Dataplot Vol 2 Vol 1

# COEFFICIENT OF DISPERSION

Name:
COEFFICIENT OF DISPERSION (LET)
Type:
Let Subcommand
Purpose:
Compute the coefficient of dispersion of a variable.
Description:
The coefficient of dispersion is defined as

$$\mbox{COD} = \frac{\tau}{\eta}$$

where $$\eta$$ and $$\tau$$ are the median and median absolute deviation, respectively.

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

Syntax 1:
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.
Syntax 2:
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.
Examples:
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:
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

Another definition is

where AAD is the average absolute deviation. In this case, we are computing the AAD as

$$\mbox{AAD} = \frac{\sum_{i=1}^{n}{X_{i} - \tilde{X}}} {n}$$

where $$\tilde{X}$$ denotes the median.

That is, we compute absolute differences from the median rather than from the mean.

This statistic can be computed in Dataplot using the command

LET A = AAD TO MEDIAN Y

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.

Note:
Dataplot statistics can be used in a number of commands. For details, enter

Default:
None
Synonyms:
None
Related Commands:
 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. MEDIAN ABSOLUTE DEVIATION = Compute the median absolute deviation of a variable.
Reference:
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.

Applications:
Data Analysis
Implementation Date:
2017/01 2017/06: DIFFERENCE OF COEFFICIENT OF DISPERSION added
Program 1:
SKIP 25
LET COD = COEFFICIENT OF DISPERSION Y

Program 2:
. Step 1:   Create the data
.
skip 25
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   |           0.00300
2.00000   |           0.00250
3.00000   |           0.00251
4.00000   |           0.00301
5.00000   |           0.00402
6.00000   |           0.00702
7.00000   |           0.00400
8.00000   |           0.00200
9.00000   |           0.00251
10.00000   |           0.00201

Program 3:
    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


NIST is an agency of the U.S. Commerce Department.

Date created: 01/24/2017
Last updated: 06/30/2017