as
for vardef=DF | ||
for vardef=N |
where c_{4n} = [(n(n+1))/((n-2)(n-3))][1/((n-1))] and c_{n} = [((n-1)^{2})/((n-2)(n-3))].
When the observations are independently distributed with a common mean and unequal variances, ,where w_{i} are individual weights, weighted analyses may be appropriate. You select a Weight variable to specify relative weights for each observation in the analysis.
The following notation is used in weighted analyses:
In addition to vardef=DF and vardef=N, the variance divisor is also computed as
for vardef=WDF, sum of weights minus 1 | ||
for vardef=WGT, sum of weights |
With , and the expected value
Note | The use of vardef=WDF/WGT may not be appropriate since it is the weighted average of individual variances, , which have unequal expected values. |
For vardef=DF/N, s^{2}_{w} is the variance of observations with unit weight and may not be informative in the weighted plots of parametric normal distributions. SAS/INSIGHT software uses the weighted sample variance for an observation with average weight, , to replace s^{2}_{w} in the plots.
The weighted skewness is computed as
for DF | ||
for N |
The weighted kurtosis is computed as
for DF | ||
for N |
The formulations are invariant under the transformation w^{*}_{i} = c w_{i}, c > 0. The sample skewness and kurtosis are set to missing if vardef=WDF or vardef=WGT.
To view or change the divisor d used in the calculation of variances, or to view or change the use of observations with missing values, click on the Method button from the variables dialog to display the method options dialog.
By default, SAS/INSIGHT software uses vardef=DF, degrees of freedom to compute the variance divisor.
When multiple Y variables are analyzed,
and some Y variables have missing values,
the Use Obs with Missing Values option uses
all observations with nonmissing values for
the Y variable being analyzed.
If the option is turned off, observations with missing values
for any Y variable are not used for any analysis.
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