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| Central Tendency and Variability |
| Columns - Research to Practice | |
| Sunday, 31 March 2002 | |
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For this issue of Counselor, we will focus on summarizing, which includes describing the concepts of middleness or central tendency and variability, terms for average and spread. These concepts answer questions such as what was the typical number of drinks consumed by a client in the last month (average)? What were the heavy drinking and light drinking days in that period (spread)? If you work in an addictions program or have your own practice, you will ask these and similar kinds of questions. As much as these statistical concepts help simplify complex data, they do have a down side. By summarizing information you sacrifice uniqueness. Essentially, you end up with less than what you started with (Tal, 2001); and that suggests you cannot determine by looking at averaged information what a situation truly means for a particular client. Take caution when applying general summarized information to a single person.Central tendency or middleness These concepts are best represented by interval and ratio numbers; only describe the state of data; gives you a numerical image of your data; and do not make conclusions or draw inferences (Hayes, 2000). The numerical image revolves around the average, middle, central point, or central tendency of a distribution. For example, 1,2,3,4,5 is a distribution, and in this case the middle point is 3. Always keep in mind that there is more than one kind of average (mean, mode, median) (Kranzler & Moursund, 1999). Each average provides you with different kinds of information about the middle set of scores, such as the average of the previous scores, the most frequent score, and the middle score (Salkind, 2000). Mean The mean is the most common type of average. Most people recognize this from elementary school. It is usually represented by a number of different symbols. The most common are x or M. To calculate a mean you simply divide a set of values in a group by the number of values in that group (Salkind, 2000). The addictions field uses this frequently. For instance, perhaps an accreditation or licensing body wants to review your records and asks for data that proves you adhere to your program mission (which is to treat people with alcohol problems). You have such data, it comes in the form of valid test instrument results, in this case the Michigan Alcohol Screening Test (MAST). It also happens that you administer this test to all new clients. To satisfy the reviewing body, you pull a sample of scores from last week's admissions and find the mean score to be 12.28. (A score of 5.0 or more on the MAST usually indicates a drinking problem.) Your sample came from seven clients whose MAST scores were 6,9,9,11,13,16,22. (Add the scores and divide the sum by seven. This equals 12.28.) You present the reviewing body with data demonstrating you are serving the program mission. This is a simple example, but one that demonstrates the usefulness of statistics, especially for the mean. Mode A mode (symbol = Mo) is the value that occurs most frequently. It is the least precise and most general measure of central tendency. Using the previous example for mean, note the value "9" appears more than any other number. Therefore, "9" is the mode. An easy way to remember mode is that the letters "mo" in mode are the same as the first two letters in most. Median When scores or values are arranged in order, from lowest to highest (or visa versa), the median (symbol = Med) is the middle score. Using the example from above (6,9,9,11,13,16,22), guess what number is the median? In this example, it happens to be 11. Mean, mode, and median are methods that best characterize an entire set of scores (Kranzler & Moursund, 1999). With one number, you get a numerical visual about numerical data and values. Variability Think of the words spread or scattering and you get the idea of variability (Salkind, 2000). For our purposes, we will deal with two easy variability concepts, range and standard deviation. Range This is the simplest measure of variability. It tells you how far apart values are from one another. In any collection of numbers, you simply subtract the lowest number from the highest (Salkind, 2000). Following our example from above, subtract 6 from 22, and 16 is the range. This is not very useful except for telling the basic spread of data. Standard deviation This is the most widely used measure of variability. It is the spread of scores around the mean, or the average distance of a particular value or score from the mean (symbol = SD or s) (Jaeger, 1993; Salkind, 2000). Using our previous example, the SD of this distribution is 3.19, this number is one standard deviation from the mean, which was M = 12.28. One standard deviation from the mean on the low-end our scores equals 9.09 (12.28 minus 3.19), and one standard deviation on the high side is 15.47 (12.28 plus 3.19). Now I know that for this week, all the scores above 15.47 and below 9.09 are more than one SD from the mean of the week. This is useful program information, if not useful clinical information. One small footnote about standard deviation is in order. If you square the standard deviation you get a thing called variance. It is just another way to measure variability. You won't often see it in journals or reports, but you should be aware of it. Getting confused? An old student of mine named Betty used to complain cynically that she would never understand statistics. But, she persevered and finally "got it." She walked into my office one day and proudly announced that she had received an A- in her statistics course. So, for those of you who doubt your ability to understand all this, think of Betty. If she can "get it" anybody, and I mean anybody, can. Next time, we'll tackle organizing data through plots and graphs. References Hayes, N. (2000). Doing Psychological Research. Buckingham, England: Open University Press. Jaeger, R.M. (1993). Statistics: A Spectator Sport (2nd). Thousand Oaks, CA: Sage. Kranzler, G. & Moursund, J. (1999). Statistics for the Terrified (2nd). Upper Saddle River, NJ: Prentice Hall. Salkind, N.J. (2000). Statistics for People Who (think they) Hate Statistics. Thousand Oaks, CA: Sage. Tal, J. (2001). Reading Between the Numbers. New York: McGraw-Hill. Michael J. Taleff, PhD, CADC, MAC, is the Alcohol & Drug Education Program Coordinator for the University of Hawai'i at Manoa.              |
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