Variability

As important as measures of central tendency are, they do not provide us with a complete picture of the underlying scores.  We may know what the averages are, but what about the distribution of grades - the 'spread' of the data?   Will you feel the same about two possible grading schemes with the same average - one where there will be only five A's and five F's and a second where there will no A's or F's?  Experience suggests that most students would care, although there is no agreement as to the preferred scheme.  Those with good academic records tend to favor the first distribution while those with weak records tend to favor the second.

The differences between the two grade distributions will be captured in the variability of the scores.  As with the previous discussion of central tendency, there are a number of measures of variation. We will look at range, variance, and standard deviation.  For an on-line discussion of measures of variability you should check out the UCLA On-line Statistics Course, the Electronic Textbook,  Introductory Statistics: Concepts, Models, and Applications Copyright 1996 by David W. Stockburger,  Mathematics 220DX Statistics at the New Hampshire College, DAU the Stat refresher, and Hyperstat Online by David Lane at Rice University. 

To demonstrate the measures of variability we will use our two grading examples.  The frequency diagram of the exam scores for ECN202 and the histogram of the ECN201 grades are repeated below.

Range: This is the easy one. Once you have your data sorted to identify your median, simply look at the highest and lowest values. The range is the difference between the lowest and highest values.   In the ECN202 grade example, grades range from 70 to 100.  No one received a score higher than 100 or lower than 70.

In the ECN201 example, the scores ranged from 7 to 29 for the entire class and from 12 to 27 for the sample. 

Variance: How likely is it that we would get a score close to the average?  Did most of the students receive similar scores or were they spread out roughly equally over the entire range.  The variance can be thought of as a measure of variability derived by a two-step process.  In the first we compute a new variable that equals the test score minus the mean score (approximately 84.5).  The result appears in the second column below. If we add the deviation in the first row (7.9) to the mean (84.5) we get the score (92.3) [Note: there may be a small difference due to rounding]. 

ECN201 Grades

In the second step we square all of the deviation terms in column 2 that generates column 3. We now add these squared terms and divide by the number of observations to get the variance.  All other things equal, the greater the variance the greater the spread of the scores.

What would the distribution look like if the variance were larger?  What you would see would be fewer observations close to the mean and more observations near the upper and lower limits. 

Returning to our ECN201 grades, we find that the variance for the entire class the variance is 19.12.  You would expect that the variance is smaller in the sample given that there are no scores in the sample below 12, and this is precisely what you find.  The variance in the sample is 16.86 points. 

Standard Deviation: There is one problem with the variance - it is influenced by the size of the variable being analyzed which will make comparisons of different score distributions impossible.   For example, if one teacher used a 4-point scale and another used the 100-point scale, then there would be a real problem comparing the variability in grades for the two classes.   To allow for this comparability, we can 'normalize' the variance by taking its square root. The result is the standard deviation which in the ECN202 example is 8.74 points.  In the ECN201 example, the standard deviations for the entire class and the sample are 4.37 and 4.11. 

In the next section where we examine probability and probability distributions, the standard deviation will take on special significance.