Frequency Distributions and Their Graphs 2.1 40 CHAPTER 2 Descriptive Statistics Frequency Distributions Graphs of Frequency Distributions What You Should Learn How to construct a frequency distribution, including limits, midpoints, relative frequencies, cumulative frequencies, and boundaries How to construct frequency histograms, frequency polygons, relative frequency histograms, and ogives Frequency Distributions There are many ways to organize and describe a data set. Important characteristics to look for when organizing and describing a data set are its center, its variability (or spread), and its shape. Measures of center and shapes of distributions are covered in Section 2.3. Measures of variability are covered in Section 2.4. When a data set has many entries, it can be difficult to see patterns. In this section, you will learn how to organize data sets by grouping the data into intervals called classes and forming a frequency distribution. You will also learn how to use frequency distributions to construct graphs. A frequency distribution is a table that shows classes or intervals of data entries with a count of the number of entries in each class. The frequency f of a class is the number of data entries in the class. DEFINITION In the frequency distribution shown at the left, there are six classes. The frequencies for each of the six classes are 5, 8, 6, 8, 5, and 4. Each class has a lower class limit, which is the least number that can belong to the class, and an upper class limit, which is the greatest number that can belong to the class. In the frequency distribution shown, the lower class limits are 1, 6, 11, 16, 21, and 26, and the upper class limits are 5, 10, 15, 20, 25, and 30. The class width is the distance between lower (or upper) limits of consecutive classes. For instance, the class width in the frequency distribution shown is 6 - 1 = 5. Notice that the classes do not overlap. The difference between the maximum and minimum data entries is called the range. In the frequency table shown, suppose the maximum data entry is 29, and the minimum data entry is 1. The range then is 29 - 1 = 28. You will learn more about the range of a data set in Section 2.4. Constructing a Frequency Distribution from a Data Set 1. Decide on the number of classes to include in the frequency distribution. The number of classes should be between 5 and 20; otherwise, it may be difficult to detect any patterns. 2. Find the class width as follows. Determine the range of the data, divide the range by the number of classes, and round up to the next convenient number. 3. Find the class limits. You can use the minimum data entry as the lower limit of the first class. To find the remaining lower limits, add the class width to the lower limit of the preceding class. Then find the upper limit of the first class. Remember that classes cannot overlap. Find the remaining upper class limits. 4. Make a tally mark for each data entry in the row of the appropriate class. 5. Count the tally marks to find the total frequency f for each class. GUIDELINES Example of a Frequency Distribution Class Frequency, f 1–5 5 6–10 8 11–15 6 16–20 8 21–25 5 26–30 4 Study Tip In general, the frequency distributions shown in this text will use the minimum data entry for the lower limit of the first class. Sometimes it may be more convenient to choose a lower limit that is slightly less than the minimum data entry. The frequency distribution produced will vary slightly.
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