A Study of Error: A Summary and Evaluation of Methods Used in Six Years of Study of the Scholastic Aptitude Test of the College Entrance Examination Board

By Carl C. Brigham | Go to book overview
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APPENDIX II

STATISTICAL METHODS

Averages

Let us assume that we are dealing with first semester and second semester academic grades of the six-group, literal type: A, B, C, D, E, F. Combining marks for two semesters throws the thirty-six possible combinations into eleven intervals on a scale of merit running from F + F to A + A as follows:

A+F
B+F F+A A+E
C+F F+B B+E E+A A+D
D+F F+C C+E E+B B+D D+A A+C
E+F F+D D+E E+C C+D D+B B+C C+A A+B
F+F F+E E+E E+D D+D D+C C+C C+B B+B B+A A+A
(Lowest) (Highest)

These thirty-six possible combinations of six grades for two terms give a scale of eleven points, which may be designated by the letters above, or turned into a numerical system. If F were 1, and A were 6, the eleven combinations would be expressed on a scale from 2 to 12. To illustrate the methods of treating academic marks, we will assume that the grades given to 62 men in any full year (two term) course were distributed in the following manner:

Sum of grades . . 2 3 4 5 6 7 8 9 10 11 12
No. of cases . . . 1 3 5 7 9 12 9 7 5 3 1

The average grade may be obtained by multiplying each sum of grades by the number of cases (or frequencies) at each grade, and dividing the sum of these products by the number of cases, as follows:

12 × 1 = 12
11 × 3 = 33
10 × 5 = 50
9 × 7 = 63
8 × 9 = 72
7 × 12 = 84
6 × 9 = 54
5 × 7 = 35
4 × 5 = 20
3 × 3 = 9
2 × 1 = 2
Sum 434

This process of finding the average is simply a short-cut method of adding a list of 62 numbers: 2 + 3 + 3 + 3 + 4 + 4 + 4 + 4 + 4, etc.


Deviations

Having once found the average, we may disregard the scale of numbers from which the average was obtained by calling the average zero, and regarding the other numbers as deviations above and below the average. The distribution of grades given above may be written:

Deviations . . -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5
No. of cases 1 3 5 7 9 12 9 7 5 3 1

The Standard Deviation or Sigma (σ)

A novice thinks of a series of numbers in terms of the average, but it is readily seen that two series may be alike in respect to the mean but different in dis

-321-

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