Spearman Rank Order Correlation



Spearman Rank Order Correlation

The sixth grade creative writing contest was scored by two different teachers without the use of a scoring rubric. Of a possible 45 points, the work of the nine writers was awarded in the following manner.

                        Score A                                                          Score B

            Adams, Larry           28                                Adams, Larry           41
            Burns, Kate               35                                Burns, Kate               26
            Conners, Chuck       31                                Conners, Chuck       34
            Due, Donna               36                                Due, Donna               38
            Edwards, Ed             26                                Edwards, Ed             28
            Hall, Mary                 25                                Hall, Mary                 30
            Jones, Betty               37                                Jones, Betty               39
            Miller, Art                 40                                Miller, Art                 36
            Smith, Dan                27                                Smith, Dan                29
            
            N = 9                                                              N = 9
                   Range = 15                                                    Range = 15

How closely do the two scores seem to be in agreement about the merits of the group’s creative writing skills?

Developing a rank order correlation will help us to find the “degree” of agreement.






















Perfect                                                                                                           Perfect
Negative                                                                                                        Positive
Correlation                                                                                                   Correlation
-1.0         .80       .60         .20          ??             .20     .40       .60       .80       +1.0     

Rank order correlations are useful for teachers using single class and a pair of variables… be very careful … the correlation may not mean cause & effect!

We would expect certain variables to usually be positively correlated. 

We would expect a usually negative correlation with certain variables; as one increases, the other decreases.                                                             

  • academic achievement + hours watching TV
  • time practicing keyboarding & typing errors

We would expect some traits are not correlated. 































Spearman Rank Order Correlation

Student ranking highest in I.Q. also highest in math.

                        Rank IQ         Rank Math
            Alex         1     Ã§Ã¨      1                  a perfect positive
            Bill           2     Ã§Ã¨      2                  correlation p = +1.0
            Chuck     3       Ã§Ã¨     3                  highest I.Q. always 
            Dan         4       Ã§Ã¨     4                  gets highest math score.
            Ed           5       Ã§Ã¨     5
            
Student spending most practice time has fewest number of missed baskets. 

                        Time shooting          Number of  
                        Baskets                       Misses

            Tom                1                      5                      a perfect negative correlation
            Sue                  2                      4                      p = -1.0
            Jeff                  3                      3                      Tom had most practice and least
            Diane             4                      2                      misses. Bob least practice…most
            Bob                 5                      1                      misses.

            We can compare almost any two discrete measures.

                        Height            I.Q.
                        Rank               Rank

            Alice   1                      3                      no clear correlation between ranks but 
            Sue      2                      4                      Ann happens to have lowest I.Q. and is
            Mary  3                      2                      shortest
            Beth    4                      1
            Ann    5                      5          














Correlations

                        Scorer             Scorer             Difference                  Sum Difference
                           A                      B                     in Rank                        Squared
                   ------------------------------------------------------------------------------------------
Miller               1                   _______          _________                  ________________
Jones                2                   _______          _________                  ________________
Doe                    3                   _______          _________                  ________________
Burns                4                   _______          _________                  ________________
Conners           5                   _______          _________                  ________________
Adams             6                   _______          _________                  ________________
Smith               7                   _______          _________                  ________________
Edwards          8                   _______          _________                  ________________
Hall                   9                   _______          _________                  ________________

Spearman Rank Order Correlation

            P = 1 – 6ED2
                        N(N2 – 1)

            1 - ____________

            1 - ____________

            1 – 

            P = 
Use this “rule of thumb” correlation guide to measure 
the “degree” or extent of agreement

Minus                                                      No                                                             Plus
-1.0      .80       .60       .40       .20      Correlation    .20       .40       .60       .80       +1.0                      

















Answers for Spearman Rank Order Correlation Exercise
                                    Scorer             Scorer          Difference        Sum Difference 
                                   B                   in Rank              Squared
                                    _____              ____                _______          ____________
Miller, Art                    1                      4                      3                      9
Jones, Betty                 2                      2                      0                      0       
Doe, Donna                 3                      3                      0                      0
Burns, Kate                 4                      9                      5                     25
Conners, Chuck         5                      5                      0                      0
Adams, Larry             6                      1                      5                     25
Smith, Don                   7                      7                      0                      0
Edwards, Ed               8                      8                      0                      0
Hall, Mary                    9                      6                      3                      9
   N = 9                                                                                               _____   ED2                                                                                                                             68

Spearman Rank Order Correlation

P = 1 –    6 ED2
            N (N2 – 1)

       1 – 6 (68)            6 x 68 = 408
             9 (80)             9 x 80 = 720

       1 408                 408                         .567
             720                 - 720         720  408.000          
   
       1 - .567                 1,000
-      .567
______
   .433
P = .433          


















Correlations

                        Judge              Judge              Difference                 Sum Difference 
 A                       B                    in Rank                           Squared

Bill                  1                        4                         3                                        9
Sue                  2                        3                         1                                        1
Tom                3                        1                         2                                        4
Jack                 4                        2                         2                                        4
Paul                5                        5                         0                                        0
Mary              6                        6                         0                                        0
Ellen               7                        8                         1                                        1
Bob                 8                        9                         1                                        1
Tami               9                        7                         2                                        4
Joe                  10                      11                        1                                        1
Fay                 11                      10                        1                                        1
N = 11                                                                                                     _______ ED2         


            F = 1 – 6ED2
N (N2 – 1)
            
                   1 – 6 x 26                                     6 x 26 = 156
                        11 (121 – 1)                            11 x 11 = 121

                   1 – 156                                         156
                        1320                                        1320  = .117

                   1 - .117                                         1.000                                       
                                      -.117
                                ______
                                 .883

            P = .88















Correlations

The first diving meet in the school’s new pool is next weekend. The P. E. Department has four experienced judges but needs one more. Which of the three new judges would be the best choice for the event. 

                                    Expert            New               New      New 
                                    Judge             Judge A      Judge B     Judge C

Angela                       1                      4                      5                      3
Susana                        2                      3                      3                      5
Graciela                     3                      1                      6                      6
Elena                          4                      2                      2                      1          
Maria                         5                      5                      1                      2
Gaby                          6                      6                      9                      7          
Victoria                     7                      8                      7                      9
Evette                         8                      9                      8                      4
Ana                             9                      7                      4                      8
Rosa                           10                    11                    11                    10
Yolanda                     11                    10                    10                    11        
N = 11

Based on the degree of correlation between the “expert” judge and each of the possible new judges, I would select Judge ____ as my first choice, Judge ____ as my next choice and Judge ____ only as a last resort. 

Perfect                                                                                                            Perfect
Negative                                                                                                       Positive
Correlation                                           No                                                  Correlation
-1.0      .80       .60       .40       .20    Correlation   .20  .40     .60     .80      +1.0



Enter each degree of correlation check P = 1 – 6ED2/N (N2-1)













Using Basic Statistics

Why … to see overall patterns … to see the “big picture” … to display overall shape of the data … see where the center lies and major deviations.

How … by constructing bar graphs, pie charts, Histograms, frequency polygons, etc.

Computing basic measurements from a distribution of test scores

31                    24                    33                    21                    27                    25
22                    26                    19                    29                    37                    38
29                    28                    34                    30                    32                    35
33                    32                    31                    29                    35                    25
26                    29                    32                    28                    25                    27
                                                                                                                 N = 30

To get a “good look” to display the “overall shape” of these data we must:
-      arrange scores in rank order
-      develop a frequency distribution
-      develop a numeric frequency
-      develop a cumulative frequency




























                                                            Numeric
Rank               Frequency                 Frequency                 Cumulative
Order             Distribution              Distribution              Frequency

38                    1                                              1                                  30
37                    1                                              1                                  29
36                    0                                              0                                  28
35                    11                                            2                                  28
34                    1                                              1                                  26
33                    11                                            2                                  25
32                    111                                          3                                  23
31                    11                                            2                                  20
30                    1                                              1                                  18
29                    1111                                        4                                  17        
28                    11                                            2                                  13        
27                    11                                            2                                  11
26                    11                                            2                                  9
25                    111                                          3                                  7
24                    1                                              1                                  4
23                    0                                              0                                  3
22                    1                                              1                                  3          
21                    1                                              1                                  2
20                    0                                              0                                  1
19                    1                                              1                                  1          
                  N = 30                                    N = 30

o  Rank order lists scores from high to low.
o  Frequency distribution accounts for all scores.
o  Numeric frequency distribution substitutes “workable” numerals from hashmarks.
o  Cumulative frequency accumulates scores bottom to top. 


















Experimental Research Designs

One Shot
Case Study                            X                     O1
            Least adequate design
            Baseline ??                 Control group??
            We can’t “generalize” from results.

One Group
Pre-Test         Post-Test        O1       X         O2
            An improved design
            Can compute - + difference between scores 
            Cannot compare to similar group not involved in the experiment.

Quasi-experimental
Non-equivalent group        O1       X         O2
Pre-test           Post-test         O3       C         O4
            Often used by teachers 
            Real randomness not possible
            Interpret results accordingly
            Do not generalize too much

Post-test Only                      R         X         O1
Equivalent Groups              R         C         O2
Compares randomly selected X and C groups…can analyze difference in scores … was difference result of sampling error or result of X treatment. This is a true experimental design. 

Pre-test Post-test                  R         O1       X         O2
Equivalent Groups              R         O3       C         O4
Similar to above, but both groups tested pre and post treatment … it is possible to determine if treatment yielded significantly different scores. This is the strongest experimental design. 

Static Group                        X                     O1
Comparison                         C                     O2
            Compare groups after the event no evidence groups were ever alike cannot claim equivalence, so we cannot claim difference in scores can be attributed to the treatment. 




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