12.2 Frequency Distributions and Statistical Graphs 775 Frequency polygons are line graphs with scales the same as those of the histogram; that is, the horizontal scale indicates observed values and the vertical scale indicates frequency. To construct a frequency polygon, place a dot at the corresponding frequency above each of the observed values. Then connect the dots with straight-line segments. When constructing frequency polygons, always put in two additional class marks, one at the lower end and one at the upper end on the horizontal scale (values for these added class marks are not needed on the frequency polygon). Since the frequency at these added class marks is 0, the end points of the frequency polygon will always be on the horizontal scale. Profile in Mathematics Ronald Fisher (1890–1962) Sir Ronald Aylmer Fisher (1890– 1962), often referred to as “the father of statistics,” was born in London, England, and educated at Cambridge University. In addition to being a statistician, he was an evolutionary biologist and a geneticist. The historian Anders Hald said, “Fisher was a genius who almost single-handedly created the foundation for modern statistical science.” Fisher’s sister wrote that, “Because of his poor eyesight, he was tutored in mathematics without the aid of paper and pen, which developed his ability to visualize problems in geometrical terms.” Fisher wrote a number of books on statistics, in which he introduced many new statistical ideas and refined others. He made important contributions to the statistical areas of design of experiments and the analysis of variance. He also introduced the null hypothesis and Fisher’s z -distribution, and was the first to use the term parameter. Solution The vertical scale must extend at least to the number 18, since that is the greatest recorded frequency. The horizontal scale must include the numbers 0–9, the number of children observed. Eight families have no children. We indicate that by constructing a bar above the number 0, centered at 0, on the horizontal scale extended up to 8 on the vertical scale (Fig. 12.7). Eleven families have one child, so we construct a bar extending to 11 above the number 1, centered at 1, on the horizontal scale. We continue this procedure for each observed value. Both the horizontal and vertical scales should be labeled, the bars should be the same width and centered at the observed value, and the histogram should have a title. In a histogram, the bars should always touch. Now try Exercise 25 Number of Families Number of Children per Family Number of Children 0123456789 5 10 15 20 Figure 12.7 7 Example 5 Construct a Frequency Polygon Construct a frequency polygon of the frequency distribution in Example 1. Solution Since eight families have no children, place a mark above the 0 at 8 on the vertical scale, as shown in Fig. 12.8. Because there are 11 families with one child, place a mark above the 1 on the horizontal scale at the 11 on the vertical scale, and so on. Connect the dots with straight-line segments and bring the end points of the graph down to the horizontal scale, as shown. Now try Exercise 27 Number of Families Number of Children per Family Number of Children 0123456789 5 10 0 15 20 Figure 12.8 7 Timely Tip When constructing a histogram or frequency polygon, be sure to label both scales of the graph.
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