# Extended Statistics

Description | Usage | Output | Method | See Also

## Description

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:

# Extended Statistics Results Selection: x
Arithmetic Mean | 11.30475
Geometric Mean | 11.26326
Harmonic Mean | 11.21715
Standard Deviation | 0.92432
Coefficient of Variation | 0.08176
Variance | 0.85436
Skewness | -1.53203
Kurtosis | 5.43447
Draco Plugin System Copyright © 2008-2009 Approximatrix, LLC

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.

## See Also

Plugin System Overview

Copyright © 2009 Approximatrix, LLC
Text licensed under the Creative Commons Attribution-Share Alike 3.0 License
DracoTM and the Approximatrix logo are trademarks of Approximatrix, LLC
Other trademarks are property of their respective owners