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Chapter 4

Statistical Concepts: Making Meaning Out of Scores

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Raw Scores

Jeremiah scores a 47 on one test and Elise scores a 95 on a different test. Who did better? Depends on:

How many items there are on the tests (47 or 950?)

Average score of everyone who took the tests.

Is higher or lower a better score?

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Rule #1: Raw Scores are Meaningless!

Raw scores tell us little, if anything, about how an individual did on a test

Must take those raw scores and do something to make meaning of them

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Making Raw Scores Meaningful:Norm Group Comparisons

However, norm group comparisons are helpful:

They tell us the relative position, within the norm group, of a person’s score

They allow us to compare the results among test-takers who took the same

They allow us to compare test results on two or more different tests taken by the same individual

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Frequency Distributions

Using a frequency distribution helps to make sense out of a set of scores

A frequency distribution orders a set of scores from high to low and lists the corresponding frequency of each score

See Table 4.1, p. 69

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Frequency Distributions (cont’d)

Make a frequency distribution:

1 2 4 6 12 16 14 4 7 21 4 3 11 4 10 12 7 9 3 21 3 6 1 3 6 5 10 3

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Histograms and Frequency Polygons

Use a graph to make sense out your frequency distribution

Two types of graphs: Histograms (bar graph) Frequency Polygons

Must determine class intervals to draw a histogram or frequency polygon Class intervals tell you how many people

scored within a grouping of scores

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Class Intervals

Making Class Intervals (Numbers from Table 4.1, p. 69)Subtract lowest number in series of scores from highest number: 66 - 32 = 34Divide number by number of class intervals you want (e.g., 7). 34/7 = 4.86Round off number obtained: 4.86 becomes 5Starting with lowest number, use number obtained (e.g., 5) for number of scores in each interval: 32–36, 37–41,42–46, 47–51, 52–56, 57–61, 62–66See Table 4.2, p. 70; then Fig 4.1 and 4.2, p. 71

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Creating Class Intervals

Make a distribution that has class intervals of 3 from the same set of scores:

1 2 4 6 12 16 14 4

7 21 4 3 11 4 10 12 7 9 3 2 1 3 6 1 3 6 5 10 3

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Making a Frequency Polygon and a Histogram

From your frequency distribution of class intervals (done on last slide), place each interval on a graph.

Then, make a frequency polygon and then a histogram using your answers.

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Another Frequency Distribution

Make a frequency distribution from the following scores:

15, 18, 25, 34, 42, 17, 19,

20, 15, 33, 32, 28, 15, 19,

30, 20, 24, 31, 16, 25, 26

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Make a Class Interval

Make a distribution that has class Intervals of 4 from the same set of scores: 15, 18, 25, 34, 42, 17, 19, 20, 15, 33, 32, 28, 15, 19,

30, 20, 24, 31, 16, 25, 26

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Create A Frequency Polygon and Histogram

From your frequency distribution of class intervals (done on last slide), place each interval on a graph

Then, make a frequency polygon and then a histogram using your answers

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Cumulative Distributions

Also called Ogive curve

Gives information about the percentile rank

Convert frequency of each class interval into a percentage and add it to previous cumulative percentage (see Table 4.3, p. 72)

Graph class intervals with their cumulative percentages (see Figure 4.3, p. 73)

Helps to determine percentage on any point in the distribution

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Normal and Skewed Curves

The Normal Curve Follows Natural Laws of the Universe Quincunx (see Fig. 4.4, p. 73):www.stattucino.com/berrie/dsl/Galton.html

Rule Number 2: God does not play dice with the universe.”

(Einstein)Contrast with Skewed Curves (See Fig. 4.5, p. 74)

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Measures of Central Tendency

Helps to put more meaning to scores

Tells you something about the “center” of a series of scores

Mean, Median, Mode

Compare means, medians, and modes on skewed and normal curves (see page 76, Figure 4.6)

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Measures of Central Tendency

Median: Middle Score odd number of scores—exact middle even number: average of two middle

scores.

Mode: Most frequent score

Median: Add scores and divide by number of scores

See Table 4.4, p. 75

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Measures of Variability

Tells you even more about a series of scores

Three types: Range: Highest score - Lowest score

+1 Standard Deviation Interquartile Range

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Interquartile Range

(middle 50% of scores--around median)see Figure 4.7, p. 77Using numbers from previous example: (3/4)N - (1/4)N (where N = number of

scores) 2 then, round off and find

that specific score See Table 4.5, p. 78

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Standard Deviation

Can apply S.D. to the normal curve (see Fig. 4.9, p. 80)Many human traits approximate the normal curveFinding Standard Deviation (see Table 4.6, p. 80)

Find another SD (next two slides)SD

(X M)2

N

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Figuring Out SD

X X - M (X - M)2 12 12-8 42 = 16 11 11-8 32 = 9 10 10-8 22 = 4 10 10-8 22 = 4 10 10-8 22 = 4 8 8-8 02 = 0 7 7-8 12 = 1 5 5-8 32= 9 4 4-8 42 = 16 3 3-8 52 = 25 80 88

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Figuring Out SD (Cont’d)

SD = 88/10 =8.8 = 2.96

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Remembering the Person

Understanding measures of central tendency and variability helps us understand where a person falls relative to his or her peer group, but….Don’t forget, that how a person FEELS about where he or she falls in his or her peer group is always critical