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October 17, 2011 (Regular Meeting) Page 763 impact on the ratio than smaller values. As a general rule, this measure is, therefore, somewhat less useful for sales ratio work than the un- weighted mean. A highly useful statistic is the MEDIAN. It is a measure which is least influenced by extreme values as it is based upon position rather than on level. That is, it is the value half -way from either end of a list of values when the list is arrayed in ascending (or descending) order. If the list contains an odd number of sales then the median is the middle value in the list. However, if there is an even number of sales in the list then it is the average of the two values on either side of the theoretical mid point in the list. Using our example it is: MEDIAN = (TOTAL NUMBER OF SALES + 1) / 2 + (10 + 1) / 2 + 5.5th item in the list That is in our list: 1 2 3 4 5 Median 5.5 Sales ---------- > 6 7 8 9 10 Sales Sales Ratio 95% 92 90 86 86 80 75 72 64 60 The median is, therefore, halfway between the ratio 86 and 80 or: MEDIAN = (86 + 80) / 2 = 166 / 2 = 830 This statistic is generally is the one normally used in judging uniformity and level of assessment. (Note: you may also calculate a median sales value as well as a median appraised value.) MODE The mode is a measure of central tendency that is easy to understand. It is the value in the set of observations which occurs most frequently. In our example, the mode of sales ratios would be 86% (occurs 2 times). MEASURES OF VARIABILITY A classic example of reliance on the use of the mean only as a method of description may be rather graphically illustrated by the following: If you were fired upon one time and were missed by 100 yards and were fired upon a second time and were hit, you could conclude that you were missed by an average of 50 yards. The point is the mean does not tell the whole story about the data. Other tools are needed to better describe the data. These tools are measures of how much you miss the mean (in general) or in more technical terms, measures of dispersion. RANGE The range is simply the lowest and highest value in your set of observations subtracted from one another; although it may be reported as the minimum and maximum values themselves. In our example, you could say the range (for the sales ratios) is: 350 or from 60% to 95% As a general statement it is not too useful in analysis due to its obvious dependence on extreme values. MEAN DEVIATION & MEDIAN DEVIATION This measure is the average of the difference between the mean (or median) and the individual observations. MD = [d] / N or [x] / N That is, the mean or median deviation is the sum of the absolute value of the differences between the mean (or median) and each observation divided by the