City Number Pct. 1.2 STEMS AND LEAVES

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1 1.2 STEMS AND LEAVES Think back on the preceding example. We dealt with a list of cities giving their populations and areas. Usually the science of statistics does not concern itself with identifying the individuals that were measured in our example, we might have a list of city areas or populations, but the names of the cities might not even appear. In statistics we are not interested in such questions as which city is the largest, the most populous, or the most densely populated. Such questions may be important or interesting, but they are dealt with outside of statistics. In statistics we are interested in the general aspects of the measured data. Are the population densities of the world s largest cities all more or less the same, or do they vary widely? What is the general shape, or distribution, of the city population densities? Is there some value of population density that we can consider typical or representative for the most populated cities? In this book we will learn a variety of descriptive and inferential statistical techniques, some based on computing interesting statistics, such as the average population among the most populated cities, and others based on graphical techniques. A thorough statistical analysis usually includes both the computation of interesting statistics and the production of informative graphs. Such analysis is usually partly descriptive of the data and partly inferential, in that conclusions concerning the data are usually drawn. The stem-and-leaf approach is a graphical method used to describe the general shape, or distribution, of the data. It is distinguished by the fact that

2 Table 1.6 Notable Buildings of the United States and Canada City Building Number of stories 1. Atlanta Peachtree Center Austin One American Center Boston John Hancock Tower Calgary Petro Canada Center Chicago Sears Tower Cleveland Society Center Detroit Westin Hotel Edmonton Manulife Place Honolulu Imperial Plaza Los Angeles First Interstate World Center New York City World Trade Center I Seattle Columbia Seafirst Center Toronto First Canada Place Vancouver Royal Center Tower 36 the value of every piece of data used to construct the graph can in fact be read off the graph. When we have unorganized pieces of numerical information, it is very helpful (and sometimes necessary) to organize them so that the reader can quickly see certain aspects of the data. Table 1.6 provides the number of stories in 14 of the tallest buildings in the United States and Canada. These data are ordered alphabetically by city. Thus, the table is not designed to enable us to see important patterns in the data. For example, if we want to see at a glance the largest number and the smallest number of stories, we would have to reorder the data, as we did with the city population data in Section 1.1. Nor can we easily see from such a table how the building heights are grouped. Are they tightly clustered around one value, or are they spread out over a wide range of values? Is one height typical? What is the general shape of the data? We cannot easily tell from the table. There is another way of organizing data that is easy to do, and it makes the data easier to understand and discuss. It is the stem-and-leaf plot, one of the standard graphical techniques used in statistics. The stem-and-leaf plot will be seen below to be a special kind of histogram, which is perhaps the most widely used graphical display of tables of numbers, such as city populations or building heights. So it is important to realize that introducing the stem-and-leaf plot is this textbook s way of introducing histograms. The stem-and-leaf plot s main virtue is that no information about the data is lost while it carries out its desired role of showing the statistician the shape of the data, that is, how frequently the data occur in various ranges of values. This easy way of organizing data was developed by the well-known American statistician John Tukey of Princeton University.

3 Here is how the stem-and-leaf plot works for the above data set. The first step is to group the data by 10s, 100s, or 1000s, and so on, so that we obtain a reasonable number of stems, say, 5 to 15 of them. Thus in this case we see from the data in Table 1.6 that we can group them by 10s. That is, we can group the 30s together, the 40s together, and so on, using a list such as the following. We start with a 1, which stands for the group of buildings having from 10 to 19 stories. The 2 stands for the group from 20 to 29 stories high. These numbers are used to form the first column in the stem-and-leaf plot below. These numbers that stand for 10, 20, 30, and so on, are called stems. Now we start writing down the data on the number of stories in the buildings. Let s start with these two buildings: Peachtree Center, Atlanta One American Center, Austin 63 stories 32 stories We list those two data points as follows: For 63, we have separated the 6, a stem (which in this table stands for 60), from 3, which is called a leaf. In order to read this table, we need a key. Here is a key to this table: Key: 6 3 in the above stem-and-leaf plot stands for 63. The key tells us how to read the table. Here the key reminds us that in this table the stems are 10s and the leaves are units. When you make a stem-and-leaf table, you should always provide a key. We now add the rest of the data. The completed stem-and-leaf plot is presented in Table 1.7. A stem-and-leaf plot has two very attractive properties. First, it contains all the numerical data that were in the original table. That is, we can write down exactly each data point from the stem-and-leaf diagram. In some

4 Table 1.7 -and- Plot of Number of Stories of Notable Buildings ,6, ,7 6 3,0 7 1,3,2, ,0 12 Key: 6 3 stands for 63. other kinds of plots the numbers that were used to make the plot are not recoverable from the plot if you want to read the actual numbers, you must go back to the original list of data (which may not be possible if, for example, you have come across the graph in a publication). Second, and powerfully, we now see the general centering, spread, and shape of the data at a glance. For example, they seem to center around 50 stories, and they are mostly spread between 30 and 70 stories but have two extreme values (later to be called outliers) at 110 stories. We sometimes need to use certain tricks to construct stem-and-leaf plots for special sets of data. For example, sometimes every choice of stem size (10s, 100s, etc.) produces either too many or too few stems to adequately show the general shape of the data set. Consider the following data. Example 1.3 Newly Minted Coins A sample of 25 pennies from the U.S. mint were weighed on a balance. Here are the results, in centigrams ( of a gram): A stem-and-leaf can be used to display these weights in a useful manner. As in the example above, we can group the data by 10s, using , , and Thus the stems become 30, 31, 32. See Table 1.8.

5 Table 1.8 -and- Plot of Weights of Pennies 30 1,8,4,9,7,5,9,2 31 4,6,0,1,7,6,0,2,0,3,2,2,4,3,1, Key: 30 1 stands for 301. We see that the range of measurements is rather small from a low weight of 301 centigrams to a high weight of 320 centigrams. Thus, there is a problem with this table. The data are scrunched together with only three stems, so much so that we find it hard to get a good idea about the general shape of the data, for example. We can overcome this difficulty by stretching out the table. But first, let s put the leaves of the table in order. If time permits, that is usually a desirable step, even if we don t plan to split the stem intervals. 30 1,2,4,5,7,8,9,9 31 0,0,0,1,1,2,2,2,3,3,4,4,6,6,7, We can divide each interval into two parts. (Clearly, the only other choice is to split the interval into five parts: , , and so on. This seems to create too many stems.) For example, for the first row we can write the following: 30 1,2,4 30 5,7,8,9,9 The asterisk (*) means that the interval is continued from the previous row; that is, we now have two rows with the same stem. So the top row includes values from 300 to 304, and the second row includes the remaining values, in the interval from 305 to 309 (intervals intentionally chosen to be of equal length). Table 1.9 is the expanded table. We now get a much better look at the spread of the data, and in particular we find that the quite narrow interval from 310 to 314 contains about one-half of all the weights (12 out of 25). We also see a pattern, very typical in data sets, in which many of the data pile up in the middle, providing a somewhat bell-shaped stem-and-leaf plot. Note that this bell-shaped nature of the data was not evident from Table 1.8. Most standard statistical computer programming packages for practitioners have a stem-and-leaf program. Clearly, the user has to decide on the number of stem intervals. We will not get into a detailed discussion of

6 SECTION 1.2 EXERCISES Table 1.9 Expanded -and- Plot of Data of Table ,2,4 30 5,7,8,9,9 31 0,0,0,1,1,2,2,2,3,3,4,4 31 6,6,7, Key: 32 0 stands for 320. such judgments, but some of the exercises that follow help with this. The important thing is to be familiar with this powerful and often used tool. 1. The following scores were obtained by 30 d. For the plot in (b), which stem has the most students on a final exam in statistics. leaves? The stem-and-leaf plot that follows gives stop ping distances (in meters) of 10 test cars go ing 50 kilometers per hour when the brakes were applied. What are the original data from a. Construct a stem-and-leaf plot of these data. Be sure to give the key. b. Which stem has the most leaves? c. Which of the following scores is closest to which the table was made? the center of the data? 27, 36, 46, 55, ,1,4,4,8 d. Is the value in part (c) in the stem from part 7 1,3,5 (b)? 8 0,2 2. The following weights of a group of rats used Key: 6 0 stands for 60. in an agricultural research project are reported to the nearest gram: a. Construct a stem-and-leaf plot using the hundreds place for the stems and the tens place for the leaves (so that there are two stems). b. Construct another stem-and-leaf plot with the tens as the stems, so that there are five stems. c. Which plot gives a better picture of the data? 4. The following stem-and-leaf plot gives final scores on a statistics test for 20 students. From the table, write down each student s score. 7 2,3,4,5,9 8 0,2,3,4,4,5,7,8 9 0,1,2,3,5,6,6 Key: 8 2 stands for Using the stem-and-leaf plot of Exercise 4, answer the following questions: a. What was the lowest score obtained? b. What was the highest score obtained?

7 c. How far apart were the lowest and highest a. Construct a stem-and-leaf plot of the high scores? (This statistic is called the range of temperatures. the data.) b. How far apart were the lowest and highest 6. The following table gives the high temperatures temperatures for these cities? ( F) for 20 cities on April 3, Refer to Table 1.5 on population densities. Asheville, North Carolina 72 a. What stems would you use to make a stem- Champaign, Illinois 68 and-leaf plot? Indianapolis, Indiana 69 b. Draw a stem-and-leaf plot for the data. Abilene, Texas 67 c. What does this plot tell you about what the Los Cruces, New Mexico 72 data looks like? Los Angeles, California 71 For additional exercises, see page 713. Billings, Montana 56 Chicago, Illinois 65 Albany, New York 53 Charleston, South Carolina 77 Miami, Florida 77 Birmingham, Alabama 74 Iowa City, Iowa 63 Detroit, Michigan 58 Molokai, Hawaii 77 St. Louis, Missouri 76 Lincoln, Nebraska 63 Boston, Massachusetts 53 Baltimore, Maryland 66 Boise, Idaho 60