All Terrain Thinking

A Compendium of things I think are Important

Statistics: It's not just whats' in your wallet

As important as measures of central tendency are, they do not provide us with a complete picture of the underlying scores. We know what the averages are, but what about the distribution of grades-the 'spread' of the data? Once again there are a number of measures of variation. We will look at range, variance, and standard deviation.

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 this example the grades range from 70 to 100. No one received a score higher than 100 or lower than 70.

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. Variance is a measure of variability designed to provide us with this information. The variance can be thought of as derived by a two-step process. In the first we compute a new variable which 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]. In the second step we square all of the deviation terms in column 2 which generates column 3. We now add these 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.

Standard Deviation: There is, however, one problem with the variance-it is influenced by the size of the variable being analyzed which would make comparisons of different score distributions impossible. To allow for this comparability, we can 'normalize' the variance by taking its square root. The result is the standard deviation.

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