CHAPTER II – MEASURES OF VARIABILITY
Measures of Variability Objectives After reading this chapter, the student should be able to: 1 Define what is meant by the terms measures of variability, and give two examples of each. 2 Construct a frequency polygon for a set of scores 3 Calculate the standard deviation for grouped data 4 Calculate the standard deviation for ungrouped data 5 Calculate z-scores 6 Calculate T-scores 7 Convert raw scores to percentiles and T scores and z-scores 8 Calculate percentile scores
Key Terms Statistics: A means by which a set of data may be described and interpreted in a meaningful manner; also a method by which data may be analyzed and inferences and conclusions drawn Ungrouped data: Raw scores presented as they were recorded, no attempt being made to arrange them into a more meaningful or convenient form Grouped data: Scores that have been arranged in some manner such as from high to low or into classes or categories to give more meaning to the data or to facilitate further calculations Variability: Variability may be defined as the scatter or spread of scores from a point of central tendency. The range: Is the difference between the highest and lowest scores. Bell curve: A frequency distribution statistics. Normal distribution is shaped like a bell. Standard Deviation: A measure of the variability, or spread, of a set of scores around the mean. Variance: A measure of variation within a distribution, determined by averaging the squared deviations from the mean of a distribution. Variation: The dispersion of data points around the mean of a distribution. z-Score: A standard score used to combine different tests together with mean 0 and standard deviation.
Introduction Are we still having fun? Of course we are and it is going to get better. Now here is something you need to know. There are a number of psychological factors that prevent a lot of people from starting or sticking with a statistics class. They include; Fear of change: This has nothing to do with your socks. It’s a little more cosmic than that. Change in any form causes stress….a new class, a new job, a new relationship, a new you. Wow, what's going to happen if you make yourself more attractive to the opposite sex by learning Stats? You know what women say about intelligent men…they have money! You might have to deal with people socially and sexually in a whole different way just because of stats. And that is serious stress. Guilt: In real life, there is no such thing as one hundred and one percent. So if your dance card is already full, you're going to have to rearrange priorities. Rob Peter, to pay Paul. Ask yourself these questions and answer them honestly: Can you take an hour off from playing Wii to read your stats book every day? Will your video performance really suffer? Will studying after school really wreck your home life? Do you want to repeat this course again next semester? There is that fear thingy again. Intimidation: You think: "All those people in class know more than I do. I am going to walk in there and not know which way to turn. I'll be the only one there who's out of sync and not in statistical shape. They're all going to point at my cerebral love handles and laugh.'' And you don't even want to think about what's going to happen when the class starts. Had enough? Okay, now that we've triggered all your fears, let's try to tackle some of these problems. First, you have to realize that it's fine to do something for yourself. You know like learning something of value now and then. It's not a selfish act. Just the opposite. By making time for your statistics class and getting a descent education, you'll not only feel better about yourself, you'll feel better about life, your job, and your relationships. Taking a couple hours a week to achieve this is a good move. Playing Wii six hours a day now that's selfish. Second, make sure you give statistics a fair chance. Give it time. Make yourself give it a good shot. I know I said this before but I am saying it again. Set a time limit. I will recommend twelve weeks. That's generally considered the time period needed for the psychological changes in that six inch square on top of your shoulders to really get in gear. No excuses. You're going to gut it out for three months. Either that or take the entire course over again. Trust me; it's not as bad as boot camp. (You pacifists and non-Israeli women can ignore that analogy.) Third, make it easy on yourself. Who sang that? Hmmm. Oh, the Righteous Brothers. Remember blue-eyed soul? Sorry, back to the book. The greatest danger in starting a stats course is overdoing it. That's how you defeat yourself mentally. If your mind easily wanders . . . "You never close your eyes anymore when I kiss your lips ..." Wait, where were we? Right, if you're one of those weird people whose mind easily wanders, then give your brain some places to wander with it. If reading the text every day bores you to tears, (which it won’t) then work the formulas, or surf the internet for statistical games. They are out there and they are fun to play. Fourth, make studying a habit. Get yourself into a ritual. Make it part of your regular routine—like breakfast or sex. You can play little tricks on yourself to make this happen. If you're not a terribly organized person, don't sign up for an online class, where they tell you to drop in any old time. You'll never do it. There will always be some excuse—you're too tired, you have too much work. You need a class where you can go at your own pace and study when you need too. I found that it is easy to make it early in the morning when there's little else to distract you. If your only choice is between studying and watching reruns of Captain Kangaroo you're going to be in pretty good shape…I hope!
Five, if all else fails, use guilt to your advantage. Get a friend to sign up for the class who's even more insecure about it than you (there is still time before drop and add). The two of you going together will make for positive reinforcement. If you don't show up, your friend's going to be a little annoyed. Now would you do that to a friend? Better yet sign up with your mother. One last point…don’t get all psyched up about the class. I think all that psyching stuff might have started with Ronald Reagan. Win one for the Gipper, and all that. In those days, psyching up meant a lot of screaming and banging of heads against lookers. Over the years it got worse. Vomiting was added to the ritual…nice-touch, eh? This foolishly macho system of psyching up was designed to raise an individual’s adrenaline level by initiating the "fight or flight" response. After all these years, some folks are finally starting to realize that there are some problems with that. First, it scares the heck out of your teacher and that is going to send your grade right into the red. Also, confusion and uncertainty is often a product of over psyching. So forget every coach or psychology teacher you ever had. The old methods of psyching up for your classes are just that—old methods. Current thought favors stress reduction and conservation of energy. Low-key mental preparation is what will get you to the library and have you doing well once you're there. Now we can get started.
Measures of Variability Variability may be defined as the scatter or spread of scores from a point of central tendency. When we measure the variability of a group of scores, we are determining the amount of scatter or spread in the scores. This information tells us how homogeneous or heterogeneous a group is. Don’t you just love these technical terms… they go better than wrapped weenies at a cocktail party. While two groups may have the same mean or median, they may differ considerably in variability. For example, if five students score 84, 80, 78, 75, and 73 on a test, their mean score is 78. Another group of five students score 98, 95, 78, 65, and 54, and their mean score is also 78. However, there is an obvious difference between the two groups in the variability of their scores. You do see the difference…right? Figure 2-1 Distribution of archery scores
Perhaps the best way to develop an understanding of variability is through examples. Let’s say that Jim and Tom were enrolled in an archery course for which each shot 100 rounds. Although both boys made a mean score of 20, you can see from Figure 2-1 that their performances were quite
different. The Figure that Tom, whose scores range from 15 to 25, was much more consistent in his shooting. Jim, with a range from 10 to 30, was more erratic. Jim was all over the freaken place with his shots. Figure 2-2 Heights of two physical education classes
Let me give you a better example… to see if I can confuse you a little more. Suppose in one middle school, the first period physical education class consists entirely of eighth grade boys. The fourth period class is a mixture of 7th, 8th, and 9th grades. Let’s say that the teacher was interested in comparing the heights of the two groups. Although he found the mean heights for the two groups to be similar, Figure 2-2 reveals that the 8th grade class was more homogeneous. In other words there was less variance in that group. Now let’s say a teacher wished to compare the performance of a group of physical education majors on their anatomy and physiology finals. Exam scores in physiology were: 70, 71, 75, 78, 79, 80, 81, 82, 85, 89, and 90. Exam scores in anatomy were: 60, 70, 72, 73, 75, 80, 85, 87, 90, 98, and 100. The mean score on each exam is 80. Would you say the performances were about equal? If you said, “NO!” you get a red star. If you said, “YES!” I am taking that Ph. D. away from you that I gave you earlier. See, I told you degrees don’t mean anything…Bozo! Anywho, these three examples show the need for some way of describing the variability as we previously defined as the "spread" or "scatter" of scores. A simple measure of variability is the range. You recall that the range is the difference between the highest and lowest scores. In Table 1-9, the range of Jim's scores is 20 (30-10=20) while the range of Tom's scores is 10 (25-15 = 10). Note, that for this example the larger range is associated with the more variable set of scores. In Table 1-10 the range for the 8th grade class is 8 (70-62=8) and the range of the mixed class is figured at 16 (74-58=16). Again, the group which has more scattered scores has the larger range. This is also true for the anatomy and physiology example. The physiology scores have a range of 20 (70–90=20), but the anatomy scores, which are more variable, have a range of 40 (60-100= 40). Tell me all of this is sinking into to six inch square sitting on top of your shoulders. Good, because now I am going test your cognitive clearance a little. Consider, what would happen to the range of the 8th grade class in Table 1-10 if one 9th grade basketball player (76" tall) was put into the class because of a scheduling conflict. Instantly, the range jumps from 8"(70-62= 8) to 14"(76-62=14) because of one individual. (See Table 1-11) In this instance the range would be a deceptive measure of variability.
The best and most commonly used measure of variability is called the standard deviation (SD). Like the range, it is an indicator of the spread of the scores and, in general, larger values of the standard deviation indicate greater spread in the data. Figure 2-3 Distributions of squat-thrust scores of two classes
Okay, get ready for a little analytical gymnastics. Get back here… it’s not like I am going to ask you to do a triple cerebral somersault. It is not even going to be as difficult as an intellectual forward roll. Now, suppose two classes were tested for the number of squat-trusts they could do. (Squatthrusts….sounds gross doesn’t it) Class 1 had a mean of 30 and a standard deviation of 10. Class 2 had a mean of 30 and a standard deviation of 5. From this information we can say that Class 1’s scores are more spread than those from Class 2. Generally, about 2/3 of the scores in a distribution will fall between the score one standard deviation above the mean (M + 1 SD) and one standard deviation below the mean (M - 1 SD). Knowing that M=30 and that SD=10 tells us that about 2/3 of the scores in Class 1 fell between 20 and 40 (30-10 and 30+10). For Class 2, knowing that M=30 and SD=5 tells us that about 2/3 of the scores fell between 25 and 35 (30-5 and 30+5). This will let us predict that their respective distribution curves look something like Figure 2-3. Figure 2-4 Effect of one extreme score on the range
In general, about 2/3 of the scores of a group will fall in the interval between the score 1 standard deviation below the mean (M-1 SD) and the score 1 standard deviation above the mean (M+1 SD). In simple notation, 2/3 of the scores fall in the interval: M ± 1 SD
Standard Deviation for Large Groups Now that you have some idea of what a standard deviation represents, you need to learn how to compute one. Don’t worry! This is so simple your little brother could do it… if he happens to be Steven Hawkins. I am just kidding; it really is simple. We will again use the sit-up data for the class of 9th grade boys and expand the table by adding one new column. That column is ƒd2. (See Table 2-1). To form the fd 2 column you simply multiple the d column and the fd. In other words, each entry in this column is the product of the corresponding entries in the fd and d columns. Note, this is not obtained by squaring the entries in the fd column. Let’s do it this way…the steps for computing the Standard Deviation for grouped data are as follows: 1. Complete the first 5 steps for computing the mean. 2. Form the fd 2 column. Each entry in this column is the product of the corresponding entries in the ƒd and d columns. Again, note that this is not obtained by squaring the entries in the ƒd column. I said that so many times it is strating to sound like a religion. 3. Find fd 2 . In other words, find the sum of the fd 2 column. 4. Use the formula to find the standard deviation. fd 2 fd SD I N N
2
Table 2-1 Procedure for computing the standard deviation of the sit-up scores of the class of 9th grade boys (Grouped Data)
S.I. 78-82 73-77 68-72 63-67 58-62 53-57 48-52 43-47 38-42 33-37 28-32 23-27
Tally
ƒ 1 2 4 3 7 6 10 6 4 3 2 2
d 6 5 4 3 2 1 0 -1 -2 -3 -4 -5
ƒd ƒd2 6 36 10 50 16 64 9 27 14 28 6 (+61) 6 0 0 -6 6 -8 16 -9 27 -8 32 -10 (-41) 50 N=50 Σ ƒd = 20
Σ ƒd2 = 342
SD I
fd 2 fd N N
342 20 SD I 50 50
SD 5 6.84 .40
2
2
2
SD 5 6.84 .1600
SD 5 6.68 SD 5 2.6 SD 13.0
By looking back at Table 1-2, we see that only 2 of the scores do not fall within 2 standard deviations above and below the mean ( X ±2 SD): X =52 + 26 = 78 ( X + 2 SD); X =52 – 26 = 26 ( X – 2 SD).
Standard Deviation for Small Groups Sometimes, we have relatively few scores to consider. It would be inconvenient to group these scores and use the formula for standard deviation for grouped data. There is a method for computing the standard deviation of a small group of scores. We use this when the number (N) is 15 or less; if N is larger than 15, the procedure for grouped data is more convenient. Table 2-2 illustrates the procedure for finding the standard deviation of ungrouped data. The steps for computing a standard deviation for ungrouped data are as follows: 1. Form the X column. List the number of pushups each boy completed in order, from largest to smallest. 2. Compute the mean. 3. Form the d column. This is done by subtracting the mean from each score and entering the difference on the proper line. 4. Compute Σd. (This will be zero if you haven't rounded off.) 5. Form the d2 column. This is done by squaring each value in the d column. 6. Compute Σd2. 7. Using this formula SD
d2 , compute the standard deviation. N
Table 2-2 Procedure for computing the standard deviation of the push-up scores of the class of 9th grade girls (Ungrouped Data)
Number of Push-ups X d 35 11 27 3 25 1 22 -2 22 -2 13 -11 ΣX = 144 Σd = 0 M = 24
d2 121 9 1 4 4 121 Σd2 = 260
SD
d2 N
260 6 SD 43.33 SD 6.58
SD
Note once again, that you use this procedure only when N is 15 or less. Keeping it simple is important for every part of a statistics regimen. When you first open up your textbook and see all those gleaming formulas, there's a great tendency to plug-in all the formulas at once, making a smorgasbord out it. Don’t do that because it can overwhelm you. And you know what that can lead to…bwain damage, as Bugs Bunny would say. Just take it one formula at a time at a pace that seems most comfortable to you. This isn't competition. The idea isn't to keep up with the person next door. I want you to go at your own pace here. And realize, as you take on each new statistical exercise, that it may well take a few attempts before you begin to get it right and feel the true benefits. It's a process, and processes take time. Just because you don't get it the first time out doesn't mean it's wrong for you. You'll find that the more you work at statistics, the more it will work for you. You are doing GREAT! Keep up it up. Okay, enough motivational horsehockey.