The Extended Statistics analysis calculates extended descriptive statistics for a variable of interest. These statistics include:
Arithmetic Mean
Geometric Mean
Harmonic Mean
Standard Deviation
Coefficient of Variation
Variance
Skewness
Kurtosis
Usage
After selecting Extended Statistics... from the Data Menu, the user will be
presented with the window above. The user can then select exactly one variable for which to compute the descriptive statistics
listed above.
Output
Once the regression is complete, the user will be presented with
results similar to the following:
If the analysis fails for some reason, the failure message will be
displayed in the results window rather than results. Descriptive statistics that do not apply to the selected
data may not appear in output.
Method
The extended statistics analysis is implemented in pure JRuby
using the Draco Plugin System. Each of the parameters is described below:
Arithmetic Mean
The arithmetic mean, often refered to simply as the mean, is calculated as:
Geometric Mean
The geometric mean is calculated using the product of the elements as follows:
The actual algorithm first takes the nth root of each element before multiplying by the running product
in order to avoid floating point overruns.
Harmonic Mean
The harmonic mean is only computed for series that do not contain zeros. It is computed as:
Standard Deviation
The standard deviation, as computed by Draco, is:
Coefficient of Variation
The coefficient of variation represents the ratio of the standard deviation to the mean:
Variance
The variance, the square of the standard deviation, is nominally calculated as:
Skewness
The skewness represents a measure of the assymmetry of a variable:
The value mu is defined as the third moment about the mean.
Kurtosis
The kurtosis is a measure of the peakedness of a variable:
The value mu is defined as the fourth moment about the mean.
kth Moment about the Mean
The skewness and kurtosis calculations rely on the third and fourth moments about the mean. The kth moment about the
mean is defined as: Because this analysis is implemented as a plugin, the source code is available in the scripts/ruby/plugin directory.