 Dataplot Vol 2 Vol 1

# SHORTEST HALF MIDRANGE

Name:
SHORTEST HALF MIDRANGE (LET)
Type:
Let Subcommand
Purpose:
Compute the shortest half midrange for a variable.
Description:
The midrange of a variable is the mean of the minimum and maximum values. The shortest half midrange is based on the most compact half of the data. This is essentially an asymetric version of the mean of the lower quartile and the upper quartile. Although it has rather low efficiency (lower than the median), it is less sensitive to asymmetrically distributed outliers. The formula for the shortest half midrange is

$$\mbox{Sh/mid} = \frac{x_{k} + x_{k+m}} {2} \hspace{0.3in} \mbox{for the minimum} \hspace{0.1in} (x_{k+m} - x_{k})$$

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

Syntax 1:
LET <par> = SHORTEST HALF MIDRANGE <y>             <SUBSET/EXCEPT/FOR qualification>
where <y> is the response variable;
<par> is a parameter where the computed shortest half midrange is stored;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.
Syntax 2:
LET <par> = DIFFERENCE OF SHORTEST HALF MIDRANGE <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 midranges is stored;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.
Examples:
LET A = SHORTEST HALF MIDRANGE Y1
LET A = SHORTEST HALF MIDRANGE Y1 SUBSET TAG > 2

LET A = DIFFERENCE OF SHORTEST HALF MIDRANGE Y1 Y2

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

Default:
None
Synonyms:
SHORTEST HALF MID RANGE is a synonym for SHORTEST HALF MIDRANGE
Related Commands:
 SHORTEST HALF MIDMEAN = Compute the 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.
References:
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.

Applications:
Robust Data Analysis
Implementation Date:
2017/02
2017/06: Added DIFFERENCE OF SHORTEST HALF MIDRANGE
Program 1:

SKIP 25
LET SHMM = SHORTEST HALF MIDRANGE Y

Program 2:

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

The following output is generated

Cross Tabulate SHORTEST HALF MIDRANGE

(Response Variables: Y        )
---------------------------------------------
X          |   SHORTEST HALF M
---------------------------------------------
1.00000   |           0.99700
2.00000   |           0.99800
3.00000   |           0.99700
4.00000   |           0.99600
5.00000   |           0.99400
6.00000   |           1.00250
7.00000   |           0.99900
8.00000   |           0.99800
9.00000   |           0.99700
10.00000   |           0.99600 Program 3:

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

Cross Tabulate DIFFERENCE OF SHORTEST HALF MIDRANGE

(Response Variables: Y1       Y2      )
---------------------------------------------
X          |   DIFFERENCE OF S
---------------------------------------------
1.0000   |            1.7000
2.0000   |            2.9500
3.0000   |            3.5000

.
XTIC OFFSET 0.2 0.2
X1LABEL GROUP ID
Y1LABEL DIFFERENCE OF SHORTEST HALF MIDRANGE
CHAR X
LINE BLANK
DIFFERENCE OF SHORTEST HALF MIDRANGE PLOT Y1 Y2 X CHAR X ALL
LINE BLANK ALL
BOOTSTRAP DIFFERENCE OF SHORTEST HALF MIDRANGE PLOT Y1 Y2 X NIST is an agency of the U.S. Commerce Department.

Date created: 07/03/2017
Last updated: 07/03/2017