56 Chapter 2 Graphical Summaries of Data Table 2.12 Women with a Given Number of Children Number of Children Frequency 0 435 1 175 2 222 3 112 4 38 5 9 6 7 7 0 8 2 0 1 2 3 4 5 6 7 8 500 400 300 200 100 0 Frequency Number of Children Figure 2.9 Histogram for data in Table 2.12 Objective 3 Determine the shape of a distribution from a histogram Shapes of Histograms The purpose of a histogram is to give a visual impression of the “shape” of a data set. Statisticians have developed terminology to describe some of the commonly observed shapes. A histogram is symmetric if its right half is a mirror image of its left half. Very few histograms are perfectly symmetric, but many are approximately symmetric. A histogram is skewed if one side, or tail, is longer than the other. A histogram with a long right-hand tail is said to be skewed to the right, or positively skewed. A histogram with a long left-hand tail is said to be skewed to the left, or negatively skewed. These terms apply to both frequency histograms and relative frequency histograms. Figure 2.10 presents some histograms for hypothetical samples. As another example, the histogram for particulate concentration, shown in Figure 2.5, is skewed to the right. 0 1 2 3 4 5 6 (a) 7 8 9 10 11 80 60 40 20 0 Frequency 0 1 2 3 4 5 6 7 8 9 10111213 (b) 0.20 0.15 0.10 0.05 0 Relative Frequency 0 1 2 3 4 5 6 7 8 9 101112 (c) 200 150 100 50 0 Frequency Figure 2.10 (a) A histogram skewed to the left. (b) An approximately symmetric histogram. (c) A histogram skewed to the right. The examples in Figure 2.10 are straightforward to categorize. In real life, the classification is not always clear-cut, and people may sometimes disagree on how to describe the shape of a particular histogram. Modes A peak, or high point, of a histogram is referred to as a mode. A histogram is unimodal if it has only one mode, and bimodal if it has two clearly distinct modes. In principle, a histogram can have more than two modes, but this does not happen often in practice. The histograms in Figure 2.10 are all unimodal. Figure 2.11 presents a bimodal histogram for a hypothetical sample. As another example, it is reasonable to classify the histogram for particulate emissions, shown in Figure 2.5, as unimodal, with the rectangle above the class 1–2 as the only mode. While some might say that the rectangle above the class 3–4 is another mode, most would agree that it is too small a peak to count as a second mode.
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