EXAMPLE 1.14

Distribution of IQ scores. You have probably heard that the distribution of scores on IQ tests is supposed to be roughly “bell-shaped.” Let’s look at some actual IQ scores. Table 1.1 displays the IQ scores of 60 fifth-grade students chosen at random from one school.

  1. 1. Divide the range of the data into classes of equal width. Let’s use

    75 ≤ IQ score < 85

    85 ≤ IQ score < 95

    ⋮

    145 ≤ IQ score < 155

    Table : TABLE 1.1 IQ Test Scores for 60 Randomly Chosen Fifth-Grade Students
    145 139 126 122 125 130 96 110 118 118
    101 142 134 124 112 109 134 113 81 113
    123 94 100 136 109 131 117 110 127 124
    106 124 115 133 116 102 127 117 109 137
    117 90 103 114 139 101 122 105 97 89
    102 108 110 128 114 112 114 102 82 101

    15

    Be sure to specify the classes precisely so that each individual falls into exactly one class. A student with IQ 84 would fall into the first class, but IQ 85 falls into the second.

  2. 2. Count the number of individuals in each class. These counts are called frequenciesfrequency, and a table of frequencies for all classes is a frequency tablefrequency table.

    Class Count Class Count
    75 ≤ IQ score < 85 2 115 ≤ IQ score < 125 13
    85 ≤ IQ score < 95 3 125 ≤ IQ score < 135 10
    95 ≤ IQ score < 105 10 135 ≤ IQ score < 145 5
    105 ≤ IQ score < 115 16 145 ≤ IQ score < 155 1

  3. 3. Draw the histogram. First, on the horizontal axis mark the scale for the variable whose distribution you are displaying. That’s the IQ score. The scale runs from 75 to 155 because that is the span of the classes we chose. The vertical axis contains the scale of counts. Each bar represents a class. The base of the bar covers the class, and the bar height is the class count. There is no horizontal space between the bars unless a class is empty, so its bar has height zero. Figure 1.7 is our histogram. It does look roughly “bell-shaped.”