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# Chapter 4

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### Chapter 4

1. 1. Chapter 4 Statistical Concepts: Making Meaning Out of Scores
2. 2. Raw Scores <ul><li>Jeremiah scores a 47 on one test and Elise scores a 95 on a different test. Who did better? </li></ul><ul><ul><li>Depends on: </li></ul></ul><ul><ul><ul><li>How many items there are on the tests (47 or 950?) </li></ul></ul></ul><ul><ul><ul><li>Average score of everyone who took the tests. </li></ul></ul></ul><ul><ul><ul><li>Is higher or lower a better score? </li></ul></ul></ul>
3. 3. Rule #1: Raw Scores are Meaningless ! <ul><li>Raw scores tell us little, if anything, about how an individual did on a test </li></ul><ul><li>Must take those raw scores and do something to make meaning of them </li></ul>
4. 4. Making Raw Scores Meaningful: Norm Group Comparisons <ul><li>However, norm group comparisons are helpful: </li></ul><ul><ul><li>They tell us the relative position, within the norm group, of a person’s score </li></ul></ul><ul><ul><li>They allow us to compare the results among test-takers who took the same </li></ul></ul><ul><ul><li>They allow us to compare test results on two or more different tests taken by the same individual </li></ul></ul>
5. 5. Frequency Distributions <ul><li>Using a frequency distribution helps to make sense out of a set of scores </li></ul><ul><li>A frequency distribution orders a set of scores from high to low and lists the corresponding frequency of each score </li></ul><ul><li>See Table 4.1, p. 69 </li></ul>
6. 6. Frequency Distributions (cont’d) <ul><li>Make a frequency distribution: </li></ul><ul><li>1 2 4 6 12 16 14 4 </li></ul><ul><li>7 21 4 3 11 4 10 </li></ul><ul><li>12 7 9 3 2 1 3 </li></ul><ul><li>6 1 3 6 5 10 3 </li></ul>
7. 7. Histograms and Frequency Polygons <ul><li>Use a graph to make sense out your frequency distribution </li></ul><ul><li>Two types of graphs: </li></ul><ul><ul><li>Histograms (bar graph) </li></ul></ul><ul><ul><li>Frequency Polygons </li></ul></ul><ul><li>Must determine class intervals to draw a histogram or frequency polygon </li></ul><ul><ul><li>Class intervals tell you how many people scored within a grouping of scores </li></ul></ul>
8. 8. Class Intervals <ul><li>Making Class Intervals (Numbers from Table 4.1, p. 69) </li></ul><ul><li>Subtract lowest number in series of scores from highest number: 66 - 32 = 34 </li></ul><ul><li>Divide number by number of class intervals you want (e.g., 7). 34/7 = 4.86 </li></ul><ul><li>Round off number obtained: 4.86 becomes 5 </li></ul><ul><li>Starting with lowest number, use number obtained (e.g., 5) for number of scores in each interval: </li></ul><ul><li>32–36, 37–41,42–46, 47–51, 52–56, 57–61, 62–66 </li></ul><ul><li>See Table 4.2, p. 70; then Fig 4.1 and 4.2, p. 71 </li></ul>
9. 9. Creating Class Intervals <ul><li>Make a distribution that has class intervals of 3 from the same set of scores: </li></ul><ul><li>1 2 4 6 12 16 14 4 </li></ul><ul><li>7 21 4 3 11 4 10 </li></ul><ul><li>12 7 9 3 2 1 3 </li></ul><ul><li>6 1 3 6 5 10 3 </li></ul>
10. 10. Making a Frequency Polygon and a Histogram <ul><li>From your frequency distribution of class intervals (done on last slide), place each interval on a graph. </li></ul><ul><li>Then, make a frequency polygon and then a histogram using your answers. </li></ul>
11. 11. Another Frequency Distribution <ul><li>Make a frequency distribution from the following scores: </li></ul><ul><li>15, 18, 25, 34, 42, 17, 19, </li></ul><ul><li>20, 15, 33, 32, 28, 15, 19, </li></ul><ul><li>30, 20, 24, 31, 16, 25, 26 </li></ul>
12. 12. Make a Class Interval <ul><li>Make a distribution that has class Intervals of 4 from the same set of scores: </li></ul><ul><li> 15, 18, 25, 34, 42, 17, 19, 20, 15, 33, 32, 28, 15, 19, </li></ul><ul><li>30, 20, 24, 31, 16, 25, 26 </li></ul>
13. 13. Create A Frequency Polygon and Histogram <ul><li>From your frequency distribution of class intervals (done on last slide), place each interval on a graph </li></ul><ul><li>Then, make a frequency polygon and then a histogram using your answers </li></ul>
14. 14. Cumulative Distributions <ul><li>Also called Ogive curve </li></ul><ul><li>Gives information about the percentile rank </li></ul><ul><li>Convert frequency of each class interval into a percentage and add it to previous cumulative percentage (see Table 4.3, p. 72) </li></ul><ul><li>Graph class intervals with their cumulative percentages (see Figure 4.3, p. 73) </li></ul><ul><li>Helps to determine percentage on any point in the distribution </li></ul>
15. 15. Normal and Skewed Curves <ul><li>The Normal Curve </li></ul><ul><ul><li>Follows Natural Laws of the Universe </li></ul></ul><ul><ul><li>Quincunx (see Fig. 4.4, p. 73): </li></ul></ul><ul><ul><li>www.stattucino.com/berrie/dsl/Galton.html </li></ul></ul><ul><li>Rule Number 2: </li></ul><ul><ul><li>God does not play dice with the universe.” (Einstein) </li></ul></ul><ul><ul><li>Contrast with Skewed Curves (See Fig. 4.5, p. 74) </li></ul></ul>
16. 16. Measures of Central Tendency <ul><li>Helps to put more meaning to scores </li></ul><ul><li>Tells you something about the “center” of a series of scores </li></ul><ul><li>Mean, Median, Mode </li></ul><ul><li>Compare means, medians, and modes on skewed and normal curves (see page 76, Figure 4.6) </li></ul>
17. 17. Measures of Central Tendency <ul><li>Median: Middle Score </li></ul><ul><ul><li>odd number of scores—exact middle </li></ul></ul><ul><ul><li>even number: average of two middle scores. </li></ul></ul><ul><li>Mode: Most frequent score </li></ul><ul><li>Median: Add scores and divide by number of scores </li></ul><ul><li>See Table 4.4, p. 75 </li></ul>
18. 18. Measures of Variability <ul><li>Tells you even more about a series of scores </li></ul><ul><li>Three types: </li></ul><ul><ul><li>Range: Highest score - Lowest score +1 </li></ul></ul><ul><ul><li>Standard Deviation </li></ul></ul><ul><ul><li>Interquartile Range </li></ul></ul>
19. 19. Interquartile Range <ul><li>(middle 50% of scores--around median) </li></ul><ul><li>see Figure 4.7, p. 77 </li></ul><ul><li>Using numbers from previous example: </li></ul><ul><ul><li>(3/4)N - (1/4)N (where N = number of scores) </li></ul></ul><ul><ul><li> 2 then, round off and find </li></ul></ul><ul><ul><li>that specific score </li></ul></ul><ul><li>See Table 4.5, p. 78 </li></ul>
20. 20. Standard Deviation <ul><li>Can apply S.D. to the normal curve (see Fig. 4.9, p. 80) </li></ul><ul><li>Many human traits approximate the normal curve </li></ul><ul><li>Finding Standard Deviation (see Table 4.6, p. 80) </li></ul><ul><li>Find another SD (next two slides) </li></ul>
21. 21. Figuring Out SD <ul><ul><ul><li>X X - M (X - M) 2 </li></ul></ul></ul><ul><ul><ul><li>12 12-8 4 2 = 16 </li></ul></ul></ul><ul><ul><ul><li>11 11-8 3 2 = 9 </li></ul></ul></ul><ul><ul><ul><li>10 10-8 2 2 = 4 </li></ul></ul></ul><ul><ul><ul><li>10 10-8 2 2 = 4 </li></ul></ul></ul><ul><ul><ul><li>10 10-8 2 2 = 4 </li></ul></ul></ul><ul><ul><ul><li>8 8-8 0 2 = 0 </li></ul></ul></ul><ul><ul><ul><li>7 7-8 1 2 = 1 </li></ul></ul></ul><ul><ul><ul><li>5 5-8 3 2 = 9 </li></ul></ul></ul><ul><ul><ul><li>4 4-8 4 2 = 16 </li></ul></ul></ul><ul><ul><ul><li>3 3-8 5 2 = 25 </li></ul></ul></ul><ul><ul><ul><li>80 88 </li></ul></ul></ul>
22. 22. Figuring Out SD (Cont’d) <ul><li>SD = 88/10 = </li></ul><ul><li> 8.8 = 2.96 </li></ul>
23. 23. Remembering the Person <ul><li>Understanding measures of central tendency and variability helps us understand where a person falls relative to his or her peer group, but…. </li></ul><ul><li>Don’t forget, that how a person FEELS about where he or she falls in his or her peer group is always critical </li></ul>