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October 17, 2011 (Regular Meeting) Page 764 number of observations. (Absolute value means the signs are ignored, that is assumed to be positive, when accumulating [x] or [d].) For our example: SALES RATIO - MEAN = [x] ([d] is used for the median) 95 - 80 = 15 92 - 80 = 12 90 - 80 = 10 86 - 80 = 6 86 - 80 = 6 80 - 80 = 0 75 - 80 = 5 72 - 80 = 8 64 - 80 = 16 60 - 80 = 20 Hence: MD = 98 / 10 = 9.8% This ratio expresses the average amount by which the data varies from the mean (or median) in a particular set of data. It is influenced by extremes as is the mean and even when computed about the median, it is likewise influenced. It also is not useful in making further statistical analysis of the data. STANDARD DEVIATION To overcome the handicaps of the mean deviation, the standard deviation is used. It is a numerical measure of the degree of dispersion, variability, or non - homogeneity of the data to which it is applied. In calculation, it is similar to the average deviation but differs in its method of averaging differences from the mean. It does this by squaring each difference and eventually summing all squared differences averaging them and taking the square root thereof giving an "average deviation" from the mean. In practice it is quite easy to compute using a handy "working formula" to make the task easier. First the formal formula: STANDARD DEVIATION = ❑ -U) or ❑ -u) Where u = "mu" N N -1 (arithmetic mean) Sum of the individual differences squared Number of observations The second formula using N -1 is most often used when dealing with sample data and is used in our sales ratio reports. In our example, using sales ratios it would be: Observation X (X -U) (X -U) 1 95% 15 225 2 92 12 144 3 90 10 100 4 86 6 36 5 86 6 36 6 80 0 0 7 75 5 25 8 72 8 64 9 64 16 256 10 60 20 400 X = 800% (X -U) = 1286 Arithmetic Mean (u) Sales Ratio = 800 / 10 = 80% Hence: SD = (X -u) OR SD = i (X -u) N N -1 1286 = 1286 10 10 -1