2-1 Frequency Distributions for Organizing and Summarizing Data 47 5. List the lower class limits in a vertical column and then determine and enter the upper class limits. 6. Take each individual data value and put a tally mark in the appropriate class. Add the tally marks to find the total frequency for each class. When constructing a frequency distribution, be sure the classes do not overlap. Each of the original values must belong to exactly one class. Include all classes, even those with a frequency of zero. Try to use the same width for all classes, although it is sometimes impossible to avoid open-ended intervals, such as “65 years or older.” Using the daily commute times in Los Angeles in Table 2-1 from the Chapter Problem, follow the above procedure to construct the frequency distribution shown in Table 2-2. Use seven classes. CP YOUR TURN. Do Exercise 13 “Chicago Commute Time.” EXAMPLE 1 Daily Commute Time in Los Angeles SOLUTION Step 1: Select 7 as the number of desired classes. Step 2: Calculate the class width as shown below. Note that we round 12 up to 15, which is a more convenient number for commute time measurements. Class width ≈ 1maximum data value2 - 1minimum data value2 number of classes = 90 - 5 7 = 12.1 ≈ 151rounded up to a more convenient number2 Step 3: Let’s select 0 as the first lower class limit because it is below the value of 5 and is a very convenient starting point for commute time. Step 4: Add the class width of 15 to the starting value of 0 to get the second lower class limit of 15. Continue to add the class width of 15 until we have seven lower class limits. The lower class limits are therefore 0, 15, 30, 45, 60, 75, and 90. Step 5: List the lower class limits vertically, as shown in the margin. From this list, we identify the corresponding upper class limits as 14, 29, 44, 59, 74, 89, and 104. Step 6: Enter a tally mark for each data value in the appropriate class. Then add the tally marks to find the frequencies shown in Table 2-2. 0– 15– 30– 45– 60– 75– 90– Causes of Fatal Plane Crashes EXAMPLE 2 Table 2-3 on the next page lists data for the causes of fatal plane crashes from 1960 until a recent year. The causes are categorical data at the nominal level of measurement, but we can create the frequency distribution as shown. We can see that pilot error is the major cause of fatal plane crashes. Such information is helpful to regulatory agencies, such as the Federal Aviation Administration, as they develop strategies for reducing such crashes. continued Categorical Data So far we have discussed frequency distributions using only quantitative data sets, but frequency distributions can also be used to summarize categorical (or qualitative or attribute) data, as illustrated in Example 2. e Authors Identified In 1787–88 Alexander Hamilton, John Jay, and James Madison anonymously published the famous Federalist Papers in an attempt to convince New Yorkers that they should ratify the Constitution. The identity of most of the papers’ authors became known, but the authorship of 12 of the papers was contested. Through statistical analysis of the frequencies of various words, we can now conclude that James Madison is likely to have been the author of these 12 papers. For many of the disputed papers, the evidence in favor of Madison’s authorship is overwhelming to the degree that we can be almost certain of being correct. Coincidentally, the author of this book now lives in a town called Madison.
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