Oneway Analysis of Variance (ANOVA)

© 1998 by Dr. Thomas W. MacFarland -- All Rights Reserved


************
one_anov.doc
************
Background:  A common statistical technique for determining 
             if differences exist between between three or 
             more "groups" is Oneway Analysis of Variance 
             (ANOVA), and the associated F test:

             The F test and subsequent ANOVA methodology 
             involves the determination of differences for: 

             1.  one group with multiple (typically, three 
                 or more) variations, as well as 

             2.  one variable, compared to multiple groups.

             When using Oneway ANOVA for three or more groups, 
             an immediate concern is how to interpret findings 
             if the hypothesis is not accepted (consult an 
             appropriate statistics text to review why there are 
             those who consider it more appropriate to declare 
             "The null hypothesis was not accepted" instead of 
             "The null hypothesis was rejected").  

             When only two groups are compared and if the Null
             Hypothesis is not accepted, then you know that the 
             difference between Group #1 and Group #2 is a true 
             difference (at the declared level of significance, 
             or p level).  What happens, however, if you reject 
             the null hypothesis for a Oneway ANOVA design
             involving three groups:

             -- Is the difference between Group A and Group B the 
                reason for failure to accept the null hypothesis?

             -- Is the difference between Group A and Group C the 
                reason for failure to accept the null hypothesis?

             -- Is the difference between Group B and Group C the 
                reason for failure to accept the null hypothesis?

             There are certainly many techniques for determining 
             multiple comparisons between the means of each group.  
             The following mean comparison tests are found in 
             SPSS for the purpose of comparing differences between 
             means in a Oneway ANOVA design:

             1.  LSD ............  Least-significant difference

             2.  DUNCAN .........  Duncan's multiple range test

             3.  SNK ............  Student-Newman-Keuls

             4.  TUKEYB .........  Tukey's alternate procedure

             5.  TUKEY ..........  Honestly significant difference

             6.  LSDMOD .........  Modified LSD

             7.  SCHEFFE ........  Scheffe's test

             Be sure to remember that Oneway ANOVA methodology,
             as opposed to Student's t-test, can serve as a
             useful tool in the development of processes for
             understanding "real-world" problems.  Most "real-
             world" problems are related to complex issues.
             Statistical tests that can account for this
             complexity are needed if meaningful decisions are
             to be effected.


Scenario:    This study examines if there are differences
             in final examination test scores between four 
             groups of students in a software engineering  
             course:  

             -- The first group of students was taught by 
                traditional lecture.

             -- The second group of students was taught by 
                Computer Based Training.

             -- The third group of students was taught by 
                the use of instructional videotapes.

             -- The last group of students was enrolled 
                through independent study.

             Students were all from a university senior-
             level software engineering course who were
             assigned, through random selection, to
             placement into one of four groups:  instruction
             by traditional lecture, instruction by CBT
             (Computer Based Training), instruction by
             the use of instructional videotapes, and
             independent study.  Because the teacher was 
             confident that final examination scores 
             represented interval data (i.e., the data are 
             parametric, with the difference between "89" 
             and "90" equal to the difference between "75" 
             and "76"), Oneway ANOVA (Analysis of Variance) 
             was correctly judged to be the appropriate test 
             for this analysis of summative differences in 
             final examination scores between three or more 
             groups.

             Final examination test scores are summarized in 
             Table 1.


             Table 1

             Final Examination Test Scores in a Senior-Level
             Software Engineering Course by Instructional
             Method:  Traditional Lecture, Computer Based
             Training, Instructional Videotape, and 
             Independent Study
             ==================================================== 
  
                              Instructional
                              Method
                              =============
                              1 = Lecture
                              2 = CBT
                              3 = Video
                              4 = Independent
             Student Number       Study (IDS)   Final Score
             ---------------------------------------------------- 
  
                   01             1              089
                   02             1              081
                   03             1              073
                   04             1              084
                   05             1              070
                   06             1              056
                   07             1              070
                   08             1              081
                   09             1              078
                   10             1              069
                   11             1              089
                   12             1              088
                   13             1              045
                   14             1              083
                   15             1              095
                   16             1              077
                   17             1              069
                   18             1              080
                   19             2              093
                   20             2              086
                   21             2              089
                   22             2              095
                   23             2              089
                   24             2              088
                   25             2              098
                   26             2              089
                   27             2              094
                   28             2              095
                   29             2              095
                   30             2              098
                   31             2              087
                   32             2              085
                   33             2              098
                   34             2              093
                   35             2              087
                   36             2              095
                   37             2              093
                   38             2              093
                   39             3              095
                   40             3              096
                   41             3              083
                   42             3              089
                   43             3              088
                   44             3              087
                   45             3              094
                   46             3              097
                   47             3              095
                   48             3              093
                   49             3              085
                   50             3              095
                   51             3              092
                   52             3              082
                   53             3              086
                   54             3              087
                   55             3              089
                   56             3              097
                   57             3              100
                   58             3              093
                   59             3              096
                   60             4              084
                   61             4              085
                   62             4              073
                   63             4              092
                   64             4              057
                   65             4              063
                   66             4              069
                   67             4              073
                   68             4              091
                   69             4              065
                   70             4              074
                   71             4              071
                   72             4              068
                   73             4              062
                   74             4              056
                   75             4              085
             ---------------------------------------------------- 
  
             
             Note.  Notice how the N (i.e., number of subjects or
                    group members) for each instructional group
                    does not have to be equal.

      
Ho:          Null Hypothesis:  There is no difference in the 
             final examination test scores of students enrolled
             in a university senior-level software engineering 
             course after students were assigned, through random 
             selection, to placement into one of four groups:  
             instruction by traditional lecture, instruction by 
             CBT (Computer Based Training), instruction by the 
             use of instructional videotapes, and independent 
             study (p <= .01).

             Note.  The p (i.e., probability or alpha level) 
                    value is declared as p <= .01 instead of
                    the more liberal p <= .05.


Files:       1.  one_anov.doc

             2.  one_anov.dat

             3.  one_anov.r01

             4.  one_anov.o01

             5.  one_anov.con

             6.  one_anov.lis


Command:     At the Unix prompt (%), key:

             %spss -m < one_anov.r01 > one_anov.o01


************
one_anov.dat
************
                   01             1              089
                   02             1              081
                   03             1              073
                   04             1              084
                   05             1              070
                   06             1              056
                   07             1              070
                   08             1              081
                   09             1              078
                   10             1              069
                   11             1              089
                   12             1              088
                   13             1              045
                   14             1              083
                   15             1              095
                   16             1              077
                   17             1              069
                   18             1              080
                   19             2              093
                   20             2              086
                   21             2              089
                   22             2              095
                   23             2              089
                   24             2              088
                   25             2              098
                   26             2              089
                   27             2              094
                   28             2              095
                   29             2              095
                   30             2              098
                   31             2              087
                   32             2              085
                   33             2              098
                   34             2              093
                   35             2              087
                   36             2              095
                   37             2              093
                   38             2              093
                   39             3              095
                   40             3              096
                   41             3              083
                   42             3              089
                   43             3              088
                   44             3              087
                   45             3              094
                   46             3              097
                   47             3              095
                   48             3              093
                   49             3              085
                   50             3              095
                   51             3              092
                   52             3              082
                   53             3              086
                   54             3              087
                   55             3              089
                   56             3              097
                   57             3              100
                   58             3              093
                   59             3              096
                   60             4              084
                   61             4              085
                   62             4              073
                   63             4              092
                   64             4              057
                   65             4              063
                   66             4              069
                   67             4              073
                   68             4              091
                   69             4              065
                   70             4              074
                   71             4              071
                   72             4              068
                   73             4              062
                   74             4              056
                   75             4              085


************
one_anov.r01
************
SET WIDTH      = 80
SET LENGTH     = NONE
SET CASE       = UPLOW
SET HEADER     = NO
TITLE          = Oneway Analysis of Variance (ONEWAY ANOVA)
COMMENT        = This file examines if there are differences
                 in final examination test scores between four 
                 groups of students in a software engineering  
                 course:  the first group of students was taught 
                 by traditional lecture, the second group of 
                 students was taught by Computer Based Training, 
                 the third group of students was taught by the 
                 use of instructional videotapes, and the last
                 group of students were enrolled through 
                 independent study.

                 Students were all from a university senior-
                 level software engineering course who were
                 assigned, through random selection, to
                 placement into one of four groups:  instruction
                 by traditional lecture, instruction by CBT
                 (Computer Based Training), instruction by
                 the use of instructional videotapes, and
                 independent study.  Because the teacher was 
                 confident that final examination scores 
                 represented interval data (i.e., the data are 
                 parametric, with the difference between "89" 
                 and "90" equal to the difference between "75" 
                 and "76"), Oneway ANOVA (Analysis of Variance) 
                 was correctly judged to be the appropriate test 
                 for this analysis of summative differences in 
                 final examination scores between three or more 
                 groups.
DATA LIST FILE = 'one_anov.dat' FIXED
     / Stu_Code  20-21
       Method       35
       Score     50-52 

Variable Labels
       Stu_Code   "Student Code"
     / Method     "Method:  Lecture, CBT, Video, IDS"
     / Score      "Final Examination Score"

Value Labels
       Method     1 'Lecture: Traditional Lecture'
                  2 'CBT: Computer-Based Training'
                  3 'Video: Instructional Videotape'
                  4 'IDS: Independent Study'

ONEWAY Score BY Method(1,4)
     / STATISTICS   = ALL
     / RANGES       = SCHEFFE (.01)
     / FORMAT       = LABELS


************
one_anov.o01
************
   1  SET WIDTH      = 80
   2  SET LENGTH     = NONE
   3  SET CASE       = UPLOW
   4  SET HEADER     = NO
   5  TITLE          = Oneway Analysis of Variance (ONEWAY ANOVA)
   6  COMMENT        = This file examines if there are differences
   7                   in final examination test scores between four
   8                   groups of students in a software engineering
   9                   course:  the first group of students was taught
  10                   by traditional lecture, the second group of
  11                   students was taught by Computer Based Training,
  12                   the third group of students was taught by the
  13                   use of instructional videotapes, and the last
  14                   group of students were enrolled through
  15                   independent study.
  16
  17                   Students were all from a university senior-
  18                   level software engineering course who were
  19                   assigned, through random selection, to
  20                   placement into one of four groups:  instruction
  21                   by traditional lecture, instruction by CBT
  22                   (Computer Based Training), instruction by
  23                   the use of instructional videotapes, and
  24                   independent study.  Because the teacher was
  25                   confident that final examination scores
  26                   represented interval data (i.e., the data are
  27                   parametric, with the difference between "89"
  28                   and "90" equal to the difference between "75"
  29                   and "76"), Oneway ANOVA (Analysis of Variance)
  30                   was correctly judged to be the appropriate test
  31                   for this analysis of summative differences in
  32                   final examination scores between three or more
  33                   groups.
  34  DATA LIST FILE = 'one_anov.dat' FIXED
  35       / Stu_Code  20-21
  36         Method       35
  37         Score     50-52
  38

This command will read 1 records from one_anov.dat

Variable   Rec   Start     End         Format

STU_CODE     1      20      21         F2.0
METHOD       1      35      35         F1.0
SCORE        1      50      52         F3.0

  39  Variable Labels
  40         Stu_Code   "Student Code"
  41       / Method     "Method:  Lecture, CBT, Video, IDS"
  42       / Score      "Final Examination Score"
  43
  44  Value Labels
  45         Method     1 'Lecture: Traditional Lecture'
  46                    2 'CBT: Computer-Based Training'
  47                    3 'Video: Instructional Videotape'
  48                    4 'IDS: Independent Study'
  49
  50  ONEWAY Score BY Method(1,4)
  51       / STATISTICS   = ALL
  52       / RANGES       = SCHEFFE (.01)
  53       / FORMAT       = LABELS

ONEWAY problem requires 504 bytes of memory.


                       - - - - -  O N E W A Y  - - - - -


      Variable  SCORE      Final Examination Score
   By Variable  METHOD     Method:  Lecture, CBT, Video, IDS

                                  Analysis of Variance


                           Sum of         Mean        F      F
        Source    D.F.    Squares       Squares     Ratio  Prob.

Between Groups      3     5372.3343     1790.7781  23.6178 .0000
Within Groups      71     5383.4524       75.8233
Total              74    10755.7867



                                Standard   Standard
Group       Count        Mean   Deviation  Error    95 Pct Conf Int for Mean

Lecture:    18     76.5000     12.2774     2.8938     70.3946 TO 82.6054
CBT: Com    20     92.0000      4.1675      .9319     90.0495 TO 93.9505
Video: I    21     91.3810      5.1037     1.1137     89.0578 TO 93.7041
IDS: Ind    16     73.0000     11.4601     2.8650     66.8934 TO 79.1066

Total       75     84.0533     12.0561     1.3921     81.2795 TO 86.8272

       Fixed Effects Model      8.7077     1.0055     82.0485 to 86.0582

       Random Effects Model                4.9191     68.3987 to 99.7080

Random Effects Model - estimate of between component variance 91.79



GROUP        MINIMUM     MAXIMUM

Lecture:     45.0000     95.0000
CBT: Com     85.0000     98.0000
Video: I     82.0000    100.0000
IDS: Ind     56.0000     92.0000

TOTAL        45.0000    100.0000


Levene Test for Homogeneity of Variances

    Statistic    df1    df2       2-tail Sig.
      6.5470      3     71          .001


                       - - - - -  O N E W A Y  - - - - -


      Variable  SCORE      Final Examination Score
   By Variable  METHOD     Method:  Lecture, CBT, Video, IDS

Multiple Range Tests:  Scheffe test with significance level .01

The difference between two means is significant if
  MEAN(J)-MEAN(I)  >= 6.1572 * RANGE * SQRT(1/N(I) + 1/N(J))
  with the following value(s) for RANGE: 4.94

   (*) Indicates significant differences which are shown in the
lower triangle

                          I L V C
                          D e i B
                          S c d T
                          : t e :
                            u o
                          I r : C
                          n e   o
                          d : I m
     Mean      METHOD

    73.0000    IDS: Ind
    76.5000    Lecture:
    91.3810    Video: I   * *
    92.0000    CBT: Com   * *


************
one_anov.con
************
Outcome:     Computed F  = 23.6178

             Criterion F = 4.13 (alpha = .01, df = 3,60)

             Note.  Although df = 3,71 the table values 
                    for the F distribution increase from
                    df = 3,40 (F = 4.31), to df = 3,60
                    (F = 4.13), to df = 3,120 (F = 3.95).

                    Occasionaly, it is necessary to
                    extrapolate the F statistic when
                    determining the Criterion F statistic. 
 
             Computed F (23.62) > Criterion F (4.13)

             Therefore, the null hypothesis is rejected.  
             That is to say, there are differences in final 
             examination test scores in a senior-level
             software engineering course, based on instructional
             method (p <= .01).

             The p value is another way to view differences in
             the final examination test scores:

             -- The calculated p value is .000. 

             -- The delcared p value is .01.

             The calculated p value is less than the declared 
             p value and there is, accordingly, a difference
             in test scores.

Conclusion:  Although you now know that differences exist, the F
             statistic does not tell you where the difference(s)
             exist between instructional methods.  

             Instead, review the following section of the SPSS
             output file:

   (*) Indicates significant differences which are shown in the
lower triangle

                          I L V C
                          D e i B
                          S c d T
                          : t e :
                            u o
                          I r : C
                          n e   o
                          d : I m
     Mean      METHOD

    73.0000    IDS: Ind
    76.5000    Lecture:
    91.3810    Video: I   * *
    92.0000    CBT: Com   * *

             Using traditional methodology, you could also 
             visually present on your own the mean comparisons 
             among groups by using underscores, as presented 
             below:

             IDS           Lecture           Video        CBT     

             73.00           76.50           91.38        92.00
             _____________________           __________________


             Although it is not possible at this point to 
             suggest "why" differences exist, there is sufficient 
             evidence from this one-time study to:

             -- There is no difference in final examination test
                scores between students who received instruction
                through IDS and Lecture.

             -- There is no difference in final examination test
                scores between students who received instruction
                through Video and CBT.   

             -- There is a difference in final examination test
                scores between students who received instruction
                through CBT and either IDS or Lecture, with CBT 
                students receiving a higher score.

             -- There is a difference in final examination test
                scores between students who received instruction
                through Video and either IDS or Lecture, with
                video students receiving a higher score.

             You will notice that these complex outcomes have a
             graphic representation in MINITAB that is fairly
             easy to understand, as opposed to the more complex
             graphical representation in SPSS.


************
one_anov.lis
************
% minitab

 MTB > outfile 'one_anov.lis'
 Collecting Minitab session in file: one_anov.lis
 MTB > # MINITAB Addendum to 'one_anov.dat'
 MTB > #
 MTB > read 'one_anov.dat' c1 c2 c3
 Entering data from file: one_anov.dat
      75 rows read.
 MTB > print c1 c2 c3
 
 
  ROW    C1   C2     C3
 
    1     1    1     89
    2     2    1     81
    3     3    1     73
    4     4    1     84
    5     5    1     70
    6     6    1     56
    7     7    1     70
    8     8    1     81
    9     9    1     78
   10    10    1     69
   11    11    1     89
   12    12    1     88
   13    13    1     45
   14    14    1     83
   15    15    1     95
   16    16    1     77
   17    17    1     69
   18    18    1     80
 Continue? y
   19    19    2     93
   20    20    2     86
   21    21    2     89
   22    22    2     95
   23    23    2     89
   24    24    2     88
   25    25    2     98
   26    26    2     89
   27    27    2     94
   28    28    2     95
   29    29    2     95
   30    30    2     98
   31    31    2     87
   32    32    2     85
   33    33    2     98
   34    34    2     93
   35    35    2     87
   36    36    2     95
   37    37    2     93
   38    38    2     93
   39    39    3     95
   40    40    3     96
   41    41    3     83
 Continue? y
   42    42    3     89
   43    43    3     88
   44    44    3     87
   45    45    3     94
   46    46    3     97
   47    47    3     95
   48    48    3     93
   49    49    3     85
   50    50    3     95
   51    51    3     92
   52    52    3     82
   53    53    3     86
   54    54    3     87
   55    55    3     89
   56    56    3     97
   57    57    3    100
   58    58    3     93
   59    59    3     96
   60    60    4     84
   61    61    4     85
   62    62    4     73
   63    63    4     92
   64    64    4     57
 Continue? y
   65    65    4     63
   66    66    4     69
   67    67    4     73
   68    68    4     91
   69    69    4     65
   70    70    4     74
   71    71    4     71
   72    72    4     68
   73    73    4     62
   74    74    4     56
   75    75    4     85
 
 MTB > # I'll unstack the data in c3 and c2 and then use
 MTB > # the two commands to effect the Oneway ANOVA
calculations.
 MTB > #
 MTB > # If at all possible, stack and or unstack data but
 MTB > # never re-key data.  
 MTB > #
 MTB > unstack (c2-c3) (c5-c6) (c7-c8) (c9-c10) (c11-c12);
 SUBC> subscripts c2.
 MTB > print c1-c12
 
 
  ROW    C1   C2     C3   C5    C6   C7    C8   C9    C10   C11  C12
 
    1     1    1     89    1    89    2    93    3     95     4  84
    2     2    1     81    1    81    2    86    3     96     4  85
    3     3    1     73    1    73    2    89    3     83     4  73
    4     4    1     84    1    84    2    95    3     89     4  92
    5     5    1     70    1    70    2    89    3     88     4  57
    6     6    1     56    1    56    2    88    3     87     4  63
    7     7    1     70    1    70    2    98    3     94     4  69
    8     8    1     81    1    81    2    89    3     97     4  73
    9     9    1     78    1    78    2    94    3     95     4  91
   10    10    1     69    1    69    2    95    3     93     4  65
   11    11    1     89    1    89    2    95    3     85     4  74
   12    12    1     88    1    88    2    98    3     95     4  71
   13    13    1     45    1    45    2    87    3     92     4  68
   14    14    1     83    1    83    2    85    3     82     4  62
   15    15    1     95    1    95    2    98    3     86     4  56
   16    16    1     77    1    77    2    93    3     87     4  85
   17    17    1     69    1    69    2    87    3     89          
   18    18    1     80    1    80    2    95    3     97         
  
 Continue? y
   19    19    2     93               2    93    3    100          
   20    20    2     86               2    93    3     93          
   21    21    2     89                          3     96          
   22    22    2     95                                            
   23    23    2     89                                            
   24    24    2     88
   25    25    2     98
   26    26    2     89                                           
   27    27    2     94                                           
   28    28    2     95                                           
   29    29    2     95                                           
   30    30    2     98                                           
   31    31    2     87                                           
   32    32    2     85                                           
   33    33    2     98                                           
   34    34    2     93                                           
   35    35    2     87                                           
   36    36    2     95                                           
   37    37    2     93                                           
   38    38    2     93                                           
   39    39    3     95                                           
   40    40    3     96                                           
   41    41    3     83                                           
 Continue? y
   42    42    3     89                                           
   43    43    3     88                                           
   44    44    3     87                                           
   45    45    3     94                                           
   46    46    3     97                                           
   47    47    3     95                                           
   48    48    3     93                                           
   49    49    3     85                                           
   50    50    3     95                                           
   51    51    3     92                                           
   52    52    3     82                                           
   53    53    3     86                                           
   54    54    3     87                                           
   55    55    3     89                                           
   56    56    3     97                                           
   57    57    3    100                                           
   58    58    3     93                                           
   59    59    3     96                                           
   60    60    4     84                                           
   61    61    4     85                                           
   62    62    4     73                                           
   63    63    4     92                                           
   64    64    4     57                                           
 Continue? y
   65    65    4     63                                           
   66    66    4     69                                           
   67    67    4     73                                           
   68    68    4     91                                           
   69    69    4     65                                           
   70    70    4     74                                           
   71    71    4     71                                           
   72    72    4     68                                           
   73    73    4     62                                           
   74    74    4     56                                           
   75    75    4     85                                           
  
 
 * NOTE  * One or more variables are undefined.
 
 MTB > describe c6 c8 c10 c12
 
                 N     MEAN   MEDIAN   TRMEAN    STDEV   SEMEAN
 C6             18    76.50    79.00    77.31    12.28     2.89
 C8             20   92.000   93.000   92.056    4.168    0.932
 C10            21    91.38    93.00    91.42     5.10     1.11
 C12            16    73.00    72.00    72.86    11.46     2.87
 
               MIN      MAX       Q1       Q3
 C6          45.00    95.00    69.75    85.00
 C8         85.000   98.000   88.250   95.000
 C10         82.00   100.00    87.00    95.50
 C12         56.00    92.00    63.50    84.75
 
 MTB > histogram c6 c8 c10 c12
 
 Histogram of C6   N = 18
 
 Midpoint   Count
       45       1  *
       50       0
       55       1  *
       60       0
       65       0
       70       4  ****
       75       2  **
       80       4  ****
       85       2  **
       90       3  ***
       95       1  *
 
 Continue? y
 
 Histogram of C8   N = 20
 
 Midpoint   Count
       85       1  *
       86       1  *
       87       2  **
       88       1  *
       89       3  ***
       90       0
       91       0
       92       0
       93       4  ****
       94       1  *
       95       4  ****
       96       0
       97       0
       98       3  ***
 
 Continue? y
 
 Histogram of C10   N = 21
 
 Midpoint   Count
       82       1  *
       84       1  *
       86       2  **
       88       3  ***
       90       2  **
       92       1  *
       94       3  ***
       96       5  *****
       98       2  **
      100       1  *
 
 Continue? y
 
 Histogram of C12   N = 16
 
 Midpoint   Count
       55       2  **
       60       1  *
       65       2  **
       70       3  ***
       75       3  ***
       80       0
       85       3  ***
       90       2  **
 
 MTB > # I will now use the MINITAB command for STACKED data.
 MTB > #
 MTB > oneway c3 c2
 
 ANALYSIS OF VARIANCE ON C3      
 SOURCE     DF        SS        MS        F        p
 C2          3    5372.3    1790.8    23.62    0.000
 ERROR      71    5383.5      75.8
 TOTAL      74   10755.8
                                    INDIVIDUAL 95 PCT CI'S FOR MEAN
                                    BASED ON POOLED STDEV
  LEVEL      N      MEAN     STDEV  -----+---------+---------+---------+-
      1     18    76.500    12.277        (----*----)
      2     20    92.000     4.168                           (----*----)
      3     21    91.381     5.104                          (----*----)
      4     16    73.000    11.460   (----*-----)                                            
                                    -----+---------+---------+---------+-
 POOLED STDEV =    8.708              72.0      80.0      88.0      96.0
 MTB > #
 MTB > # And I will now use the MINITAB command for UNSTACKED
data.
 MTB > aovoneway c6 c8 c10 c12
 
 ANALYSIS OF VARIANCE
 SOURCE     DF        SS        MS        F        p
 FACTOR      3    5372.3    1790.8    23.62    0.000
 ERROR      71    5383.5      75.8
 TOTAL      74   10755.8
                                    INDIVIDUAL 95 PCT CI'S FOR MEAN
                                    BASED ON POOLED STDEV
  LEVEL      N      MEAN     STDEV  -----+---------+---------+---------+-
 C6         18    76.500    12.277        (----*----)
 C8         20    92.000     4.168                           (----*----)
 C10        21    91.381     5.104                          (----*----)
 C12        16    73.000    11.460   (----*-----)                                              
                                    -----+---------+---------+---------+-
 POOLED STDEV =    8.708              72.0      80.0      88.0      96.0
 MTB > #
 MTB > # Although I'm keen on the use of SPSS, the graphic 
 MTB > # output with MINITAB on a Oneway ANOVA is very 
 MTB > # useful and easy to understand.
 MTB > #
 MTB > # Here, you can easilty see that C6 (lecture) and c12
 MTB > # (independent study) share the same pooled mean score
 MTB > # on the final examination.  Equally, you can also see
 MTB > # that c8 (CBT) and c10 (videotape instruction) also 
 MTB > # share the same pooled mean.
 MTB > #
 MTB > # Finally, you can also see that lecture and independent
 MTB > # study final examination scores are totally different,
 MTB > # with no overlap, from CBT and videotape instruction
 MTB > # final examination scores.
 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 'one_anov.ssi'

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

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