44 CHAPTER 2 Descriptive Statistics Graphs of Frequency Distributions Sometimes it is easier to discover patterns in a data set by looking at a graph of the frequency distribution. One such graph is a frequency histogram. A frequency histogram uses bars to represent the frequency distribution of a data set. A histogram has the following properties. 1. The horizontal scale is quantitative and measures the data entries. 2. The vertical scale measures the frequencies of the classes. 3. Consecutive bars must touch. DEFINITION Because consecutive bars of a histogram must touch, bars must begin and end at class boundaries instead of class limits. Class boundaries are the numbers that separate classes without forming gaps between them. For data that are integers, subtract 0.5 from each lower limit to find the lower class boundaries. To find the upper class boundaries, add 0.5 to each upper limit. The upper boundary of a class will equal the lower boundary of the next higher class. Constructing a Frequency Histogram Draw a frequency histogram for the frequency distribution in Example 2. Describe any patterns. SOLUTION First, find the class boundaries. Because the data entries are integers, subtract 0.5 from each lower limit to find the lower class boundaries and add 0.5 to each upper limit to find the upper class boundaries. So, the lower and upper boundaries of the first class are as follows. First class lower boundary = 155 - 0.5 = 154.5 First class upper boundary = 190 + 0.5 = 190.5 The boundaries of the remaining classes are shown in the table at the left. To construct the histogram, choose possible frequency values for the vertical scale. You can mark the horizontal scale either at the midpoints or at the class boundaries. Both histograms are shown below. Frequency (number of adults) 1 2 3 5 7 4 6 172.5 208.5 244.5 280.5 316.5 352.5 388.5 Times (in minutes) Cell Phone Screen Times (labeled with class midpoints) Broken axis 3 5 6 7 3 2 4 Frequency (number of adults) Times (in minutes) Cell Phone Screen Times (labeled with class boundaries) 154.5 190.5 226.5 262.5 298.5 334.5 370.5 406.5 1 2 3 5 7 4 6 3 5 6 7 3 2 4 Interpretation From either histogram, you can determine that two thirds of the adults are spending more than 262.5 minutes each day using their cell phones. EXAMPLE 3 Class Class boundaries Frequency, f 155–190 154.5–190.5 3 191–226 190.5–226.5 2 227–262 226.5–262.5 5 263–298 262.5–298.5 6 299–334 298.5–334.5 7 335–370 334.5–370.5 4 371–406 370.5–406.5 3 Study Tip It is customary in bar graphs to have spaces between the bars, whereas with histograms, it is customary that the bars have no spaces between them.
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