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Measures of average allow us to refine our analysis of group data in order to describe the group performance as a whole. The word "average" is commonly used to refer to a value obtained by adding together a set of measurements and then dividing by the number of measurements in the set. You have probably done this yourself and referred to it as the "mean." But this is only one type of average as you shall see. The Arithmetic Mean This is the most common measure of central tendency and, as has been said above, this is the one you have probably calculated yourself. The formula is simple. Add up all the observations and divide the answer (the sum) by the number of observations. Although this is the most common measure of central tendency, there is a problem connected with it. It is subject to the influence of extreme observations. If we add 2, 4, 5, 6, 7, and 6, we get the sum of 30. Divide 30 by the number of observations, which is 6, and we get 5. This number is the arithmetic mean for the series of numbers. When we look at the group of numbers as a whole, the average of 5 looks pretty good. But let's build in an extreme number in our set of numbers. If we add 2, 4, 5, 6, 7, and 42, we get the sum of 66. Divide 66 by the number of observations, which is 6, and we get 11. The "extreme" number, 46, has bumped the average up to 11. This tendency of extreme numbers to influence the average in one direction or another can generate problems with certain statistical analyses. At the right is the mathematical formula for calculating the mean. The Median The median is also an average. It is that point in a distribution where half of the observations fall above it and half of the observations fall below it. Consider this series of numbers: 2, 10, 16, 20, and 28. The median is the middle point, which is 16. There are two observations above it and two observations below it. This, of course, is easy to figure when you have a small group of numbers and an odd number of observations. Consider this series of numbers: 2, 10, 16, 18, 20, and 28. Here we have an even number of observations. What is the median? To determine this, find the two middle observations and insert a mid-point. The mid-point in this case would be between 16 and 18, which are the two middle observations. The median, therefore, would be 17. Now consider this series of numbers: 2, 10, 16, 19, 20, and 28. We still have an even number of observations. The two middle observations are 16 and 19. What is the median? The mid-point between these two would be 17.5 or 17 1/2, which is the median of the set of numbers. This works well with small groups of observations. What do you do with a large number of observations? If the distribution of observations has been organized in a frequency distribution, there is an easy way to determine the median. See the box. The median is useful for describing certain kinds of data because it is not affected by extreme scores. The Mode The mode refers to the observation occurring most often in a distribution. It is determined by simple observation. Consider these observations: 11, 11, 12, 12, 12, 13, 13, 13, 13, 13, 14, 14, 14, 15, 15, 15, 16, 16, 17, 17, and 18. Which value occurs more frequently? The value 13 occurs five times, more than any other value, and, therefore, 13 is the mode. It is possible that a distribution may have more than one mode, and such a distribution is called bimodal. For instance, if we added two more 12s to the distribution above, it would have 12 appearing five times and 13 appearing five times. Both 12 and 13 would be modes and the distribution would be bimodal. It is also possible that no mode can be calculated in a distribution. In a situation where all values occur with equal frequency, no modal value can be calculated. Consider this distribution: 2, 7, 16, 19, 20, 25, and 27. Which value occurs more frequently than any other value? Of course, none does. In this case, no mode can be calculated. The same thing occurs with this distribution: 2, 2, 2, 5, 5, 5, 7, 7, 7, 12, 12, and 12. No one value occurs more frequently than the others. No mode can be calculated. The mode is a statistic of limited practical value. It has little meaning unless the number of measurements under consideration is fairly large. Measures of Variability The measures of central tendency have some limitations with regard to describing group performance. We need to know how the data are distributed in order to get a more accurate picture. We need to know how compact or how scattered the data are from a specified point. Measures of variability provide a numerical index that measures the amount of spread (also called dispersion) of a set of data. This makes it possible to judge the amount of "sameness" or "dissimilarity" of the observations. When we need to compare one set of data with another, we need to know the amount and nature of variability contained in each set of data. The Range The simplest and most rudimentary measure of variability is the range. This is the difference between the lowest and the highest observation in the distribution. The purpose here is to provide a numerical value indicating the overall spread (or dispersion) of a set of observations. Consider this set of measurements: 10, 12, 15, 18, and 20. The lowest measurement is 10 and the highest is 20. The range is 20 minus 10, or 10. The range, however, has two disadvantages. First, for large sets of observations it is an unstable descriptive measure. Second, the range is not independent of the size of the set of observations. But the range can be effectively used with small numbers of observations. The Deviation Score This is also a simple and useful measure of variability. The deviation score provides a means for determining the distance of an individual observation from the mean. The formula for determining the deviation score is: The Standard Deviation The standard deviation is the most precise measure of variability to be included here. It takes into account the variability of all the observations in a distribution. For those of you who want to learn more about the advantages obtained by using the standard deviation and want to know how to calculate the standard deviation, it is suggested that you consult an introductory text in statistics. The big advantage with the standard deviation is that with standard deviations calculated for two sets of data, we can more accurately interpret the means of the two sets of data since we have some common basis for comparison. To Page 3 Enrich Your Life With a Philosophy Book... Enrich Your Life With a Philosophy Magazine... 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