The Use of the Range and Mean Deviation in Interpreting the Standard Deviation

The Use of the Range and Mean Deviation in Interpreting the Standard Deviation With the emergence of sophisticated MIS systems, today's businessmen and women have more access to statistical data than ever before. Many of these individuals have difficulty interpreting and, consequently, using many of the statistics provided on computer printouts. The standard deviation(1) is one of these statistics. Although it is commonly used, it is difficult to interpret because of its mathematical complexity. Lay people, in particular, have a difficult time understanding the standard deviation.(2) For example, the manager of a credit union who received reports that contain the mean and standard deviation of loan amounts may have problems explaining these statistics to members of the credit union governing board. If in a particular report the mean and standard deviation of the loans were $5000 and $400, respectively, the average loan of $5000 could be easily understood by the board members, but the standard deviation of $400 may not - even though it may be very important in analyzing the overall loan situation. One can imagine many situations in business similar to this where an intuitive understanding of the standard deviation may be useful. One method to facilitate the interpretation of the standard deviation is to relate it to the range. For an infinite, normal population, three standard deviation units above and below the mean encompass 99.73 percent of the distribution, which results in the range equaling approximately six standard deviations. A similar technique for interpreting the standard deviation for a sample would be very useful, since the sample standard deviation is commonly found on business reports. According to McNemar [2], for samples from the normal distribution, the range is equal to five standard deviations when n (the sample size) is 50, six standard deviations when n is 200, and seven standard deviation units when n is 1000. Pearson and Stephens [4] provide 12 ratios for the normal distribution (mathematically derived) for sample sizes from three to 100. For a sample size of fifty, Pearson and Stephens' ratio is 4.4212 compared with McNemar's 5.000. Information on the relationship (or ratio) of the range to standard deviation for samples from non-normal distributions is scarce. Baker [1] compared ratios from the normal distribution to ratios from the platykurtic-bimodal and skewed-bimodal distributions for sample sizes 64 and 100. He found that the ratios for the platykurtic-bimodal distribution are smaller than those for the normal distribution: 4.1349 to 4.8272 when n is 64 and 4.4864 to 5.1214 when n is 100. For samples from the skewed-bimodal distribution, the ratios differ minimally from those for the normal distribution. Additional research to determine the extent to which these ratios vary when the population shape becomes non-normal may be critical in establishing the range as an aid in explaining the standard deviation. Another approach to interpreting or explaining the standard deviation (S) is to describe it in terms of the mean deviation (MD). The mean deviation is easily understood since it is the average of the absolute deviations about the mean or, more plainly stated, the average amount the observations vary from the mean. Because of its simplicity, if the sample mean deviation is approximately equal to the sample standard deviation, regardless of the sample size or population shape, the meaning of the standard deviation to lay people would be clearer. McNemar [2] states that the relationship of the mean deviation to the standard deviation for the normal distribution is MD = .798S. However, no research is available to substantiate whether this is true for all sample sizes, whether from normal or non-normal distributions. The purpose of this study is to determine the relationship of the range (W) and the mean deviation (MD) to the standard deviation (S) for various sample sizes from various shaped distributions. …