GINI MEAN DIFFERENCE

GINI MEAN DIFFERENCE (LET)

Let Subcommand

Compute the Gini mean difference for a variable.

Given a response variable \( X \) with observations \( x_1, x_2, ..., x_n \) sorted from low value to high value, the Gini mean difference is defined as:
\( \begin{array}{lcl} \mbox{GMD} & = & \frac{2}{n(n-1)} \sum_{i=1}^{n} {\sum_{j=1}^{i-1}{|x_{i} - x_{j}|}} \\ & = & \frac{2}{n(n-1)} \sum_{i=1}^{n-1} {i(n-i)x_{i+1} - x_{i}} \end{array} \)

where \( n \) is the sample size.

The Gini mean difference was proposed by Gini (1912) and is the average absolute differences in all pairs of observations. Note that this is a measure of dispersion that does not depend on a measure of location.

LET <par> = GINI MEAN DIFFERENCE <y>
                  <SUBSET/EXCEPT/FOR qualification>
where <y> is the response variable;
            <par> is a parameter where the computed Gini mean difference is stored;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

LET A = GINI MEAN DIFFERENCE Y1
LET A = GINI MEAN DIFFERENCE Y1 SUBSET TAG > 2

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

None

None

GINI MEAN DIFFERENCE LOG RATIO	=	Compute the log of the ratio of the Gini mean differences of two samples.
MEDIAN ABSOLUTE DEVIATION	=	Compute the median absolute deviation of a variable.
AVERAGE ABSOLUTE DEVIATION	=	Compute the average absolute deviation of a variable.
STANDARD DEVIATION	=	Compute the standard deviation of a variable.
VARIANCE	=	Compute the variance of a variable.
RANGE	=	Compute the range of a variable.

Tena (2009), "Test Procedures for Equality of Two Variances in Delta Distributions," Dissertations, 695, https://scholarworks.wmich.edu/dissertations/695.

Gini (1921), "Measurement of Inequality in Incomes," The Economic Journal, 31, pp. 124-126.

Data Analysis

2023/07

 
SKIP 25
READ ROSNER.DAT Y
.
BOOTSTRAP SAMPLES 10000
TITLE CASE ASIS
LABEL CASE ASIS
CASE ASIS
.
TITLE Bootstrap Plot for Gini Mean Difference
X1LABEL Bootstrap Sample
X2LABELDataset: ROSNER.DAT
Y1LABEL Gini Mean Difference
.
BOOTSTRAP GIN MEAN DIFFERENCE PLOT Y
.
LET LCL = ROUND(B025,3)
LET UCL = ROUND(B975,3)
JUSTIFICATION CENTER
MOVE 50 5
TEXT Lower 95% Confidence Limit: ^LCL, Upper 95% Confidence Limit: ^UCL
LINE DOTTED
LINE COLOR RED
DRAWSDSD 15 ^LCL 85 ^LCL
DRAWSDSD 15 ^UCL 85 ^UCL
    

The following output is generated

 
             Bootstrap Analysis for the GINI MEAN DIFFERENCE
  
 Response Variable One: Y
  
 Number of Bootstrap Samples:                      10000
 Number of Observations:                              50
 Mean of Bootstrap Samples:                      1.13362
 Standard Deviation of Bootstrap Samples:        0.16878
 Median of Bootstrap Samples:                    1.13333
 MAD of Bootstrap Samples:                       0.11439
 Minimum of Bootstrap Samples:                   0.55771
 Maximum of Bootstrap Samples:                   1.71342
  
  
  
 Percent Points of the Bootstrap Samples
 -----------------------------------
   Percent Point               Value
 -----------------------------------
             0.1    =        0.63803
             0.5    =        0.71597
             1.0    =        0.75432
             2.5    =        0.81257
             5.0    =        0.85786
            10.0    =        0.91725
            20.0    =        0.98941
            50.0    =        1.13333
            80.0    =        1.27518
            90.0    =        1.35585
            95.0    =        1.41879
            97.5    =        1.47064
            99.0    =        1.52013
            99.5    =        1.56189
            99.9    =        1.63985
  
  
             Percentile Confidence Interval for Statistic
  
 ------------------------------------------
   Confidence          Lower          Upper
  Coefficient          Limit          Limit
 ------------------------------------------
        50.00        1.01674        1.24562
        75.00        0.93664        1.33103
        90.00        0.85786        1.41879
        95.00        0.81257        1.47064
        99.00        0.71597        1.56189
        99.90        0.61387        1.65573
 ------------------------------------------
    

 
SKIP 25
READ GEAR.DAT Y X
.
TITLE CASE ASIS
LABEL CASE ASIS
CASE ASIS
.
CHARACTER CIRCLE BLANK
CHARACTER HW 1 0.75
CHARACTER FILL ON
LINE BLANK SOLID
.
XLIMITS 1 10
MAJOR X1TIC MARK NUMBER 10
MINOR X1TIC MARK NUMBER 0
TIC MARK OFFSET UNITS DATA
X1TIC MARK OFFSET 0.5 0.5
.
Y1LABEL Gini Mean Difference
X1LABEL Batch ID
TITLE Gini Mean Difference Plot for GEAR.DAT
.
GINI MEAN DIFFERENCE PLOT Y X