Computes basic statistics from grouped data.
TABLE — Vector of
length NGROUP
containing the frequencies within the groups. (Input)
The
entries in TABLE
are interpreted as counts. They must be nonnegative.
CLOW — The center (class mark) of the lowest class interval. (Input)
CWIDTH — The
class width. (Input)
CWIDTH must be
positive.
STAT — Vector of length 13 containing the statistics. (Output)
I STAT(I)
1 The sum of the frequencies in TABLE.
2 Mean (arithmetic mean, first moment).
3 Sample standard deviation. (Uses STAT(1) − 1 as divisor).
4 Second moment about the mean, uncorrected for grouping. (Uses STAT(1) as divisor.)
5 Second moment about the mean, adjusted using Sheppard’s correction.
6 Third moment about the mean, uncorrected for grouping.
7 Third moment about the mean, adjusted using Sheppard’s correction.
8 Fourth moment about the mean, uncorrected for grouping.
9 Fourth moment about the mean, adjusted using Sheppard’s correction.
10 Median.
11 Geometric mean; defined only if CLOW − CWIDTH/2 is nonnegative.
12 Harmonic mean; defined only if CLOW − CWIDTH/2 is nonnegative.
13 Mode; defined only if one element of TABLE is strictly greater than all other elements of TABLE.
NGROUP — Number
of groups. (Input)
Default: NGROUP = size (TABLE,1).
IPRINT — Printing
option. (Input)
If IPRINT = 0, no
printing is performed; and if IPRINT = 1, the
statistics in STAT
are printed.
Default: IPRINT = 0.
Generic: CALL GRPES (TABLE, CLOW, CWIDTH, STAT [,…])
Specific: The specific interface names are S_GRPES and D_GRPES.
Single: CALL GRPES (NGROUP, TABLE, CLOW, CWIDTH, IPRINT, STAT)
Double: The double precision name is DGRPES.
The routine GRPES computes various statistics using data from equally spaced groups. The second, third, and fourth moments are computed both with and without Sheppard’s corrections. These corrections for grouped data are most useful for distributions whose densities tail off smoothly (such as the normal distribution). Kendall, Stuart, and Ord (1987, Chapters 2 and 3) discuss these corrections.
The moments are computed using the sum of the frequencies as the divisor. The standard deviation (STAT(3)), on the other hand, is computed using as the divisor the sum of the frequencies minus one.
If any of the class marks are negative, the geometric and harmonic means are not computed, and NaN (not a number) is stored as the value of STAT(11). Likewise, if the mode does not exist (no group has a frequency greater than that of all other groups), NaN is stored as the value of STAT(13).
This example is taken from Conover and Iman (1983, page
119). The objective is to compute some basic statistics relating to test scores,
using the following data:
Score |
Frequency |
91−100 |
7 |
81−90 |
13 |
71−80 |
11 |
61−70 |
5 |
≤ 60 |
4 |
USE GRPES_INT
IMPLICIT NONE
INTEGER IPRINT, NGROUP
REAL CLOW, CWIDTH, STAT(13), TABLE(5)
!
NGROUP = 5
CLOW = 55.5
CWIDTH = 10.0
TABLE(1) = 4.0
TABLE(2) = 5.0
TABLE(3) = 11.0
TABLE(4) = 13.0
TABLE(5) = 7.0
IPRINT = 1
CALL GRPES (TABLE, CLOW, CWIDTH, STAT, IPRINT=IPRINT)
END
Statistics from GRPES
Sum
freqs.
40.0
Mean
79.0
Std. dev. 12.1
2nd
moment 142.8
2nd,
adj. 134.4
3rd
moment -741.8
3rd,
adj. -2716.8
4th moment
48242.3
4th, adj.
47929.0
Median
80.5
Geometric
78.0
Harmonic
77.0
Mode
85.5
In this example, there are negative values of some class
marks, and there is no modal class.
Class Marks |
Frequency |
−2.0 |
2 |
−1.0 |
5 |
0.0 |
7 |
1.0 |
7 |
2.0 |
2 |
USE GRPES_INT
IMPLICIT NONE
INTEGER NGROUP, IPRINT
REAL TABLE(5), CLOW, CWIDTH, STAT(13)
!
NGROUP = 5
CLOW = -2.0
CWIDTH = 1.0
TABLE(1) = 2.0
TABLE(2) = 5.0
TABLE(3) = 7.0
TABLE(4) = 7.0
TABLE(5) = 2.0
IPRINT = 1
CALL GRPES (TABLE, CLOW, CWIDTH, STAT, IPRINT=IPRINT)
END
Statistics from GRPES
Sum freqs.
23.0000
Mean
0.0870
Std. dev. 1.1246
2nd
moment 1.2098
2nd,
adj. 1.1265
3rd
moment -0.2293
3rd,
adj. -0.2510
4th
moment 3.3292
4th,
adj.
2.7960
Median
0.1429
The mode is not defined, since no class has higher
frequency
than all others.
The geometric and harmonic means are not defined,
since
the lower bound is negative.
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