Pearson's Product-Moment Coefficient of Correlation
© 1998 by Dr. Thomas W. MacFarland -- All Rights Reserved
************
correlat.doc
************
Background: Consider the degree of association between the
Grade Point Average of an individual student and
the family income for this student:
-- A student who comes from a wealthy family
may not have to work after school, but can
instead use this free time to study and
do extra homework problems.
-- A student who comes from a poor family
probably does not have this free time, but
instead needs to work after school and on
weekends to help the family with daily
expenses.
-- A student who comes from a wealthy family
quite possibly has their own computer and
is able to prepare programming homework
problems at any convenient time.
-- A student who comes from a poor family
probably does not have a computer at home,
but instead has to schedule "at computer"
time during open laboratory hours, which
may quite possibly limit time-on-task for
attempts at creative programming projects.
As such, it is not at all surprising that there
is historically a strong positive correlation
between Grade Point Average and family income:
-- Overally, for groups of students, Grade
Point Average increases as family income
increases.
-- Correspondingly, for groups of students,
Grade Pont Average decreases as family
income decreases.
The key points here are that:
-- Family income does not "cause" the Grade
Point Average. Instead, there is merely
an association between the two phenonema.
-- This degree of association is for a
collective "group" of subjects. Any one
individual could have results that are
different than group observations.
With the importance of relationships between
data in mind, it is often helpful to determine if
there is a positive or negative relationship (i.e.,
association) between various phenomena.
Consider the following scenario:
1. There is a high "negative" correlation between
miles of paved roads and infant mortality:
Let X equal miles of paved roads
Let Y equal incidence of infant mortality
As X increases Y decreases.
2. That is to say, as the miles of paved roads in
an area increases the rate of infant mortality
decreases.
Why? Well, that is a totally different concern. Do
not assume that cause and effect is in place here.
Sure, a few mothers may get to the hospital faster
(and therefore have children who survive birth),
but that is hardly the reason for the association.
Other factors such as economic wealth, societal
issues, and the general infrastructure are also
prime concerns.
Pearson's Product Moment Coefficient of Correlation
is perhaps the most common test for determining if
there is an association between phenomena.
Please recall that the negative (- or decrease) and
positive (+ or increase) signs in correlation are
only used to suggest direction. The negative sign
does not mean "bad" and the positive sign does not
mean "good".
Y |*
| *
| *
| * As X increases Y decreases
| *
----------- Negative Correlation
X
Y | *
| *
| *
| * As X increases Y increases
|*
----------- Positive Correlation
X
Scenario: This study examines if there is any degree of
association (e.g., correlation) between:
-- High School Grade Point Average (GPA) and
total Scholastic Aptitude Test (SAT) scores
-- High School GPA and University GPA
-- Computer Science GPA and University GPA
Data are from the 105 students who graduated
from a local state university, earning a B.S.
in Computer Science. As entering freshmen,
these students need a Math SAT score of 550
or greater to major in Computer Science.
Because the data are all interval data (i.e.,
the data are parametric, with the difference
between a 3.87 GPA and a 3.88 GPA equal to the
difference between a 4.03 GPA and a 4.04 GPA)
Pearson's Coefficient of Correlation is the
correct test to determine the degree of
association between these variables.
Note: The SPSS operation file for this tutorial
introduces the use of:
-- COMPUTE: the SPSS command that uses existing
variables to create a new variable
-- FORMAT: the SPSS command that is used to
work with decimal numbers
-- TEMPORARY: the SPSS command that makes it
possible to perform two or more analyses in
one operation file
Data for this study are summarized in Table 1.
Table 1
Summary Statistics of Computer Science Graduates:
High School Grade Point Average (High_GPA), Math
Scholastic Aptitude Test Score (Math_SAT), Verbal
Scholastic Aptitude Test Score (Verb_SAT), Computer
Science Grade Point Average (Comp_GPA), and Overall
University Grade Point Average (Univ_GPA)
========================================================
Student
Number High_GPA Math_SAT Verb_SAT Comp_GPA Univ_GPA
--------------------------------------------------------
001 3.45 643 589 3.76 3.52
002 2.78 558 512 2.87 2.91
003 2.52 583 503 2.54 2.40
004 3.67 685 602 3.83 3.47
005 3.24 592 538 3.29 3.47
006 2.10 562 486 2.64 2.37
007 2.82 573 548 2.86 2.40
008 2.36 559 536 2.03 2.24
009 2.42 552 583 2.81 3.02
010 3.51 617 591 3.41 3.32
011 3.48 684 649 3.61 3.59
012 2.14 568 592 2.48 2.54
013 2.59 604 582 3.21 3.19
014 3.46 619 624 3.52 3.71
015 3.51 642 619 3.41 3.58
016 3.68 683 642 3.52 3.40
017 3.91 703 684 3.84 3.73
018 3.72 712 652 3.64 3.49
019 2.15 564 501 2.14 2.25
020 2.48 557 549 2.21 2.37
021 3.09 591 584 3.17 3.29
022 2.71 599 562 3.01 3.19
023 2.46 607 619 3.17 3.28
024 3.32 619 558 3.01 3.37
025 3.61 700 721 3.72 3.61
026 3.82 718 732 3.78 3.81
027 2.64 580 538 2.51 2.40
028 2.19 562 507 2.10 2.21
029 3.34 683 648 3.21 3.58
030 3.48 717 724 3.68 3.51
031 3.56 701 714 3.48 3.62
032 3.81 691 684 3.71 3.60
033 3.92 714 706 3.81 3.65
034 4.00 689 673 3.84 3.76
035 2.52 554 507 2.09 2.27
036 2.71 564 543 2.17 2.35
037 3.15 668 604 2.98 3.17
038 3.22 691 662 3.28 3.47
039 2.29 573 591 2.74 3.00
040 2.03 568 517 2.19 2.74
041 3.14 607 624 3.28 3.37
042 3.52 651 683 3.68 3.54
043 2.91 604 583 3.17 3.28
044 2.83 560 542 3.17 3.39
045 2.65 604 617 3.31 3.28
046 2.41 574 548 3.07 3.19
047 2.54 564 500 2.38 2.52
048 2.66 607 528 2.94 3.08
049 3.21 619 573 2.84 3.01
050 3.34 647 608 3.17 3.42
051 3.68 651 683 3.72 3.60
052 2.84 571 543 2.17 2.40
053 2.74 583 510 2.42 2.83
054 2.71 554 538 2.49 2.38
055 2.24 568 519 3.38 3.21
056 2.48 574 602 2.07 2.24
057 3.14 605 619 3.22 3.40
058 2.83 591 584 2.71 3.07
059 3.44 642 608 3.31 3.52
060 2.89 608 573 3.28 3.47
061 2.67 574 538 3.19 3.08
062 3.24 643 607 3.24 3.38
063 3.29 608 649 3.53 3.41
064 3.87 709 688 3.72 3.64
065 3.94 691 645 3.98 3.71
066 3.42 667 583 3.09 3.01
067 3.52 656 609 3.42 3.37
068 2.24 554 542 2.07 2.34
069 3.29 692 563 3.17 3.29
070 3.41 684 672 3.51 3.40
071 3.56 717 649 3.49 3.38
072 3.61 712 708 3.51 3.28
073 3.28 641 608 3.40 3.31
074 3.21 675 632 3.38 3.42
075 3.48 692 698 3.54 3.39
076 3.62 684 609 3.48 3.51
077 2.92 564 591 3.09 3.17
078 2.81 554 509 3.14 3.20
079 3.11 685 694 3.28 3.41
080 3.28 671 609 3.41 3.29
081 2.70 571 503 3.02 3.17
082 2.62 582 591 2.97 3.12
083 3.72 621 589 4.00 3.71
084 3.42 651 642 3.34 3.50
085 3.51 673 681 3.28 3.34
086 3.28 651 640 3.32 3.48
087 3.42 672 607 3.51 3.44
088 3.90 591 587 3.68 3.59
089 3.12 582 612 3.07 3.28
090 2.83 609 555 2.78 3.00
091 2.09 554 480 3.68 3.42
092 3.17 612 590 3.30 3.41
093 3.28 628 580 3.34 3.49
094 3.02 567 602 3.17 3.28
095 3.42 619 623 3.07 3.17
096 3.06 691 683 3.19 3.24
097 2.76 564 549 2.15 2.34
098 3.19 650 684 3.11 3.28
099 2.23 551 554 2.17 2.29
100 2.48 568 541 2.14 2.08
101 3.76 605 590 3.74 3.64
102 3.49 692 683 3.27 3.42
103 3.07 680 692 3.19 3.25
104 2.19 617 503 2.98 2.76
105 3.46 516 528 3.28 3.41
--------------------------------------------------------
Ho: Null Hypothesis: Among recent Computer Science
graduates from a local state university, there is no
association (p <= .05) between the following
variables:
-- High School GPA and Total SAT Score.
-- High School GPA and University GPA.
-- Computer Science GPA and University GPA.
Files: 1. correlat.doc
2. correlat.dat
3. correlat.r01
4. correlat.o01
5. correlat.con
6. correlat.lis
Command: At the Unix prompt (%), key:
%spss -m < correlat.r01 > correlat.o01
************
correlat.dat
************
001 3.45 643 589 3.76 3.52
002 2.78 558 512 2.87 2.91
003 2.52 583 503 2.54 2.40
004 3.67 685 602 3.83 3.47
005 3.24 592 538 3.29 3.47
006 2.10 562 486 2.64 2.37
007 2.82 573 548 2.86 2.40
008 2.36 559 536 2.03 2.24
009 2.42 552 583 2.81 3.02
010 3.51 617 591 3.41 3.32
011 3.48 684 649 3.61 3.59
012 2.14 568 592 2.48 2.54
013 2.59 604 582 3.21 3.19
014 3.46 619 624 3.52 3.71
015 3.51 642 619 3.41 3.58
016 3.68 683 642 3.52 3.40
017 3.91 703 684 3.84 3.73
018 3.72 712 652 3.64 3.49
019 2.15 564 501 2.14 2.25
020 2.48 557 549 2.21 2.37
021 3.09 591 584 3.17 3.29
022 2.71 599 562 3.01 3.19
023 2.46 607 619 3.17 3.28
024 3.32 619 558 3.01 3.37
025 3.61 700 721 3.72 3.61
026 3.82 718 732 3.78 3.81
027 2.64 580 538 2.51 2.40
028 2.19 562 507 2.10 2.21
029 3.34 683 648 3.21 3.58
030 3.48 717 724 3.68 3.51
031 3.56 701 714 3.48 3.62
032 3.81 691 684 3.71 3.60
033 3.92 714 706 3.81 3.65
034 4.00 689 673 3.84 3.76
035 2.52 554 507 2.09 2.27
036 2.71 564 543 2.17 2.35
037 3.15 668 604 2.98 3.17
038 3.22 691 662 3.28 3.47
039 2.29 573 591 2.74 3.00
040 2.03 568 517 2.19 2.74
041 3.14 607 624 3.28 3.37
042 3.52 651 683 3.68 3.54
043 2.91 604 583 3.17 3.28
044 2.83 560 542 3.17 3.39
045 2.65 604 617 3.31 3.28
046 2.41 574 548 3.07 3.19
047 2.54 564 500 2.38 2.52
048 2.66 607 528 2.94 3.08
049 3.21 619 573 2.84 3.01
050 3.34 647 608 3.17 3.42
051 3.68 651 683 3.72 3.60
052 2.84 571 543 2.17 2.40
053 2.74 583 510 2.42 2.83
054 2.71 554 538 2.49 2.38
055 2.24 568 519 3.38 3.21
056 2.48 574 602 2.07 2.24
057 3.14 605 619 3.22 3.40
058 2.83 591 584 2.71 3.07
059 3.44 642 608 3.31 3.52
060 2.89 608 573 3.28 3.47
061 2.67 574 538 3.19 3.08
062 3.24 643 607 3.24 3.38
063 3.29 608 649 3.53 3.41
064 3.87 709 688 3.72 3.64
065 3.94 691 645 3.98 3.71
066 3.42 667 583 3.09 3.01
067 3.52 656 609 3.42 3.37
068 2.24 554 542 2.07 2.34
069 3.29 692 563 3.17 3.29
070 3.41 684 672 3.51 3.40
071 3.56 717 649 3.49 3.38
072 3.61 712 708 3.51 3.28
073 3.28 641 608 3.40 3.31
074 3.21 675 632 3.38 3.42
075 3.48 692 698 3.54 3.39
076 3.62 684 609 3.48 3.51
077 2.92 564 591 3.09 3.17
078 2.81 554 509 3.14 3.20
079 3.11 685 694 3.28 3.41
080 3.28 671 609 3.41 3.29
081 2.70 571 503 3.02 3.17
082 2.62 582 591 2.97 3.12
083 3.72 621 589 4.00 3.71
084 3.42 651 642 3.34 3.50
085 3.51 673 681 3.28 3.34
086 3.28 651 640 3.32 3.48
087 3.42 672 607 3.51 3.44
088 3.90 591 587 3.68 3.59
089 3.12 582 612 3.07 3.28
090 2.83 609 555 2.78 3.00
091 2.09 554 480 3.68 3.42
092 3.17 612 590 3.30 3.41
093 3.28 628 580 3.34 3.49
094 3.02 567 602 3.17 3.28
095 3.42 619 623 3.07 3.17
096 3.06 691 683 3.19 3.24
097 2.76 564 549 2.15 2.34
098 3.19 650 684 3.11 3.28
099 2.23 551 554 2.17 2.29
100 2.48 568 541 2.14 2.08
101 3.76 605 590 3.74 3.64
102 3.49 692 683 3.27 3.42
103 3.07 680 692 3.19 3.25
104 2.19 617 503 2.98 2.76
105 3.46 516 528 3.28 3.41
************
correlat.r01
************
SET WIDTH = 80
SET LENGTH = NONE
SET CASE = UPLOW
SET HEADER = NO
TITLE = Pearson's Coefficient of Correlation
COMMENT = This file examines if there is any degree of
association (e.g., correlation) between:
-- High School Grade Point Average (GPA) and
total Scholastic Aptitude Test (SAT) scores
-- High School GPA and University GPA
-- Computer Science GPA and University GPA
Data are from the 105 students who graduated
from a local state university, earning a B.S.
in Computer Science. As entering freshmen,
these students need a Math SAT score of 550
or greater to major in Computer Science.
Because the data are all interval data (i.e.,
the data are parametric, with the difference
between a 3.87 GPA and a 3.88 GPA equal to the
difference between a 4.03 GPA and a 4.04 GPA,
Pearson's Coefficient of Correlation is the
correct test to determine the degree of
association between these variables.
Note: This operation file introduces the use
of:
-- COMPUTE: the SPSS command that uses existing
variables to create a new variable
-- FORMAT: the SPSS command that is used to
work with decimal numbers
-- TEMPORARY: the SPSS command that makes it
possible to perform two or more analyses in
one operation file
DATA LIST FILE = 'correlat.dat' FIXED
/ Stu_Code 12-14
High_GPA 22-25
Math_SAT 32-34
Verb_SAT 42-44
Comp_GPA 52-55
Univ_GPA 62-65
COMPUTE Totl_SAT = Math_SAT + Verb_SAT
COMMENT = The variable "Totl_SAT" is created by adding
together its two component scores, Math_SAT
and Verb_SAT.
FORMAT High_GPA (F4.2)
FORMAT Comp_GPA (F4.2)
FORMAT Univ_GPA (F4.2)
COMMENT = By using the "FORMAT" command in this way,
the three GPA scores are restricted to
four columns, with the last two columns to
the right of the decimal point.
Variable Labels
Stu_Code "Student Code"
/ High_GPA "High School GPA"
/ Math_SAT "Mathematics SAT Score"
/ Verb_SAT "Verbal SAT Score"
/ Totl_SAT "Total SAT Score"
/ Comp_GPA "GPA in Computer Science Courses"
/ Univ_GPA "GPA in All University Courses"
COMMENT = I will now use the "TEMPORARY" command so
that it is possible to conduct multiple
analyses in one SPSS run file.
Otherwise, it would be necessary to have
only one analysis in each separate operation
file. However, that would result in a large
set of files which would make it difficult
to keep up with file management.
TEMPORARY
FREQUENCY VARIABLES = High_GPA
Math_SAT Verb_SAT Totl_SAT
Comp_GPA Univ_GPA
/ STATISTICS = ALL
COMMENT = Before I conduct the correlation analyses,
I always like to obtain descriptive
statistics of the leading variables. This
look at central tendency (Mean, Median,
Mode, and Standard Deviation) provides a
good sense of the data.
TEMPORARY
CORRELATIONS VARIABLES = High_GPA WITH Totl_SAT
COMMENT = This analysis is a Pearson's correlation
of High School GPA to total SAT score.
TEMPORARY
CORRELATIONS VARIABLES = High_GPA WITH Univ_GPA
COMMENT = This analysis is a Pearson's correlation
of High School GPA to University GPA.
TEMPORARY
CORRELATIONS VARIABLES = Comp_GPA WITH Univ_GPA
COMMENT = This analysis is a Pearson's correlation
of GPA in Computer Science courses to
University GPA.
************
correlat.o01
************
1 SET WIDTH = 80
2 SET LENGTH = NONE
3 SET CASE = UPLOW
4 SET HEADER = NO
5 TITLE = Pearson's Coefficient of Correlation
6 COMMENT = This file examines if there is any degree of
7 association (e.g., correlation) between:
8
9 -- High School Grade Point Average (GPA) and
10 total Scholastic Aptitude Test (SAT) scores
11
12 -- High School GPA and University GPA
13
14 -- Computer Science GPA and University GPA
15
16 Data are from the 105 students who graduated
17 from a local state university, earning a B.S.
18 in Computer Science. As entering freshmen,
19 these students need a Math SAT score of 550
20 or greater to major in Computer Science.
21
22 Because the data are all interval data (i.e.,
23 the data are parametric, with the difference
24 between a 3.87 GPA and a 3.88 GPA equal to the
25 difference between a 4.03 GPA and a 4.04 GPA,
26 Pearson's Coefficient of Correlation is the
27 correct test to determine the degree of
28 association between these variables.
29
30 Note: This operation file introduces the use
31 of:
32
33 -- COMPUTE: the SPSS command that uses existing
34 variables to create a new variable
35
36 -- FORMAT: the SPSS command that is used to
37 work with decimal numbers
38
39 -- TEMPORARY: the SPSS command that makes it
40 possible to perform two or more analyses in
41 one operation file
42 DATA LIST FILE = 'correlat.dat' FIXED
43 / Stu_Code 12-14
44 High_GPA 22-25
45 Math_SAT 32-34
46 Verb_SAT 42-44
47 Comp_GPA 52-55
48 Univ_GPA 62-65
49
This command will read 1 records from correlat.dat
Variable Rec Start End Format
STU_CODE 1 12 14 F3.0
HIGH_GPA 1 22 25 F4.0
MATH_SAT 1 32 34 F3.0
VERB_SAT 1 42 44 F3.0
COMP_GPA 1 52 55 F4.0
UNIV_GPA 1 62 65 F4.0
50 COMPUTE Totl_SAT = Math_SAT + Verb_SAT
51
52 COMMENT = The variable "Totl_SAT" is created by adding
53 together its two component scores, Math_SAT
54 and Verb_SAT.
55
56 FORMAT High_GPA (F4.2)
57 FORMAT Comp_GPA (F4.2)
58 FORMAT Univ_GPA (F4.2)
59
60 COMMENT = By using the "FORMAT" command in this way,
61 the three GPA scores are restricted to
62 four columns, with the last two columns to
63 the right of the decimal point.
64
65 Variable Labels
66 Stu_Code "Student Code"
67 / High_GPA "High School GPA"
68 / Math_SAT "Mathematics SAT Score"
69 / Verb_SAT "Verbal SAT Score"
70 / Totl_SAT "Total SAT Score"
71 / Comp_GPA "GPA in Computer Science Courses"
72 / Univ_GPA "GPA in All University Courses"
73
74 COMMENT = I will now use the "TEMPORARY" command so
75 that it is possible to conduct multiple
76 analyses in one SPSS run file.
77
78 Otherwise, it would be necessary to have
79 only one analysis in each separate operation
80 file. However, that would result in a large
81 set of files which would make it difficult
82 to keep up with file management.
83
84 TEMPORARY
85 FREQUENCY VARIABLES = High_GPA
86 Math_SAT Verb_SAT Totl_SAT
87 Comp_GPA Univ_GPA
88 / STATISTICS = ALL
89
90 COMMENT = Before I conduct the correlation analyses,
91 I always like to obtain descriptive
92 statistics of the leading variables. This
93 look at central tendency (Mean, Median,
94 Mode, and Standard Deviation) provides a
95 good sense of the data.
96
Memory allows a total of 18,724 values accumulated across all variables.
There may be up to 2,340 value labels for each variable.
HIGH_GPA High School GPA
Valid Cum
Value Label Value Frequency Percent Percent Percent
2.03 1 1.0 1.0 1.0
2.09 1 1.0 1.0 1.9
2.10 1 1.0 1.0 2.9
2.14 1 1.0 1.0 3.8
2.15 1 1.0 1.0 4.8
2.19 2 1.9 1.9 6.7
2.23 1 1.0 1.0 7.6
2.24 2 1.9 1.9 9.5
2.29 1 1.0 1.0 10.5
2.36 1 1.0 1.0 11.4
2.41 1 1.0 1.0 12.4
2.42 1 1.0 1.0 13.3
2.46 1 1.0 1.0 14.3
2.48 3 2.9 2.9 17.1
2.52 2 1.9 1.9 19.0
2.54 1 1.0 1.0 20.0
2.59 1 1.0 1.0 21.0
2.62 1 1.0 1.0 21.9
2.64 1 1.0 1.0 22.9
2.65 1 1.0 1.0 23.8
2.66 1 1.0 1.0 24.8
2.67 1 1.0 1.0 25.7
2.70 1 1.0 1.0 26.7
2.71 3 2.9 2.9 29.5
2.74 1 1.0 1.0 30.5
2.76 1 1.0 1.0 31.4
2.78 1 1.0 1.0 32.4
2.81 1 1.0 1.0 33.3
2.82 1 1.0 1.0 34.3
2.83 3 2.9 2.9 37.1
2.84 1 1.0 1.0 38.1
2.89 1 1.0 1.0 39.0
2.91 1 1.0 1.0 40.0
2.92 1 1.0 1.0 41.0
3.02 1 1.0 1.0 41.9
3.06 1 1.0 1.0 42.9
3.07 1 1.0 1.0 43.8
3.09 1 1.0 1.0 44.8
3.11 1 1.0 1.0 45.7
3.12 1 1.0 1.0 46.7
3.14 2 1.9 1.9 48.6
3.15 1 1.0 1.0 49.5
3.17 1 1.0 1.0 50.5
3.19 1 1.0 1.0 51.4
3.21 2 1.9 1.9 53.3
3.22 1 1.0 1.0 54.3
3.24 2 1.9 1.9 56.2
3.28 4 3.8 3.8 60.0
3.29 2 1.9 1.9 61.9
3.32 1 1.0 1.0 62.9
3.34 2 1.9 1.9 64.8
3.41 1 1.0 1.0 65.7
3.42 4 3.8 3.8 69.5
3.44 1 1.0 1.0 70.5
3.45 1 1.0 1.0 71.4
3.46 2 1.9 1.9 73.3
3.48 3 2.9 2.9 76.2
3.49 1 1.0 1.0 77.1
3.51 3 2.9 2.9 80.0
3.52 2 1.9 1.9 81.9
3.56 2 1.9 1.9 83.8
3.61 2 1.9 1.9 85.7
3.62 1 1.0 1.0 86.7
3.67 1 1.0 1.0 87.6
3.68 2 1.9 1.9 89.5
3.72 2 1.9 1.9 91.4
3.76 1 1.0 1.0 92.4
3.81 1 1.0 1.0 93.3
3.82 1 1.0 1.0 94.3
3.87 1 1.0 1.0 95.2
3.90 1 1.0 1.0 96.2
3.91 1 1.0 1.0 97.1
3.92 1 1.0 1.0 98.1
3.94 1 1.0 1.0 99.0
4.00 1 1.0 1.0 100.0
------- ------- -------
Total 105 100.0 100.0
Mean 3.076 Std err .050 Median 3.170
Mode 3.280 Std dev .517 Variance .267
Kurtosis -.947 S E Kurt .467 Skewness -.259
S E Skew .236 Range 1.970 Minimum 2.030
Maximum 4.000 Sum 323.020
* Multiple modes exist. The smallest value is shown.
Valid cases 105 Missing cases 0
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
MATH_SAT Mathematics SAT Score
Valid Cum
Value Label Value Frequency Percent Percent Percent
516 1 1.0 1.0 1.0
551 1 1.0 1.0 1.9
552 1 1.0 1.0 2.9
554 5 4.8 4.8 7.6
557 1 1.0 1.0 8.6
558 1 1.0 1.0 9.5
559 1 1.0 1.0 10.5
560 1 1.0 1.0 11.4
562 2 1.9 1.9 13.3
564 5 4.8 4.8 18.1
567 1 1.0 1.0 19.0
568 4 3.8 3.8 22.9
571 2 1.9 1.9 24.8
573 2 1.9 1.9 26.7
574 3 2.9 2.9 29.5
580 1 1.0 1.0 30.5
582 2 1.9 1.9 32.4
583 2 1.9 1.9 34.3
591 3 2.9 2.9 37.1
592 1 1.0 1.0 38.1
599 1 1.0 1.0 39.0
604 3 2.9 2.9 41.9
605 2 1.9 1.9 43.8
607 3 2.9 2.9 46.7
608 2 1.9 1.9 48.6
609 1 1.0 1.0 49.5
612 1 1.0 1.0 50.5
617 2 1.9 1.9 52.4
619 4 3.8 3.8 56.2
621 1 1.0 1.0 57.1
628 1 1.0 1.0 58.1
641 1 1.0 1.0 59.0
642 2 1.9 1.9 61.0
643 2 1.9 1.9 62.9
647 1 1.0 1.0 63.8
650 1 1.0 1.0 64.8
651 4 3.8 3.8 68.6
656 1 1.0 1.0 69.5
667 1 1.0 1.0 70.5
668 1 1.0 1.0 71.4
671 1 1.0 1.0 72.4
672 1 1.0 1.0 73.3
673 1 1.0 1.0 74.3
675 1 1.0 1.0 75.2
680 1 1.0 1.0 76.2
683 2 1.9 1.9 78.1
684 3 2.9 2.9 81.0
685 2 1.9 1.9 82.9
689 1 1.0 1.0 83.8
691 4 3.8 3.8 87.6
692 3 2.9 2.9 90.5
700 1 1.0 1.0 91.4
701 1 1.0 1.0 92.4
703 1 1.0 1.0 93.3
709 1 1.0 1.0 94.3
712 2 1.9 1.9 96.2
714 1 1.0 1.0 97.1
717 2 1.9 1.9 99.0
718 1 1.0 1.0 100.0
------- ------- -------
Total 105 100.0 100.0
Mean 623.076 Std err 5.246 Median 612.000
Mode 554.000 Std dev 53.760 Variance 2890.186
Kurtosis -1.283 S E Kurt .467 Skewness .217
S E Skew .236 Range 202.000 Minimum 516.000
Maximum 718.000 Sum 65423.000
* Multiple modes exist. The smallest value is shown.
Valid cases 105 Missing cases 0
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
VERB_SAT Verbal SAT Score
Valid Cum
Value Label Value Frequency Percent Percent Percent
480 1 1.0 1.0 1.0
486 1 1.0 1.0 1.9
500 1 1.0 1.0 2.9
501 1 1.0 1.0 3.8
503 3 2.9 2.9 6.7
507 2 1.9 1.9 8.6
509 1 1.0 1.0 9.5
510 1 1.0 1.0 10.5
512 1 1.0 1.0 11.4
517 1 1.0 1.0 12.4
519 1 1.0 1.0 13.3
528 2 1.9 1.9 15.2
536 1 1.0 1.0 16.2
538 4 3.8 3.8 20.0
541 1 1.0 1.0 21.0
542 2 1.9 1.9 22.9
543 2 1.9 1.9 24.8
548 2 1.9 1.9 26.7
549 2 1.9 1.9 28.6
554 1 1.0 1.0 29.5
555 1 1.0 1.0 30.5
558 1 1.0 1.0 31.4
562 1 1.0 1.0 32.4
563 1 1.0 1.0 33.3
573 2 1.9 1.9 35.2
580 1 1.0 1.0 36.2
582 1 1.0 1.0 37.1
583 3 2.9 2.9 40.0
584 2 1.9 1.9 41.9
587 1 1.0 1.0 42.9
589 2 1.9 1.9 44.8
590 2 1.9 1.9 46.7
591 4 3.8 3.8 50.5
592 1 1.0 1.0 51.4
602 3 2.9 2.9 54.3
604 1 1.0 1.0 55.2
607 2 1.9 1.9 57.1
608 3 2.9 2.9 60.0
609 3 2.9 2.9 62.9
612 1 1.0 1.0 63.8
617 1 1.0 1.0 64.8
619 3 2.9 2.9 67.6
623 1 1.0 1.0 68.6
624 2 1.9 1.9 70.5
632 1 1.0 1.0 71.4
640 1 1.0 1.0 72.4
642 2 1.9 1.9 74.3
645 1 1.0 1.0 75.2
648 1 1.0 1.0 76.2
649 3 2.9 2.9 79.0
652 1 1.0 1.0 80.0
662 1 1.0 1.0 81.0
672 1 1.0 1.0 81.9
673 1 1.0 1.0 82.9
681 1 1.0 1.0 83.8
683 4 3.8 3.8 87.6
684 3 2.9 2.9 90.5
688 1 1.0 1.0 91.4
692 1 1.0 1.0 92.4
694 1 1.0 1.0 93.3
698 1 1.0 1.0 94.3
706 1 1.0 1.0 95.2
708 1 1.0 1.0 96.2
714 1 1.0 1.0 97.1
721 1 1.0 1.0 98.1
724 1 1.0 1.0 99.0
732 1 1.0 1.0 100.0
------- ------- -------
Total 105 100.0 100.0
Mean 598.600 Std err 6.145 Median 591.000
Mode 538.000 Std dev 62.964 Variance 3964.415
Kurtosis -.813 S E Kurt .467 Skewness .187
S E Skew .236 Range 252.000 Minimum 480.000
Maximum 732.000 Sum 62853.000
* Multiple modes exist. The smallest value is shown.
Valid cases 105 Missing cases 0
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
TOTL_SAT Total SAT Score
Valid Cum
Value Label Value Frequency Percent Percent Percent
1034.00 1 1.0 1.0 1.0
1044.00 1 1.0 1.0 1.9
1048.00 1 1.0 1.0 2.9
1061.00 1 1.0 1.0 3.8
1063.00 1 1.0 1.0 4.8
1064.00 1 1.0 1.0 5.7
1065.00 1 1.0 1.0 6.7
1069.00 1 1.0 1.0 7.6
1070.00 1 1.0 1.0 8.6
1074.00 1 1.0 1.0 9.5
1085.00 1 1.0 1.0 10.5
1086.00 1 1.0 1.0 11.4
1087.00 1 1.0 1.0 12.4
1092.00 1 1.0 1.0 13.3
1093.00 1 1.0 1.0 14.3
1095.00 1 1.0 1.0 15.2
1096.00 1 1.0 1.0 16.2
1102.00 1 1.0 1.0 17.1
1105.00 1 1.0 1.0 18.1
1106.00 1 1.0 1.0 19.0
1107.00 1 1.0 1.0 20.0
1109.00 1 1.0 1.0 21.0
1112.00 1 1.0 1.0 21.9
1113.00 1 1.0 1.0 22.9
1114.00 1 1.0 1.0 23.8
1118.00 1 1.0 1.0 24.8
1120.00 1 1.0 1.0 25.7
1121.00 1 1.0 1.0 26.7
1122.00 1 1.0 1.0 27.6
1130.00 1 1.0 1.0 28.6
1135.00 2 1.9 1.9 30.5
1155.00 1 1.0 1.0 31.4
1160.00 1 1.0 1.0 32.4
1161.00 1 1.0 1.0 33.3
1164.00 2 1.9 1.9 35.2
1169.00 1 1.0 1.0 36.2
1173.00 1 1.0 1.0 37.1
1175.00 2 1.9 1.9 39.0
1176.00 1 1.0 1.0 40.0
1177.00 1 1.0 1.0 41.0
1178.00 1 1.0 1.0 41.9
1181.00 1 1.0 1.0 42.9
1186.00 1 1.0 1.0 43.8
1187.00 1 1.0 1.0 44.8
1192.00 1 1.0 1.0 45.7
1194.00 1 1.0 1.0 46.7
1195.00 1 1.0 1.0 47.6
1202.00 1 1.0 1.0 48.6
1208.00 2 1.9 1.9 50.5
1210.00 1 1.0 1.0 51.4
1221.00 1 1.0 1.0 52.4
1224.00 1 1.0 1.0 53.3
1226.00 1 1.0 1.0 54.3
1231.00 1 1.0 1.0 55.2
1232.00 1 1.0 1.0 56.2
1242.00 1 1.0 1.0 57.1
1243.00 1 1.0 1.0 58.1
1249.00 1 1.0 1.0 59.0
1250.00 3 2.9 2.9 61.9
1255.00 2 1.9 1.9 63.8
1257.00 1 1.0 1.0 64.8
1261.00 1 1.0 1.0 65.7
1265.00 1 1.0 1.0 66.7
1272.00 1 1.0 1.0 67.6
1279.00 1 1.0 1.0 68.6
1280.00 1 1.0 1.0 69.5
1287.00 1 1.0 1.0 70.5
1291.00 1 1.0 1.0 71.4
1293.00 2 1.9 1.9 73.3
1307.00 1 1.0 1.0 74.3
1325.00 1 1.0 1.0 75.2
1331.00 1 1.0 1.0 76.2
1333.00 1 1.0 1.0 77.1
1334.00 3 2.9 2.9 80.0
1336.00 1 1.0 1.0 81.0
1353.00 1 1.0 1.0 81.9
1354.00 1 1.0 1.0 82.9
1356.00 1 1.0 1.0 83.8
1362.00 1 1.0 1.0 84.8
1364.00 1 1.0 1.0 85.7
1366.00 1 1.0 1.0 86.7
1372.00 1 1.0 1.0 87.6
1374.00 1 1.0 1.0 88.6
1375.00 2 1.9 1.9 90.5
1379.00 1 1.0 1.0 91.4
1387.00 1 1.0 1.0 92.4
1390.00 1 1.0 1.0 93.3
1397.00 1 1.0 1.0 94.3
1415.00 1 1.0 1.0 95.2
1420.00 2 1.9 1.9 97.1
1421.00 1 1.0 1.0 98.1
1441.00 1 1.0 1.0 99.0
1450.00 1 1.0 1.0 100.0
------- ------- -------
Total 105 100.0 100.0
Mean 1221.676 Std err 10.915 Median 1208.000
Mode 1250.000 Std dev 111.844 Variance 12509.010
Kurtosis -1.063 S E Kurt .467 Skewness .242
S E Skew .236 Range 416.000 Minimum 1034.000
Maximum 1450.000 Sum 128276.000
* Multiple modes exist. The smallest value is shown.
Valid cases 105 Missing cases 0
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
COMP_GPA GPA in Computer Science Courses
Valid Cum
Value Label Value Frequency Percent Percent Percent
2.03 1 1.0 1.0 1.0
2.07 2 1.9 1.9 2.9
2.09 1 1.0 1.0 3.8
2.10 1 1.0 1.0 4.8
2.14 2 1.9 1.9 6.7
2.15 1 1.0 1.0 7.6
2.17 3 2.9 2.9 10.5
2.19 1 1.0 1.0 11.4
2.21 1 1.0 1.0 12.4
2.38 1 1.0 1.0 13.3
2.42 1 1.0 1.0 14.3
2.48 1 1.0 1.0 15.2
2.49 1 1.0 1.0 16.2
2.51 1 1.0 1.0 17.1
2.54 1 1.0 1.0 18.1
2.64 1 1.0 1.0 19.0
2.71 1 1.0 1.0 20.0
2.74 1 1.0 1.0 21.0
2.78 1 1.0 1.0 21.9
2.81 1 1.0 1.0 22.9
2.84 1 1.0 1.0 23.8
2.86 1 1.0 1.0 24.8
2.87 1 1.0 1.0 25.7
2.94 1 1.0 1.0 26.7
2.97 1 1.0 1.0 27.6
2.98 2 1.9 1.9 29.5
3.01 2 1.9 1.9 31.4
3.02 1 1.0 1.0 32.4
3.07 3 2.9 2.9 35.2
3.09 2 1.9 1.9 37.1
3.11 1 1.0 1.0 38.1
3.14 1 1.0 1.0 39.0
3.17 7 6.7 6.7 45.7
3.19 3 2.9 2.9 48.6
3.21 2 1.9 1.9 50.5
3.22 1 1.0 1.0 51.4
3.24 1 1.0 1.0 52.4
3.27 1 1.0 1.0 53.3
3.28 6 5.7 5.7 59.0
3.29 1 1.0 1.0 60.0
3.30 1 1.0 1.0 61.0
3.31 2 1.9 1.9 62.9
3.32 1 1.0 1.0 63.8
3.34 2 1.9 1.9 65.7
3.38 2 1.9 1.9 67.6
3.40 1 1.0 1.0 68.6
3.41 3 2.9 2.9 71.4
3.42 1 1.0 1.0 72.4
3.48 2 1.9 1.9 74.3
3.49 1 1.0 1.0 75.2
3.51 3 2.9 2.9 78.1
3.52 2 1.9 1.9 80.0
3.53 1 1.0 1.0 81.0
3.54 1 1.0 1.0 81.9
3.61 1 1.0 1.0 82.9
3.64 1 1.0 1.0 83.8
3.68 4 3.8 3.8 87.6
3.71 1 1.0 1.0 88.6
3.72 3 2.9 2.9 91.4
3.74 1 1.0 1.0 92.4
3.76 1 1.0 1.0 93.3
3.78 1 1.0 1.0 94.3
3.81 1 1.0 1.0 95.2
3.83 1 1.0 1.0 96.2
3.84 2 1.9 1.9 98.1
3.98 1 1.0 1.0 99.0
4.00 1 1.0 1.0 100.0
------- ------- -------
Total 105 100.0 100.0
Mean 3.128 Std err .050 Median 3.210
Mode 3.170 Std dev .509 Variance .259
Kurtosis -.328 S E Kurt .467 Skewness -.684
S E Skew .236 Range 1.970 Minimum 2.030
Maximum 4.000 Sum 328.440
Valid cases 105 Missing cases 0
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
UNIV_GPA GPA in All University Courses
Valid Cum
Value Label Value Frequency Percent Percent Percent
2.08 1 1.0 1.0 1.0
2.21 1 1.0 1.0 1.9
2.24 2 1.9 1.9 3.8
2.25 1 1.0 1.0 4.8
2.27 1 1.0 1.0 5.7
2.29 1 1.0 1.0 6.7
2.34 2 1.9 1.9 8.6
2.35 1 1.0 1.0 9.5
2.37 2 1.9 1.9 11.4
2.38 1 1.0 1.0 12.4
2.40 4 3.8 3.8 16.2
2.52 1 1.0 1.0 17.1
2.54 1 1.0 1.0 18.1
2.74 1 1.0 1.0 19.0
2.76 1 1.0 1.0 20.0
2.83 1 1.0 1.0 21.0
2.91 1 1.0 1.0 21.9
3.00 2 1.9 1.9 23.8
3.01 2 1.9 1.9 25.7
3.02 1 1.0 1.0 26.7
3.07 1 1.0 1.0 27.6
3.08 2 1.9 1.9 29.5
3.12 1 1.0 1.0 30.5
3.17 4 3.8 3.8 34.3
3.19 3 2.9 2.9 37.1
3.20 1 1.0 1.0 38.1
3.21 1 1.0 1.0 39.0
3.24 1 1.0 1.0 40.0
3.25 1 1.0 1.0 41.0
3.28 7 6.7 6.7 47.6
3.29 3 2.9 2.9 50.5
3.31 1 1.0 1.0 51.4
3.32 1 1.0 1.0 52.4
3.34 1 1.0 1.0 53.3
3.37 3 2.9 2.9 56.2
3.38 2 1.9 1.9 58.1
3.39 2 1.9 1.9 60.0
3.40 3 2.9 2.9 62.9
3.41 4 3.8 3.8 66.7
3.42 4 3.8 3.8 70.5
3.44 1 1.0 1.0 71.4
3.47 4 3.8 3.8 75.2
3.48 1 1.0 1.0 76.2
3.49 2 1.9 1.9 78.1
3.50 1 1.0 1.0 79.0
3.51 2 1.9 1.9 81.0
3.52 2 1.9 1.9 82.9
3.54 1 1.0 1.0 83.8
3.58 2 1.9 1.9 85.7
3.59 2 1.9 1.9 87.6
3.60 2 1.9 1.9 89.5
3.61 1 1.0 1.0 90.5
3.62 1 1.0 1.0 91.4
3.64 2 1.9 1.9 93.3
3.65 1 1.0 1.0 94.3
3.71 3 2.9 2.9 97.1
3.73 1 1.0 1.0 98.1
3.76 1 1.0 1.0 99.0
3.81 1 1.0 1.0 100.0
------- ------- -------
Total 105 100.0 100.0
Mean 3.173 Std err .044 Median 3.290
Mode 3.280 Std dev .447 Variance .200
Kurtosis -.150 S E Kurt .467 Skewness -1.006
S E Skew .236 Range 1.730 Minimum 2.080
Maximum 3.810 Sum 333.150
Valid cases 105 Missing cases 0
97 TEMPORARY
98 CORRELATIONS VARIABLES = High_GPA WITH Totl_SAT
99 COMMENT = This analysis is a Pearson's correlation
100 of High School GPA to total SAT score.
101
PEARSON CORR problem requires 80 bytes of workspace.
- - Correlation Coefficients - -
TOTL_SAT
HIGH_GPA .7780
( 105)
P= .000
(Coefficient / (Cases) / 2-tailed Significance)
" . " is printed if a coefficient cannot be computed
102 TEMPORARY
103 CORRELATIONS VARIABLES = High_GPA WITH Univ_GPA
104 COMMENT = This analysis is a Pearson's correlation
105 of High School GPA to University GPA.
106
PEARSON CORR problem requires 80 bytes of workspace.
- - Correlation Coefficients - -
UNIV_GPA
HIGH_GPA .7796
( 105)
P= .000
(Coefficient / (Cases) / 2-tailed Significance)
" . " is printed if a coefficient cannot be computed
107 TEMPORARY
108 CORRELATIONS VARIABLES = Comp_GPA WITH Univ_GPA
109 COMMENT = This analysis is a Pearson's correlation
110 of GPA in Computer Science courses to
111 University GPA.
PEARSON CORR problem requires 80 bytes of workspace.
- - Correlation Coefficients - -
UNIV_GPA
COMP_GPA .9390
( 105)
P= .000
(Coefficient / (Cases) / 2-tailed Significance)
" . " is printed if a coefficient cannot be computed
************
correlat.con
************
Outcome: Before the Pearson r values are examined, to
determine the association between variables, it
may be helpful to first provide a summary of
descriptive statistics for these variables:
High_GPA -- High School GPA
N Mode Median Mean SD Range
==================================================
105 3.28 3.17 3.08 0.52 1.97: 2.03 to 4.00
Comp_GPA -- Computer Science GPA
N Mode Median Mean SD Range
==================================================
105 3.17 3.21 3.13 0.51 1.97: 2.03 to 4.00
Univ_GPA -- Overall University GPA
N Mode Median Mean SD Range
==================================================
105 3.28 3.29 3.17 0.45 1.73: 2.08 to 3.81
Math_SAT -- Math SAT Score
N Mode Median Mean SD Range
==================================================
105 554 612 623.1 53.8 202: 516 to 718
Verb_SAT -- Verbal SAT Score
N Mode Median Mean SD Range
==================================================
105 538 591 598.6 63.0 252: 480 to 732
Totl_SAT -- Total SAT Score
N Mode Median Mean SD Range
==================================================
105 1250 1208 1221.7 111.9 416: 1034 to 1450
And with this background information, it may be
somewhat easier to gain a better sense of the
association between variables declared in the
Null Hypothesis:
Null Hypothesis 1
=================
Among recent Computer Science graduates from a local
state university, there is no association (p <= .05)
between High School GPA and Total SAT Score:
TOTL_SAT
HIGH_GPA .7780
( 105)
P= .000
Computed r = .778
Criterion r = .195 (alpha = .05, n = 100)
Note. When reviewing Criterion r, be sure to notice
that an extrapolated estimate was provided
for N = 100 instead of N = 105.
Computed r (.778) > Criterion r (.195)
Therefore, the Null Hypothesis is rejected. Quite
the opposite, there is an association between
High School GPA and Total SAT Score.
The p value is another way to view the measure of
association:
-- The calculated p value is .000.
-- The delcared p value is .05.
The calculated p value is less than the declared
p value and there is, accordingly, an association
between High School GPA and Total SAT Score.
Null Hypothesis 2
=================
Among recent Computer Science graduates from a local
state university, there is no association (p <= .05)
between High School GPA and University GPA:
UNIV_GPA
HIGH_GPA .7796
( 105)
P= .000
Computed r = .780
Criterion r = .195 (alpha = .05, n = 100)
Note. When reviewing Criterion r, be sure to notice
that an extrapolated estimate was provided
for N = 100 instead of N = 105.
Computed r (.780) > Criterion r (.195)
Therefore, the Null Hypothesis is rejected. Quite
the opposite, there is an association between
High School GPA and University GPA.
The p value is another way to view the measure of
association:
-- The calculated p value is .000.
-- The delcared p value is .05.
The calculated p value is less than the declared
p value and there is, accordingly, an association
between High School GPA and University GPA.
Null Hypothesis 3
=================
Among recent Computer Science graduates from a local
state university, there is no association (p <= .05)
between Computer Science GPA and University GPA:
UNIV_GPA
COMP_GPA .9390
( 105)
P= .000
Computed r = .939
Criterion r = .195 (alpha = .05, n = 100)
Note. When reviewing Criterion r, be sure to notice
that an extrapolated estimate was provided
for N = 100 instead of N = 105.
Computed r (.939) > Criterion r (.195)
Therefore, the Null Hypothesis is rejected. Quite
the opposite, there is an association between
Computer Science GPA and University GPA.
The p value is another way to view the measure of
association:
-- The calculated p value is .000.
-- The delcared p value is .05.
The calculated p value is less than the declared
p value and there is, accordingly, an association
between Computer Science GPA and University GPA.
Note: This study certainly supports the common theme in
Educational Psychology that "Past behavior is the
best predictor of future behavior."
Generally, there is a strong association between
behavior in high school with SAT scores and later
behavior in college:
-- Students with good grades in high school tend
to do well on the SAT exam.
-- Students with good grades in high school tend
to also get good grades in college.
Further, as a general "rule of thumb," correlation
is often viewed along the following continuum:
+ .00 to + .30 = no positive correlation
between X and Y
- .00 to - .30 = no negative correlation
between X and Y
+ .40 to + .70 = mild positive correlation
between X and Y
- .40 to - .70 = mild negative correlation
between X and Y
+ .80 to + .99 = strong positive correlation
between X and Y
- .80 to - .99 = strong negative correlation
between X and Y
At the most, a correlation coefficient can only
reach -1.0 or + 1.0.
It may also be helpful to consider the sample
size when considering the efficacy or the
"practical" significance of a correlation design:
1. Correlation studies are very sensitive to n
and the declared probability level. That is to
say, a correlation of .600 is significant at p =
.05 and n = 14. But, r = .600 (with n = 14) is
not significant at p = .01 (Criterion r = .612
at p = .01, n = 14).
Refer to a table on critical values of Pearson's
Product Moment Coefficient of Correlation for
criterion r values.
2. Do not automatically think, however, that
increasing n will give you greater validity
in developing conclusions. A trivial study,
regardless of the magnitude of n, is still a
trivial study.
3. Finally, do let the notion of "cause and
effect" creep into your decisions.
************
correlat.lis
************
% minitab
MTB > outfile 'correlat.lis'
Collecting Minitab session in file: correlat.lis
MTB > # MINITAB addendum to 'correlat.dat'
MTB > #
MTB > read 'correlat.dat' c1 c2 c3 c4 c5 c6
Entering data from file: correlat.dat
105 rows read.
MTB > #
MTB > # I now need to create a new column, adding Math_SAT
MTB > # scores and Verb_SAT scores into Totl_SAT scores.
MTB > #
MTB > # Math_Sat scores are in column 3.
MTB > # Verb_SAT scores are in column 4.
MTB > #
MTB > let c8 = c3 + c4
MTB > #
MTB > print c1 c2 c3 c4 c8 c5 c6
ROW C1 C2 C3 C4 C8 C5 C6
1 1 3.45 643 589 1232 3.76 3.52
2 2 2.78 558 512 1070 2.87 2.91
3 3 2.52 583 503 1086 2.54 2.40
4 4 3.67 685 602 1287 3.83 3.47
5 5 3.24 592 538 1130 3.29 3.47
6 6 2.10 562 486 1048 2.64 2.37
7 7 2.82 573 548 1121 2.86 2.40
8 8 2.36 559 536 1095 2.03 2.24
9 9 2.42 552 583 1135 2.81 3.02
10 10 3.51 617 591 1208 3.41 3.32
11 11 3.48 684 649 1333 3.61 3.59
12 12 2.14 568 592 1160 2.48 2.54
13 13 2.59 604 582 1186 3.21 3.19
14 14 3.46 619 624 1243 3.52 3.71
15 15 3.51 642 619 1261 3.41 3.58
16 16 3.68 683 642 1325 3.52 3.40
17 17 3.91 703 684 1387 3.84 3.73
18 18 3.72 712 652 1364 3.64 3.49
Continue? y
19 19 2.15 564 501 1065 2.14 2.25
20 20 2.48 557 549 1106 2.21 2.37
21 21 3.09 591 584 1175 3.17 3.29
22 22 2.71 599 562 1161 3.01 3.19
23 23 2.46 607 619 1226 3.17 3.28
24 24 3.32 619 558 1177 3.01 3.37
25 25 3.61 700 721 1421 3.72 3.61
26 26 3.82 718 732 1450 3.78 3.81
27 27 2.64 580 538 1118 2.51 2.40
28 28 2.19 562 507 1069 2.10 2.21
29 29 3.34 683 648 1331 3.21 3.58
30 30 3.48 717 724 1441 3.68 3.51
31 31 3.56 701 714 1415 3.48 3.62
32 32 3.81 691 684 1375 3.71 3.60
33 33 3.92 714 706 1420 3.81 3.65
34 34 4.00 689 673 1362 3.84 3.76
35 35 2.52 554 507 1061 2.09 2.27
36 36 2.71 564 543 1107 2.17 2.35
37 37 3.15 668 604 1272 2.98 3.17
38 38 3.22 691 662 1353 3.28 3.47
39 39 2.29 573 591 1164 2.74 3.00
40 40 2.03 568 517 1085 2.19 2.74
41 41 3.14 607 624 1231 3.28 3.37
Continue? y
42 42 3.52 651 683 1334 3.68 3.54
43 43 2.91 604 583 1187 3.17 3.28
44 44 2.83 560 542 1102 3.17 3.39
45 45 2.65 604 617 1221 3.31 3.28
46 46 2.41 574 548 1122 3.07 3.19
47 47 2.54 564 500 1064 2.38 2.52
48 48 2.66 607 528 1135 2.94 3.08
49 49 3.21 619 573 1192 2.84 3.01
50 50 3.34 647 608 1255 3.17 3.42
51 51 3.68 651 683 1334 3.72 3.60
52 52 2.84 571 543 1114 2.17 2.40
53 53 2.74 583 510 1093 2.42 2.83
54 54 2.71 554 538 1092 2.49 2.38
55 55 2.24 568 519 1087 3.38 3.21
56 56 2.48 574 602 1176 2.07 2.24
57 57 3.14 605 619 1224 3.22 3.40
58 58 2.83 591 584 1175 2.71 3.07
59 59 3.44 642 608 1250 3.31 3.52
60 60 2.89 608 573 1181 3.28 3.47
61 61 2.67 574 538 1112 3.19 3.08
62 62 3.24 643 607 1250 3.24 3.38
63 63 3.29 608 649 1257 3.53 3.41
64 64 3.87 709 688 1397 3.72 3.64
Continue? y
65 65 3.94 691 645 1336 3.98 3.71
66 66 3.42 667 583 1250 3.09 3.01
67 67 3.52 656 609 1265 3.42 3.37
68 68 2.24 554 542 1096 2.07 2.34
69 69 3.29 692 563 1255 3.17 3.29
70 70 3.41 684 672 1356 3.51 3.40
71 71 3.56 717 649 1366 3.49 3.38
72 72 3.61 712 708 1420 3.51 3.28
73 73 3.28 641 608 1249 3.40 3.31
74 74 3.21 675 632 1307 3.38 3.42
75 75 3.48 692 698 1390 3.54 3.39
76 76 3.62 684 609 1293 3.48 3.51
77 77 2.92 564 591 1155 3.09 3.17
78 78 2.81 554 509 1063 3.14 3.20
79 79 3.11 685 694 1379 3.28 3.41
80 80 3.28 671 609 1280 3.41 3.29
81 81 2.70 571 503 1074 3.02 3.17
82 82 2.62 582 591 1173 2.97 3.12
83 83 3.72 621 589 1210 4.00 3.71
84 84 3.42 651 642 1293 3.34 3.50
85 85 3.51 673 681 1354 3.28 3.34
86 86 3.28 651 640 1291 3.32 3.48
87 87 3.42 672 607 1279 3.51 3.44
Continue? y
88 88 3.90 591 587 1178 3.68 3.59
89 89 3.12 582 612 1194 3.07 3.28
90 90 2.83 609 555 1164 2.78 3.00
91 91 2.09 554 480 1034 3.68 3.42
92 92 3.17 612 590 1202 3.30 3.41
93 93 3.28 628 580 1208 3.34 3.49
94 94 3.02 567 602 1169 3.17 3.28
95 95 3.42 619 623 1242 3.07 3.17
96 96 3.06 691 683 1374 3.19 3.24
97 97 2.76 564 549 1113 2.15 2.34
98 98 3.19 650 684 1334 3.11 3.28
99 99 2.23 551 554 1105 2.17 2.29
100 100 2.48 568 541 1109 2.14 2.08
101 101 3.76 605 590 1195 3.74 3.64
102 102 3.49 692 683 1375 3.27 3.42
103 103 3.07 680 692 1372 3.19 3.25
104 104 2.19 617 503 1120 2.98 2.76
105 105 3.46 516 528 1044 3.28 3.41
MTB > name c1 'Stu_Code'
MTB > name c2 'High_GPA'
MTB > name c3 'Math_SAT'
MTB > name c4 'Verb_SAT'
MTB > name c8 'Totl_SAT'
MTB > name c5 'Comp_GPA'
MTB > name c6 'Univ_GPA'
MTB > describe c2 c3 c4 c8 c5 c6
N MEAN MEDIAN TRMEAN STDEV SEMEAN
High_GPA 105 3.0764 3.1700 3.0825 0.5166 0.0504
Math_SAT 105 623.08 612.00 622.29 53.76 5.25
Verb_SAT 105 598.60 591.00 597.73 62.96 6.14
Totl_SAT 105 1221.7 1208.0 1219.7 111.8 10.9
Comp_GPA 105 3.1280 3.2100 3.1431 0.5090 0.0497
Univ_GPA 105 3.1729 3.2900 3.1938 0.4472 0.0436
MIN MAX Q1 Q3
High_GPA 2.0300 4.0000 2.6650 3.4800
Math_SAT 516.00 718.00 572.00 677.50
Verb_SAT 480.00 732.00 545.50 646.50
Totl_SAT 1034.0 1450.0 1119.0 1328.0
Comp_GPA 2.0300 4.0000 2.8650 3.5000
Univ_GPA 2.0800 3.8100 3.0100 3.4750
MTB > #
MTB > # Before I conduct the Pearson's test, I will first plot
MTB > # the two variables to gain a sense of the association
MTB > # between X and Y.
MTB > #
MTB > plot 'Totl_SAT' 'High_GPA'
-
Totl_SAT- * *
- *2 **
- 2* 2 * * *
1350+ * * * * **
- 2 * ** 2
- 2 2 **
- * *3 * 2 2
- * * 2 2*
1200+ * * *** * * **
- * * * * * 2** ** * *
- * * *
- *2 ** 2 **** 3
- * *** * 2**
1050+ * ** * *
- *
-
----+---------+---------+---------+---------+---------+--High_GPA
2.00 2.40 2.80 3.20 3.60 4.00
MTB > # And from this graphic, you can see that as High_GPA
MTB > # increases, there is a corresponding increase in
MTB > # Totl_SAT.
MTB > #
MTB > correlation 'High_GPA' 'Totl_SAT'
Correlation of High_GPA and Totl_SAT = 0.778
MTB > plot 'Univ_GPA' 'High_GPA'
- *
- * * ***
3.60+ * *2** * ** *2
- * * *22 * 42 ***
Univ_GPA- * *3 *2* * 23* *
- * * * ** * * * *3* * 2 *
- * 2* * * * *
3.00+ * * * * *
- **
- * *
-
- * *
2.40+ * * ** * 2* 2
- * * * **
- *
- *
-
----+---------+---------+---------+---------+---------+--High_GPA
2.00 2.40 2.80 3.20 3.60 4.00
MTB > # And in this graphic, you can see that there is a positive
MTB > # association between High_GPA and Univ_GPA. Generally, as
MTB > # High_GPA increases, there is a corresponding increase in
MTB > # Univ_GPA.
MTB > #
MTB > correlation 'High_GPA' 'Univ_GPA'
Correlation of High_GPA and Univ_GPA = 0.780
MTB > plot 'Univ_GPA' 'Comp_GPA'
- *
- * 2 2
3.60+ * * * * 24**
- * 422* ** *2 * *
Univ_GPA- * * 24* 2**3*
- * 2*63 * 2 *
- * 2**2 *
3.00+ *2* *
- * *
- * *
-
- * *
2.40+ * 3* *** * *
- *2 2
- *
- *
-
----+---------+---------+---------+---------+---------+--Comp_GPA
2.00 2.40 2.80 3.20 3.60 4.00
MTB > # And finally, this graphic clearly shows that there is a
MTB > # strong association between Comp_GPA and Univ_GPA. For
MTB > # each increase in Comp_GPA there is a corresponding
MTB > # increase in Univ_GPA.
MTB > #
MTB > correlation 'Comp_GPA' 'Univ_GPA'
Correlation of Comp_GPA and Univ_GPA = 0.939
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 'correlat.ssi'