Descriptive Statistics and Measures of Central Tendency and Dispersion
© 1998 by Dr. Thomas W. MacFarland -- All Rights Reserved
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cent_tnd.doc
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Background: Quite often when examining data and relationships
between data, it is useful to offer a general
view of the data. Imagine an array of data,
representing final examination test scores in a
computer science education class:
-- How many students sat for the test?
-- What is the average test score and are
there multiple definitions of the term
"average?"
-- Did most test scores come close to the
average, or was there a wide degree of
variance in test scores?
-- What was the range of test scores, from
the lowest test score to the highest test
score?
The following listing identifies a series of
statistical measures of central tendency (closeness
to the "average" score) and dispersion (spread or
variance in the range of scores away from average
score) typically used in the social sciences:
1. Measures of central tendency or closeness
to the average score:
A. Mode ...... most frequent score
B. Median .... mid-point of an array
C. Mean ...... arithmetic average (Sum/N)
In the "perfect" bell-shaped curve, all three
measures of central tendency would be equivalent.
2. Measures of dispersion, spread, or variance in
the range of scores away from the average
score:
A. Variance ... the sum of squared deviations
from the mean
B. SD ......... the standard deviation, or the
square root the variance
C. Range ...... the spread from the lowest score
to the highest score
It is common to present in summary statistics a
listing of these descriptive statistics, to give
the reader a general view of the data. In our
current example, you would typically identify:
-- N or number of valid final examination test
scores
-- Average final examination test score
-- Mode
-- Median
-- Mean
-- Variance in final examination test scores
-- SD (Std dev) or standard deviation
-- Range of scores from minimum score to
highest test score
This information gives a far more complete
description of test results than merely stating
that "the average test score was 80 out of 100."
Scenario: A computing technology teacher administered a final
examination at the end of a nine-week term. In an
attempt to better understand the progress of her
students, she prepared a data file and then used
leading software products to examine final
examination outcomes. Scores (potentially ranging
from 000 to 100) for her 23 students are presented
in Table 1:
Table 1
Scores for a Computing Technology Final Examination
===================================================
Student Number Score
---------------------------------------------------
01 089
02 092
03 073
04 083
05 056
06 082
07 077
08 092
09 100
10 067
11 071
12 076
13 083
14 086
15 077
16 049
17 071
18 084
19 091
20 088
21 082
22 077
23 097
___________________________________________________
Files: 1. cent_tnd.doc
2. cent_tnd.dat
3. cent_tnd.r01
4. cent_tnd.o01
5. cent_tnd.con
6. cent_tnd.lis
Command: At the UNIX prompt (%), key:
%spss -m < cent_tnd.r01 > cent_tnd.o01
Contact your system administrator if you need
to use another command to effect SPSS-X in
batch mode. Of course, slight modifications
may be necessary if you use SPSS on a PC.
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cent_tnd.dat
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01 089
02 092
03 073
04 083
05 056
06 082
07 077
08 092
09 100
10 067
11 071
12 076
13 083
14 086
15 077
16 049
17 071
18 084
19 091
20 088
21 082
22 077
23 097
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cent_tnd.r01
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SET WIDTH = 80
SET LENGTH = NONE
SET CASE = UPLOW
SET HEADER = NO
TITLE = Descriptive Statistics and Central Tendency
COMMENT = This file examines scores on a computing
technology final examination
DATA LIST FILE = 'cent_tnd.dat' FIXED
/ Stu_Code 20-21
Score 39-41
Variable Labels
Stu_Code "Student Code"
/ Score "Exam Score "
FREQUENCIES VARIABLES = Score
/ STATISTICS = All
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cent_tnd.o01
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1 SET WIDTH = 80
2 SET LENGTH = NONE
3 SET CASE = UPLOW
4 SET HEADER = NO
5 TITLE = Descriptive Statistics and Central Tendency
6 COMMENT = This file examines scores on a computing
7 technology final examination
8 DATA LIST FILE = 'cent_tnd.dat' FIXED
9 / Stu_Code 20-21
10 Score 39-41
11
This command will read 1 records from cent_tnd.dat
Variable Rec Start End Format
STU_CODE 1 20 21 F2.0
SCORE 1 39 41 F3.0
12 Variable Labels
13 Stu_Code "Student Code"
14 / Score "Exam Score "
15
16
17 FREQUENCIES VARIABLES = Score
18 / STATISTICS = All
SCORE Exam Score
Valid Cum
Value Label Value Frequency Percent Percent Percent
49 1 4.3 4.3 4.3
56 1 4.3 4.3 8.7
67 1 4.3 4.3 13.0
71 2 8.7 8.7 21.7
73 1 4.3 4.3 26.1
76 1 4.3 4.3 30.4
77 3 13.0 13.0 43.5
82 2 8.7 8.7 52.2
83 2 8.7 8.7 60.9
84 1 4.3 4.3 65.2
86 1 4.3 4.3 69.6
88 1 4.3 4.3 73.9
89 1 4.3 4.3 78.3
91 1 4.3 4.3 82.6
92 2 8.7 8.7 91.3
97 1 4.3 4.3 95.7
100 1 4.3 4.3 100.0
------- ------- -------
Total 23 100.0 100.0
Mean 80.130 Std err 2.546 Median 82.000
Mode 77.000 Std dev 12.211 Variance 149.119
Kurtosis .914 S E Kurt .935 Skewness -.813
S E Skew .481 Range 51.000 Minimum 49.000
Maximum 100.000 Sum 1843.000
Valid cases 23 Missing cases 0
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cent_tnd.con
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Conclusion: Descriptive statistics and measures of central
tendency for final examination test scores
follow:
N Mode Median Mean SD Range
============================================
23 77 82 80.1 12.2 51: 49 to 100
Far greater detail (perhaps too much detail) on
descriptive statistics for final examination test
scores can be found at the end of the output file
(cent_tnd.o01).
As you examine this section of the output file, be
sure to notice that:
-- N = 23, which is to say that there were 23
students who had scores for this examination.
-- Three separate values were provided for the
"average" score:
-- Mode (most frequent) was 77
-- Median (mid-point of the array of all final
examination scores) was 82
-- Mean (arithmetic average, or Sum of all
final examination scores / Number of final
examination scores) was 80.1
-- Variance is expressed by two leading statistics:
-- Standard Deviation (Std dev or SD,
representing dispersion of final examination
scores away from the mean) was 12.2
-- Range in final examination scores was 51,
from a minimum score of 49 to a maximum
score of 100
Each statistic is useful in our attempt to place
context to outcomes. Although it is very common
to only see N, Mean, and SD presented in the
literature, the other statistics presented above
give a more complete picture of outcomes.
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cent_tnd.lis
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% minitab
MTB > outfile 'cent_tnd.lis'
Collecting Minitab session in file: cent_tnd.lis
MTB > # MINITAB addendum to cent_tnd.dat
MTB > read 'cent_tnd.dat' c1 c2
Entering data from file: cent_tnd.dat
23 rows read.
MTB > print c1
C1
1 2 3 4 5 6 7 8 9 10 11 12 13
14 15 16 17 18 19 20 21 22 23
MTB > print c2
C2
89 92 73 83 56 82 77 92 100 67 71
76 83 86 77 49 71 84 91 88 82 77
97
MTB > histogram c2
Histogram of C2 N = 23
Midpoint Count
50 1 *
55 1 *
60 0
65 1 *
70 2 **
75 5 *****
80 2 **
85 4 ****
90 5 *****
95 1 *
100 1 *
MTB > stem-and-leaf c2
Stem-and-leaf of C2 N = 23
Leaf Unit = 1.0
1 4 9
1 5
2 5 6
2 6
3 6 7
6 7 113
10 7 6777
(5) 8 22334
8 8 689
5 9 122
2 9 7
1 10 0
MTB > dotplot c2
.
. . . : . .: ::. . .. .: . .
---+---------+---------+---------+---------+---------+---C2
50 60 70 80 90 100
MTB > tally c2
C2 COUNT
49 1
56 1
67 1
71 2
73 1
76 1
77 3
82 2
83 2
84 1
86 1
88 1
89 1
91 1
92 2
97 1
100 1
N= 23
MTB > describe c2
N MEAN MEDIAN TRMEAN STDEV SEMEAN
C2 23 80.13 82.00 80.67 12.21 2.55
MIN MAX Q1 Q3
C2 49.00 100.00 73.00 89.00
MTB > stop
--------------------------
Disclaimer: All care was used to prepare the information in this
tutorial. Even so, the author does not and cannot guarantee the
accuracy of this information. The author disclaims any and all
injury that may come about from the use of this tutorial. As
always, students and all others should check with their advisor(s)
and/or other appropriate professionals for any and all assistance
on research design, analysis, selected levels of significance, and
interpretation of output file(s).
The author is entitled to exclusive distribution of this tutorial.
Readers have permission to print this tutorial for individual use,
provided that the copyright statement appears and that there is no
redistribution of this tutorial without permission.
Prepared 980316
Revised 980914
end-of-file 'cent_tnd.ssi'