COEFFICIENT OF DISPERSION

COEFFICIENT OF DISPERSION (LET)

Let Subcommand

Compute the coefficient of dispersion of a variable.

The coefficient of dispersion is defined as
\( \mbox{COD} = \frac{\tau}{\eta} \)

with \( \eta \) and \( \tau \) denoting the median and mean absolute deviation from the median, respectively.

This statistic has been suggested as a robust alternative to the coefficient of variation.

LET <par> = COEFFICIENT OF DISPERSION <y>
                        <SUBSET/EXCEPT/FOR qualification>
where <y> is a response variable;
            <par> is a parameter where the coefficient of dispersion value is saved;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

LET <par> = DIFFERENCE OF COEFFICIENT OF DISPERSION <y1> <y2>
                        <SUBSET/EXCEPT/FOR qualification>
where <y1> is the first response variable;
            <y2> is the second response variable;
            <par> is a parameter where the difference of the coefficient of dispersion values is saved;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

LET COD = COEFFICIENT OF DISPERSION Y1
LET COD = COEFFICIENT OF DISPERSION Y1 SUBSET TAG > 2


LET DIFFCOD = DIFFERENCE OF COEFFICIENT OF DISPERSION Y1 Y2

Note that there are various other definitions for the coefficient of dispersion.

It is sometimes defined as

COD = VARIANCE/MEAN

To compute this statistic in Dataplot, use the command

LET A = INDEX OF DISPERSION Y

Note that 11/2017 version of Dataplot modified the scale factor to be the average absolute deviation from the median rather than the median absolute deviation from the median. This was done to be consistent with the Bonett paper.

Basically, you can define a dispersion index based on a scale statistic (standard deviation, average absolute deviation, median absolute deviation, etc.) divided by a location statistic (mean, median, etc.). So there are additional possibilities not discussed here.

Dataplot statistics can be used in a number of commands. For details, enter
HELP STATISTICS

None

None

COEFFICIENT OF VARIATION	=	Compute the coefficient of variation of a variable.
QUARTILE COEFFICIENT OF DISPERSION	=	Compute the quartile coefficient of dispersion of a variable.
INDEX OF DISPERSION	=	Compute the index of dispersion of a variable.
RELATIVE STANDARD DEVIATION	=	Compute the standard deviation of a variable.
MEAN	=	Compute the mean of a variable.
MEDIAN	=	Compute the median of a variable.
STANDARD DEVIATION	=	Compute the standard deviation of a variable.
AVERAGE ABSOLUTE DEVIATION FROM THE MEDIAN	=	Compute the average absolute deviation from the median of a variable.
MEDIAN ABSOLUTE DEVIATION	=	Compute the median absolute deviation of a variable.

Bonett and Seier (2005), "Confidence interval for a coefficient of dispersion in nonnormal distributions", Biometrical Journal, Vol. 48, pp. 144-148.

Gastwirth (1982), "Statistical properties as a measure of tax assessment uniformity", Journal of Statistical Planning Inference, Vol. 6, pp. 1-12.

Data Analysis

2017/01
2017/06: DIFFERENCE OF COEFFICIENT OF DISPERSION added
2017/11: Modified the definition to use the average absolute
                deviation from the median rather than the median absolute
                deviation from the median

SKIP 25
READ ZARR13.DAT Y
LET COD = COEFFICIENT OF DISPERSION Y
    

. Step 1:   Create the data
.
skip 25
read gear.dat y x
skip 0
set write decimals 6
.
. Step 2:   Define plot control
.
title case asis
title offset 2
label case asis
.
y1label Coefficient of Dispersion
x1label Group
title Coefficient of Dispersion for GEAR.DAT
let ngroup = unique x
xlimits 1 ngroup
major x1tic mark number ngroup
minor x1tic mark number 0
tic mark offset units data
x1tic mark offset 0.5 0.5
.
character X
line blank
.
set statistic plot reference line average
coefficient of dispersion plot y x
.
set write decimals 5
tabulate coefficient of dispersion y x
    



            Cross Tabulate COEFFICIENT OF DISPERSION
 
(Response Variables: Y        )
---------------------------------------------
       X          |    COEFFICIENT OF
---------------------------------------------
        1.00000   |           0.00341
        2.00000   |           0.00370
        3.00000   |           0.00281
        4.00000   |           0.00321
        5.00000   |           0.00613
        6.00000   |           0.00742
        7.00000   |           0.00550
        8.00000   |           0.00280
        9.00000   |           0.00311
       10.00000   |           0.00382
    

    SKIP 25
    READ IRIS.DAT Y1 TO Y4 X
    .
    LET A = DIFFERENCE OF COEFFICIENT OF DISPERSION Y1 Y2
    SET WRITE DECIMALS 4
    TABULATE DIFFERENCE OF COEFFICIENT Y1 Y2 X
    
    
            Cross Tabulate DIFFERENCE OF COEFFICIENT OF DISPERSION
 
(Response Variables: Y1       Y2      )
---------------------------------------------
       X          |   DIFFERENCE OF C
---------------------------------------------
         1.0000   |           -0.0335
         2.0000   |           -0.0121
         3.0000   |           -0.0051
    
    
    .
    XTIC OFFSET 0.2 0.2
    X1LABEL GROUP ID
    Y1LABEL DIFFERENCE OF COEFFICIENT OF DISPERSION
    CHAR X
    LINE BLANK
    DIFFERENCE OF COEFFICIENT OF DISPERSION PLOT Y1 Y2 X
    

    
    

    CHAR X ALL
    LINE BLANK ALL
    BOOTSTRAP DIFFERENCE OF COEFFICIENT OF DISPERSION PLOT Y1 Y2 X