46 CHAPTER 2 Descriptive Statistics A histogram and its corresponding frequency polygon are often drawn together, as shown at the left using Excel. To do this by hand, first construct the frequency polygon by choosing appropriate horizontal and vertical scales. The horizontal scale should consist of the class midpoints, and the vertical scale should consist of appropriate frequency values. Then plot the points that represent the midpoint and frequency of each class. After connecting the points with line segments, finish by drawing the bars for the histogram. A relative frequency histogram has the same shape and the same horizontal scale as the corresponding frequency histogram. The difference is that the vertical scale measures the relative frequencies, not frequencies. Constructing a Relative Frequency Histogram Draw a relative frequency histogram for the frequency distribution in Example 2. SOLUTION The relative frequency histogram is shown. Notice that the shape of the histogram is the same as the shape of the frequency histogram constructed in Example 3. The only difference is that the vertical scale measures the relative frequencies. 0.05 154.5 190.5 226.5 262.5 298.5 334.5 370.5 406.5 0.10 0.15 0.20 0.25 Times (in minutes) Relative frequency (portion of adults) Cell Phone Screen Times Interpretation From this graph, you can quickly see that 0.2, or 20%, of the adults have screen times between 262.5 minutes and 298.5 minutes, which is not immediately obvious from the frequency histogram in Example 3. TRY IT YOURSELF 5 Use the frequency distribution in Try It Yourself 2 to construct a relative frequency histogram that represents the points scored by the 55 winning teams listed on page 39. Answer: Page A36 To describe the number of data entries that are less than or equal to a certain value, construct a cumulative frequency graph. A cumulative frequency graph, or ogive (pronounced o′jive), is a line graph that displays the cumulative frequency of each class at its upper class boundary. The upper boundaries are marked on the horizontal axis, and the cumulative frequencies are marked on the vertical axis. DEFINITION EXAMPLE 5 Picturing the World Old Faithful, a geyser at Yellowstone National Park, erupts on a regular basis.The time spans of a sample of eruptions are shown in the relative frequency histogram. (Source: Yellowstone National Park) 0.40 0.30 0.20 0.10 Duration of eruption (in minutes) Relative frequency 2.0 2.6 3.2 3.8 4.4 Old Faithful Eruptions About 50% of the eruptions last less than how many minutes? EXCEL Frequency 1 0 3 2 4 5 6 7 136.5 172.5 208.5 244.5 280.5 316.5 352.5 388.5 Times (in minutes) 424.5
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