CHAPTER I – DATA ANALYSIS
Data Analysis Objectives After reading this chapter, the student should be able to: 1 Define what is meant by the terms measures of central tendency and give two examples of each. 2 Construct a frequency polygon for a set of scores 3 Calculate the mean 4 Calculate the mode. 5 Calculate the median
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 Frequency distribution: A method of grouping data; a table that presents the raw scores or intervals of scores and the frequencies with which the raw scores occur Measure of central tendency: Is a score which best represents all the scores of a group. The mean: A measure of central tendency, or average; obtained by dividing the sum of the scores by the number of score. The mode: Is the score that occurs most frequently. The median: Is the middle score of a set of scores Skewed distribution: When scores cluster at the upper or lower ends, rather than the middle, the distribution is said to be skewed.
Fundamentals of Statistics Mention the word statistics and most prospective teachers usually freak out. In laymen’s terms it scares the living heck out of them. There may be a number of reasons for this attitude… a poor background in mathematics, intimidation by the terminology and formulas, or horror stories that they have been told by other students. Whatever the reason, this is extremely unfortunate; because if you can do basic addition, subtraction, and multiplication, statistics is breeze. Seriously, anyone can learn how to compute, read and use statistics. Heck, I can do those things, so obviously it is not all that hard… actually it is EASY. Unfortunately, when students are faced with the part of teacher preparation that includes statistics many students simply endure the course and are quite sure that they will never use the computations in their teaching. Thus, they enter the field minus a valuable tool. This is a mistake of significant magnitude especially if your goal is to be the very best teacher you can be. And that should be your goal…to be the very best you can be. You may be asking yourself, "Why do I need statistics? I'm going to teach I am not going to do research?” This is probably true; however, there will be times when you need to or want to read and interpret a research article in a journal. Also, in your teaching you will probably give tests in order to grade or evaluate your students. If you don’t understand how to read statistics there is a good chance that you won’t be able to determine which tests are the best to use in a given situation. And there is always the possibility that one day you may go completely out of your mind and do some research. In that case, you will need statistics to conduct your study and report the results. Okay, so you may not go completely crazy, and you may never do a day of research in your entire life. Still, as a teacher you will need a cursory understanding of statistics in order to construct test and help you with grading. Two essential steps in evaluation are the analysis and the presentation of the results of tests and measurements. Physical educators can spend hours carefully and skillfully measuring, but if they are unable to organize and analyze the data collected, they cannot effectively use the information to evaluate the program. This is why statistical analysis is so important. In other words, you really do need to understand statistics if your goal is to be the very best teacher you possible can be. If that is not your goal, maybe you are in the wrong profession. I know K-mart is hiring and the Marines are looking for a few good men. Heck, from what I have noticed lately the Marines will take just about anyone they can get. Yea, K-Mart drop outs even. Don’t even think about that though. Keep your butt right where it is, because I am going to get you through this without causing you any type of brain damage. Hey, all I am asking is that you give it a fair chance. It’s sort of like eating Brussels sprouts. You can’t just put them in your mouth and chew them twice and spit them out… actually with Brussels sprouts you can do that. Just don’t be so quick to judge statistics. Give it a little time and effort; give it a good shot. You need this stuff. I am willing to prove that very point to you in this chapter. I am also going to demonstrate to you that statistics is nothing to get your feathers all rustled up about. So stay tuned. Better yet, let’s get started. Basic Statistics Okay, to start things off I am going to promote you to a higher level. You are now a full pledged high school physical education instructor. Ah, what the heck, I will even throw in a Ph. D. to
put behind your name. Are you feeling smart Table 1-1 and confident now? Well don’t let that Ph. D. Sit-up scores for a class of 50 ninth grade boys swell your head to much, because degrees and 80 74 58 40 61 58 50 76 27 61 titles mean absolutely nothing in America. 68 69 69 59 46 63 47 50 70 60 Some of the dumbest people I have ever met 60 35 52 41 45 47 48 42 55 43 have a Ph.D. behind their name, and some of 37 49 65 55 50 38 49 65 55 53 the smartest people I ever met don’t even 48 33 58 50 53 48 47 23 28 32 know what a Ph. D. is. Believe me, anyone can get a Ph.D. Heck, I got one. Okay, enough philosophizing. Let’s get down to business. Now let’s suppose you administer a sit-up test to your class of 50 ninth grade boys. You have to use your imagination here. Me, I would never give a freaken sit-up test to anyone. To start off with the only thing they tell you is how many sit-ups you can do. That’s about it which isn’t very much. What is so athletic about sit-ups anyway…my dog can do Table 1-2 them. Anyway, you are the one doing the study not me. I Tally table for sit-up scores am just here for the ride. Score ƒ Score ƒ Okay, after you administer the test, you record the 80 51 scores in your roll book. The recorded scores are given in 79 50 78 49 Table 1-1. Obviously, I did the spreadsheet for you… you can thank me later. 77 48 Does just looking at Table 1-1 tell you very much 76 47 about your class's performance? Suppose a boy runs up and 75 46 says, “I did 42 sit-ups. Is that good?” What can you tell 74 45 him from just a glance at the aforementioned data? If you 73 44 studied the scores carefully, you would eventually be able to 72 43 give him some idea of how he compared with the rest of the 71 42 class. 70 41 In an effort to make these data more meaningful, 69 40 you might construct a tally table like the one in Table 1-2. 68 39 Yea, I did that chart for you too; I know I am a great guy. 67 38 Notice that all possible scores between the highest 66 37 65 36 and lowest are included even if no one made that score. This helps you to see the groupings and the breaks in the 64 35 distribution at a glance. However, there are still too many 63 34 categories for easy interpretation. 62 33 In most cases, rather than make a tally table, we 61 32 would condense the data by grouping the scores. For 60 31 example, our first category might include the five smallest 59 30 scores… 23, 24, 25, 26, and 27 to form the step interval 58 29 (S.I.) 23-27. A step interval is constructed simply by 57 28 grouping scores into intervals. Let me warn you right now, 56 27 when I give you abbreviations memorize them ASAP. Here 55 26 is a little quiz… what does SI mean? If you said step 54 25 53 24 interval, you get a big red star. If you didn’t say step interval, we are in big trouble here. You are either not 52 23 paying attention to what you are reading or you are
intellectually constipated. Okay, looking at Table 1-2, you can see that two persons’ scores fell in that interval, ƒ=2. The next interval is made up of the next five largest scores…28, 29, 30, 31, and 32. Table 1-2 shows that the scores of two persons fell in this step interval. This process is continued until all the scores are grouped into step intervals (see Table 1-3). Note, that once the scores are placed in a step interval they lose their individual identity. For instance, in Table 1-3 the step interval 58-62 has seven scores recorded. However, you don’t know specifically what those scores are. For all you know all seven scores could be 59 or for that matter 62. Consequently, when scores are put into intervals they are called group data. Steps for Construct a Frequency Distribution
Table 1-3 Frequency distribution of sit-up scores
S.I. 78-82 73-77 68-72 63-67
Tally
ƒ 1 2 4 3
58-62 53-57 48-52 43-47
7 6 10 6
38-42 33-37 28-32 23-27
4 3 2 2 N=50
The following steps can be used to construct a frequency distribution: 1. Find the range. The range is the difference between the highest and lowest scores. In this case the range is 57 (80 - 23 = 57). 2. Decide on the size and number of step intervals. There should be between 10 to 20 step intervals. It is usually more convenient if the interval size is an odd number (1, 3, 5, 7, or 9) or a multiple of 10 (10, 20, 30...) The number of step intervals is approximately equal to the quotient of the range divided by the interval size. For this example, an interval size of 1 gives us 57 (57 ÷ 1 = 57) step intervals. An interval size of 3 gives 19 (57 ÷ 3 = 19) intervals and an interval size of 5 would give us 11 (57 ÷ 5 = 11) intervals. An interval of 7 would give 8 (57 ÷ 7 = 8) intervals. We stated that the acceptable number of step intervals was between 10 and 20. We can see that intervals of either size 3 or size 5 meet this requirement. For ease of computation we select the larger interval size because it gives us fewer step intervals. 3. Choose the smallest score (23) as the lower boundary of the lowest step interval (23-27). 4. Do not let the same score fall in two step intervals. For example, the interval 23-27 is followed by 28-32, not 27-31. That only makes sense… right? 5. Using the data in Table 1-1, tally each score into the proper step interval. For example, 80 goes in the interval 78-82; 58 in the interval 58-62; etc, etc, etc. Time for a little quiz. Where does a score of 38 go? If you said in the interval 38-42 you get another red star. 6. The total number of tallies for a step interval is the frequency (ƒ) for that interval. Notice that the sum of the ƒ column is the same as the total number of scores. You can use this as a check on your tallying. After preparing a frequency distribution, you can tell the boy who did 42 sit-ups that most of the class did better than he did…be gentle though when you tell him that. We don’t want anyone putting a gun to their head. In fact, 39 of the 50 boys did more sit-ups. Yea! My dog could do better, but don’t tell him that either.
We gain ease in interpretation by using a frequency distribution, but as indicated when the step interval size is greater than one, we lose a little information. For example, by looking at Table 1-3, we can see that the interval with the highest frequency is 48-52, but we cannot tell how many boys performed exactly 49 sit-ups or if any of them did 52 sit-ups.