Measures of Variability (Dispersion)

by

Adam J. McKee, Ph.D.

January, 2004

Variability refers to the "spread-out-ness" of scores in the distribution.  The greater the difference between scores, the more spread out the distribution is.  The more tightly the scores group together, the less variability there is in the distribution.  One such measure of variability or spread is the range.

Range

                The range is the highest score in the distribution minus the lowest. 

The range of the test scores used in the mean example above (80, 90, 75, and 60) would be calculated as follows:

Standard Deviation

                The most commonly used measure of variation in a distribution of scores is the standard deviation.  The greater the spreadoutness of scores is, the greater the standard deviation is.  The range is less useful than the standard deviation because it is based on only two extreme values that might not reflect the general spreadoutness of scores in the distribution.  A better way would be to look at how much each score differs from the central tendency of the distribution.  This is easy enough to do in practice.  If we subtract each score from the mean, we get a deviation score.  Deviation scores tell us how far each score is from the center of the distribution.

                When we wanted to get a summary of all scores, we chose to look at a measure of central tendency, which we consider the "typical" value.  The same logic holds true for getting an idea of what the spread of scores is like.  We can look at the central tendency of the deviation scores.  This is a great idea, but the rules of algebra just wont let us do it.  Because the mean is the exact center of the distribution, half the deviations will be positive and the other half will be negative. The deviation scores will always add to zero.  According to the rules of algebra, we cannot divide by zero.  This means that the mean of deviation scores cannot be calculated!  The following example will help illustrate the point.

                Score      Mean                      Deviation (X - Mean)

                7              6                              +1                                           

                6              6                                0

                7              6                              +1

                8              6                              +2

                2              6                              -4

                ___________________________________

Sum (∑)= 30                                          ∑=0

                Note that the ∑ = 0 will always occur because of how the mean is defined.  Since we cannot divide by zero, we need to get rid of those annoying negatives so the result will not come to zero.  We could take the absolute value of the deviation scores.  This has been done, but there is another way that makes the results much more useful in later statistical tests.  If we simply square the deviation scores, that nasty negative signs go away.  A negative number multiplied by a negative number is always positive after all.  The following example shows the results of squaring the deviation scores and applying them to the mean formula:

 

                Score                      Mean                      X - Mean                               (X - Mean)2

                7                              6                              1                                             1

                6                              6                              0                                             0

                7                              6                              1                                             1

                8                              6                              2                                             4

                2                              6                              -4                                             16_________

Sum     30                                                               0                                              22

 

 

Score 

Mean of Squared Deviation Scores  

                This seems to have worked nicely except for one small problem.  We are not interested in the spread of squared deviation scores; we just want to know about plain old scores.  If we want to get rid of a square, all we have to do is take the square root of it and we get back to just deviation from the mean scores.  Therefore, we add the square root to the formula to get:

Mean of Deviation Scores

                One small adjustment must still be made for reasons of statistical theory.  Because of this is a sample statistic that we want to generalize to the population, we must subtract one from N to get the Sample Standard Deviation (SD) formula as follows: 

SD

                The term N - 1 is know as degrees of freedom (df).  Degrees of freedom refers to the number of scores that are free to vary.  In the above example there were 5 scores.  Once 4 of these values are set, the other can not vary and still produce the same result.  Thus the number of scores that are free to vary is 5 - 1 = 4.  In general, the degrees of freedom for sample statistics are based on the sample size (N) and the number of sample statistics used in the calculation.  Recall that we used N - 2 degrees of freedom in the Pearson r formula because sample statistics for both X and Y were used in the calculation. 

                This is a good place to note that another measure of variation is called simply variance.  Variance is the standard deviation squared (SD2 or S2).  From this example, we can conclude that the average spread of scores is a little over 2 points.  

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