2-2 Distributions 39 2-2 Distributions When working with large data sets, it is often helpful to organize and summarize the data by constructing a table that lists the different possible data values (either individually or by groups) along with the corresponding frequencies, which represent the number of times those values occur. Definition A frequency distribution lists data values (either individually or by groups of intervals), along with their corresponding frequencies (or counts). Table 2-2 is a frequency distribution summarizing the measured cotinine levels of the 40 smokers listed in Table 2-1. The frequency for a particular class is the number of original values that fall into that class. For example, the first class in Table 2 2 has a frequency of 11, indicating that 11 of the original data values are between 0 and 99 inclusive. We first present some standard terms used in discussing frequency distributions, and then we describe how to construct and interpret them. Definitions Lower class limits are the smallest numbers that can belong to the different classes. (Table 2-2 has lower class limits of 0, 100, 200, 300, and 400.) Upper class limits are the largest numbers that can belong to the different classes. (Table 2-2 has upper class limits of 99, 199, 299, 399, and 499.) Class boundaries are the numbers used to separate classes, but without the gaps created by class limits. They are obtained as follows: Find the size of the gap between the upper class limit of one class and the lower class limit of the next class. Add half of that amount to each upper class limit to find the upper class boundaries; subtract half of that amount from each lower class limit to find the lower class boundaries. (Table 2-2 has gaps of exactly 1 unit, so 0.5 is added to the upper class limits and subtracted from the lower class limits. The first class continued Table 2-2 Distribution of Cotinine Levels of Smokers Cotinine 0 99 11 100 199 12 200 299 14 300 399 1 400 499 2
40 C HAPTER 2 Describing, Exploring, and Comparing Data has boundaries of 20.5 and 99.5, the second class has boundaries of 99.5 and 199.5, and so on. The complete list of boundaries used for all classes is 20.5, 99.5, 199.5, 299.5, 399.5, and 499.5. Class midpoints are the midpoints of the classes. (Table 2-2 has class midpoints of 49.5, 149.5, 249.5, 349.5, and 449.5.) Each class midpoint can be found by adding the lower class limit to the upper class limit and dividing the sum by 2. Class width is the difference between two consecutive lower class limits or two consecutive lower class boundaries. (Table 2-2 uses a class width of 100.) Growth Charts Updated Pediatricians typically use standardized growth charts to compare their patient s weight and height to a sample of other children. Children are considered to be in the normal range if their weight and height fall between the 5th and 95th percentiles. If they fall outside of that range, they are often given tests to ensure that there are no serious medical problems. Pediatricians became increasingly aware of a major problem with the charts: Because they were based on children living between 1929 and 1975, the growth charts were found to be inaccurate. To rectify this problem, the charts were updated in 2000 to reflect the current measurements of millions of children. The weights and heights of children are good examples of populations that change over time. This is the reason for including changing characteristics of data over time as an important consideration for a population. The definitions of class width and class boundaries are tricky. Be careful to avoid the easy mistake of making the class width the difference between the lower class limit and the upper class limit. See Table 2-2 and note that the class width is 100, not 99. You can simplify the process of finding class boundaries by understanding that they basically fill the gaps between classes by splitting the difference between the end of one class and the beginning of the next class. Procedure for Constructing a Distribution distributions are constructed for these reasons: (1) Large data sets can be summarized, (2) we can gain some insight into the nature of data, and (3) we have a basis for constructing important graphs (such as histograms, introduced in the next section). Many uses of technology allow us to automatically obtain frequency distributions without manually constructing them, but here is the basic procedure: 1. Decide on the number of classes you want. The number of classes should be between 5 and 20, and the number you select might be affected by the convenience of using round numbers. 2. Calculate Class width < shighest valued 2 slowest valued number of classes Round this result to get a convenient number. (Usually round up.) You might need to change the number of classes, but the priority should be to use values that are easy to understand. 3. Starting point: Begin by choosing a number for the lower limit of the first class. Choose either the lowest data value or a convenient value that is a little smaller. 4. Using the lower limit of the first class and the class width, proceed to list the other lower class limits. (Add the class width to the starting point to get the second lower class limit. Add the class width to the second lower class limit to get the third, and so on.) 5. List the lower class limits in a vertical column and proceed to enter the upper class limits, which can be easily identified.
2-2 Distributions 41 6. Go through the data set putting a tally in the appropriate class for each data value. Use the tally marks to find the total frequency for each class. When constructing a frequency distribution, be sure that classes do not overlap so that 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. EXAMPLE Cotinine Levels of Smokers Using the 40 cotinine levels for the smokers in Table 2-1, follow the above procedure to construct the frequency distribution shown in Table 2-2. Assume that you want 5 classes. SOLUTION Step 1: Step 2: class width < Step 3: Step 4: Step 5: Step 6: Begin by selecting 5 as the number of desired classes. Calculate the class width. In the following calculation, 98.2 is rounded up to 100, which is a more convenient number. shighest valued 2 slowest valued number of classes We choose a starting point of 0, which is the lowest value in the list and is also a convenient number. Add the class width of 100 to the starting point of 0 to determine that the second lower class limit is 100. Continue to add the class width of 100 to get the remaining lower class limits of 200, 300, and 400. List the lower class limits vertically, as shown in the margin. From this list, we can easily identify the corresponding upper class limits as 99, 199, 299, 399, and 499. After identifying the lower and upper limits of each class, proceed to work through the data set by entering a tally mark for each value. When the tally marks are completed, add them to find the frequencies shown in Table 2-2. Relative Distribution An important variation of the basic frequency distribution uses relative frequencies, which are easily found by dividing each class frequency by the total of all frequencies. A relative frequency distribution includes the same class limits as a frequency distribution, but relative frequencies are used instead of actual frequencies. The relative frequencies are sometimes expressed as percents. relative frequency 5 5 491 2 0 5 class frequency sum of all frequencies 5 98.2 < 100 In Table 2-3 the actual frequencies from Table 2-2 are replaced by the corresponding relative frequencies expressed as percents. The first class has a relative frequency of 11> 40 5 0.275, or 27.5%, which is often rounded to 28%. The 02 100 200 300 400 Table 2-3 Relative Distribution of Cotinine Levels in Smokers Cotinine Relative 0 99 28% 100 199 30% 200 299 35% 300 399 3% 400 499 5%
42 C HAPTER 2 Describing, Exploring, and Comparing Data second class has a relative frequency of 12> 40 5 0.3, or 30.0%, and so on. If constructed correctly, the sum of the relative frequencies should total 1 (or 100%), with some small discrepancies allowed for rounding errors. Because 27.5% was rounded to 28% and 2.5% was rounded to 3%, the sum of the relative frequencies in Table 2-3 is 101% instead of 100%. Because they use simple proportions or percentages, relative frequency distributions make it easier for us to understand the distribution of the data and to compare different sets of data. 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 the likely 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. Cumulative Distribution Another variation of the standard frequency distribution is used when cumulative totals are desired. The cumulative frequency for a class is the sum of the frequencies for that class and all previous classes. Table 2-4 is the cumulative frequency distribution based on the frequency distribution of Table 2-2. Using the original frequencies of 11, 12, 14, 1, and 2, we add 11 1 12 to get the second cumulative frequency of 23, then we add 11 1 12 1 14 5 37 to get the third, and so on. See Table 2-4 and note that in addition to using cumulative frequencies, the class limits are replaced by less than expressions that describe the new range of values. Critical Thinking: Interpreting Distributions The transformation of raw data to a frequency distribution is typically a means to some greater end. The following examples illustrate how frequency distributions can be used to describe, explore, and compare data sets. (The following section shows how the construction of a frequency distribution is often the first step in the creation of a graph that visually depicts the nature of the distribution.) EXAMPLE Describing Data Refer to Data Set 1 in Appendix B for the pulse rates of 40 randomly selected adult males. Table 2-5 summarizes the last digits of those pulse rates. If the pulse rates are measured by counting the number of heartbeats in 1 minute, we expect that those last digits should occur with frequencies that are roughly the same. But note that the frequency distribution shows that the last digits are all even numbers; there are no odd numbers present. This suggests that the pulse rates were not counted for 1 minute. Perhaps they were counted for 30 seconds and the values were then doubled. (Upon further examination of the original pulse rates, we can see that every original value is a multiple of four, suggesting that the number of heartbeats was counted for 15 seconds, then that count was multiplied by four.) It s fascinating to learn something about the method of data collection by simply describing some characteristics of the data.
2-2 Distributions 43 Table 2-4 Cumulative Distribution of Cotinine Levels in Smokers Table 2-5 Last Digits of Male Pulse Rates Cotinine Cumulative Last Digit Less than 100 11 Less than 200 23 Less than 300 37 Less than 400 38 Less than 500 40 0 7 1 0 2 6 3 0 4 11 5 0 6 9 7 0 8 7 9 0 EXAMPLE Exploring Data In studying the behavior of the Old Faithful geyser in Yellowstone National Park, geologists collect data for the times (in minutes) between eruptions. Table 2-6 summarizes actual data that were obtained. Examination of the frequency distribution reveals unexpected behavior: The distribution of times has two different peaks. This distribution led geologists to consider possible explanations. EXAMPLE Comparing Data Sets The Chapter Problem given at the beginning of this chapter includes data sets consisting of measured cotinine levels from smokers, nonsmokers exposed to tobacco smoke, and nonsmokers not exposed to tobacco smoke. Table 2-7 shows Table 2-7 Cotinine Levels for Three Groups Nonsmokers Nonsmokers Not Cotinine Smokers Exposed to Smoke Exposed to Smoke Table 2-6 Times (in minutes) Between Old Faithful Eruptions Time 40 49 8 50 59 44 60 69 23 70 79 6 80 89 107 90 99 11 100 109 1 0 99 28% 85% 95% 100 199 30% 5% 0% 200 299 35% 3% 3% 300 399 3% 3% 3% 400 499 5% 0% 0% 500 599 0% 5% 0% continued
44 C HAPTER 2 Describing, Exploring, and Comparing Data the relative frequencies for the three groups. By comparing those relative frequencies, it should be obvious that the frequency distribution for smokers is very different from the frequency distributions for the other two groups. Because the two groups of nonsmokers (exposed and not exposed) have such high frequency amounts for the first class, it might be helpful to further compare those data sets with a closer examination of those values. 2-2 Basic Skills and Concepts In Exercises 1 4, identify the class width, class midpoints, and class boundaries for the given frequency distribution based on Data Set 1 in Appendix B. 1. Systolic Blood 2. Pressure of Men Systolic Blood Pressure of Women Table for Exercise 13 Outcome 1 27 2 31 3 42 4 40 5 28 6 32 Table for Exercise 14 Digit 0 18 1 12 2 14 3 9 4 17 5 20 6 21 7 26 8 7 9 16 90 99 1 100 109 4 110 119 17 120 129 12 130 139 5 140 149 0 150 159 1 3. Cholesterol of Men 4. 0 199 13 200 399 11 400 599 5 600 799 8 800 999 2 1000 1199 0 1200 1399 1 80 99 9 100 119 24 120 139 5 140 159 1 160 179 0 180 199 1 Body Mass Index of Women 15.0 20.9 10 21.0 26.9 15 27.0 32.9 11 33.0 38.9 2 39.0 44.9 2 In Exercises 5 8, construct the relative frequency distribution that corresponds to the frequency distribution in the exercise indicated. 5. Exercise 1 6. Exercise 2 7. Exercise 3 8. Exercise 4 In Exercises 9 12, construct the cumulative frequency distribution that corresponds to the frequency distribution in the exercise indicated. 9. Exercise 1 10. Exercise 2 11. Exercise 3 12. Exercise 4 13. Loaded Die The author drilled a hole in a die and filled it with a lead weight, then proceeded to roll it 200 times. (Yes, the author has too much free time.) The results are given in the frequency distribution in the margin. Construct the corresponding relative frequency distribution and determine whether the die is significantly different from a fair die that has not been loaded. 14. Lottery The frequency distribution in the margin is based on the Win Four numbers from the New York State Lottery, as listed in Data Set 26 in Appendix B. Construct the corresponding relative frequency distribution and determine whether the results appear to be selected in such a way that all of the digits are equally likely.
2-2 Distributions 45 15. Bears Refer to Data Set 9 in Appendix B and construct a frequency distribution of the weights of bears. Use 11 classes beginning with a lower class limit of 0 and use a class width of 50 lb. 16. Body Temperatures Refer to Data Set 4 in Appendix B and construct a frequency distribution of the body temperatures for midnight on the second day. Use 8 classes beginning with a lower class limit of 96.5 and use a class width of 0.4 F. Describe two different notable features of the result. 17. Head Circumferences Refer to Data Set 3 in Appendix B. Construct a frequency distribution for the head circumferences of baby boys and construct a separate frequency distribution for the head circumferences of baby girls. In both cases, use the classes of 34.0 35.9, 36.0 37.9, and so on. Then compare the results and determine whether there appears to be a significant difference between the two genders. 18. Animated Movies for Children Refer to Data Set 7 in Appendix B. Construct a frequency distribution for the lengths of time that animated movies for children contain tobacco use and construct a separate frequency distribution for the lengths of time for alcohol use. In both cases, use the classes of 0 99, 100 199, and so on. Compare the results and determine whether there appears to be a significant difference. 19. Marathon Runners Refer to Data Set 8 in Appendix B. Construct a relative frequency distribution for the ages of the sample of males who finished the New York City marathon, then construct a separate relative frequency distribution for the ages of the females. In both cases, start the first class with a lower class limit of 19 and use a class width of 10. Compare the results and determine whether there appears to be any notable difference between the two groups. 20. Regular Coke> Diet Coke Refer to Data Set 17 in Appendix B. Construct a relative frequency distribution for the weights of regular Coke by starting the first class at 0.7900 lb and use a class width of 0.0050 lb. Then construct another relative frequency distribution for the weights of diet Coke by starting the first class at 0.7750 lb and use a class width of 0.0050 lb. Then compare the results and determine whether there appears to be a significant difference. If so, provide a possible explanation for the difference. 2-2 Beyond the Basics 21. Interpreting Effects of Outliers Refer to Data Set 20 in Appendix B for the axial loads of aluminum cans that are 0.0111 in. thick. The load of 504 lb is called an outlier because it is very far away from all of the other values. Construct a frequency distribution that includes the value of 504 lb, then construct another frequency distribution with the value of 504 lb excluded. In both cases, start the first class at 200 lb and use a class width of 20 lb. Interpret the results by stating a generalization about how much of an effect an outlier might have on a frequency distribution. 22. Number of Classes In constructing a frequency distribution, Sturges guideline suggests that the ideal number of classes can be approximated by 1 1 (log n) > (log 2), where n is the number of data values. Use this guideline to complete the table for determining the ideal number of classes. Table for Exercise 22 Number Ideal Number of Values of Classes 16 22 5 23 45 6 7 8 9 10 11 12