SHORTEST HALF MIDMEAN

SHORTEST HALF MIDMEAN (LET)

Let Subcommand

Compute the shortest half midmean for a variable.

The midmean of a variable is the mean of the observations between the 25th and 75th percentiles. The shortest half midmean uses the most compact half of the data rather than the middle half. This is essentially an asymetric version of the midmean. Although it has rather low efficiency (lower than the median), it is less sensitive to asymmetrically distributed outliers. The formula for the shortest half midmean is
\( \mbox{Sh/mean} = \sum_{i=k}^{k+m}{\frac{x_{i}}{m}} \hspace{0.3in} \mbox{for the minimum} \hspace{0.1in} (x_{k+m} - x_{k}) \)

\( m = n/2 \hspace{1.35in} n \mbox{ odd} \)
\( m = \mbox{INT}(n/2) + 1 \hspace{0.5in} n \mbox{ even} \)

LET <par> = SHORTEST HALF MIDMEAN <y>             <SUBSET/EXCEPT/FOR qualification>
where <y> is the response variable;
            <par> is a parameter where the computed shortest half midmean is stored;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

LET <par> = DIFFERENCE OF SHORTEST HALF MIDMEAN <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 computed difference of shortest half midmeans is stored;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

LET A = SHORTEST HALF MIDMEAN Y1
LET A = SHORTEST HALF MIDMEAN Y1 SUBSET TAG > 2


LET A = DIFFERENCE OF SHORTEST HALFMIDMEAN Y1 Y2

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

None

SHORTEST HALF MID MEAN is a synonym for SHORTEST HALF MIDMEAN

MIDMEAN	=	Compute the midmean.
SHORTEST HALF MIDRANGE	=	Compute the shortest half midrange.
MEAN	=	Compute the mean.
MEDIAN	=	Compute the median.
STANDARD DEVIATION	=	Compute the standard deviation.

David Duewer (2008), "A Comparison of Location Estimators for Interlaboratory Data Contaminated with Value and Uncertainty Outliers", Accredited Quality Assurance, Vol. 13, pp. 193-216.

Andrews, Bickel, Hampel, Huber, Rogers, and Tukey (1972), "Robust Estimates of Location", Princeton University Press, Princeton.

Rousseeuw (1985), "Multivariate Estimation with High Breakdown Point", in Grossman, Pflug, Nincze, Wetrz (eds), "Mathematical Statistics and Applications", Reidel, Dordrecht, The Netherlands, pp. 283-297.

Robust Data Analysis

2017/02
2017/06: Added DIFFERENCE OF SHORTEST HALF MIDMEAN

 
SKIP 25
READ LGN.DAT Y
LET SHMM = SHORTEST HALF MIDMEAN Y
    

 
. Step 1:   Create the data
.
skip 25
read gear.dat y x
skip 0
.
char X
line blank
y1label Shortest Half Midmean
x1label Group
x1tic mark offset 0.5 0.5
label case asis
title case asis
title Shortest Half Midmean of GEAR.DAT
title offset 2
.
set statistic plot reference line average
shortest half midmean plot y x
.
set write decimals 5
tabulate shortest half midmean y x
    

The following output is generated

 
            Cross Tabulate SHORTEST HALF MIDMEAN
 
(Response Variables: Y        )
---------------------------------------------
       X          |   SHORTEST HALF M
---------------------------------------------
        1.00000   |           0.99783
        2.00000   |           0.99817
        3.00000   |           0.99700
        4.00000   |           0.99550
        5.00000   |           0.99517
        6.00000   |           1.00183
        7.00000   |           0.99883
        8.00000   |           0.99800
        9.00000   |           0.99700
       10.00000   |           0.99633
    

 
SKIP 25
READ IRIS.DAT Y1 TO Y4 X
.
LET A = DIFFERENCE OF SHORTEST HALF MIDMEAN Y1 Y2
SET WRITE DECIMALS 4
TABULATE DIFFERENCE OF SHORTEST HALF MIDMEAN Y1 Y2 X

            Cross Tabulate DIFFERENCE OF SHORTEST HALF MIDMEAN
 
(Response Variables: Y1       Y2      )
---------------------------------------------
       X          |   DIFFERENCE OF S
---------------------------------------------
         1.0000   |            1.7346
         2.0000   |            2.8769
         3.0000   |            3.4962


.
XTIC OFFSET 0.2 0.2
X1LABEL GROUP ID
Y1LABEL DIFFERENCE OF SHORTEST HALF MIDMEAN
CHAR X
LINE BLANK
DIFFERENCE OF SHORTEST HALF MIDMEAN PLOT Y1 Y2 X



CHAR X ALL
LINE BLANK ALL
BOOTSTRAP DIFFERENCE OF SHORTEST HALF MIDMEAN PLOT Y1 Y2 X