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'

Please send comments or suggestions to Dr. Thomas W. MacFarland

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