2.5 Stem and leaf displays  (Page 2/2)

The with stem and leaf split in two is more revealing than the simpler before because the simpler table lumps too many values into a single row. Whetheryou should split stems in a display depends on the exact form of your data. If rows get too long with single stems, you mighttry splitting them into two or more parts.

There is a variation of stem and leaf displays that is useful for comparing distributions. The two distributions are placedback to back along a common column of stems. The result is a back to back stem and leaf graph . The below shows such a graph. It compares the numbers of TD passes in the 1998 and2000 seasons. The stems are in the middle, the leaves to the left are for the 1998 data, and the leaves to the right are forthe 2000 data. For example, the second-to-last row shows that in 1998 there were teams with 11, 12, and 13 TD passes, and in2000 there were two teams with 12 and three teams with 14 TD passes.

This helps us see that the two seasons were similar, but that only in 1998 didany teams throw more than 40 TD passes.

There are two things about the football data that make them easy to graph with stems and leaves. First, the data are limited towhole numbers that can be represented with a one-digit stem and a one-digit leaf. Second, all the numbers are positive. If thedata include numbers with three or more digits, or contain decimals, they can be rounded to two-digit accuracy. Negativevalues are also easily handled. Let us look at another example.

shows data from a study on aggressive thinking. Each value is the mean difference over a series of trials between the time it took an experimentalsubject to name aggressive words (like "punch") under two conditions. In one condition the words were preceded by anon-weapon word like "rabbit" or "bug." In the second condition, the same words were preceded by a weapon word such as"gun" or "knife." The issue addressed by the experiment was whether a preceding weapon word would speed up (or prime)pronunciation of the aggressive word, compared to a non-weapon priming word. A positive difference implies greater priming ofthe aggressive word by the weapon word. Negative differences imply that the priming by the weapon word was less than for aneutral word.

You see that the numbers range from 43.2 to -27.4. The first value indicates that one subject was 43.2 milliseconds fasterpronouncing aggressive words when they were preceded by weapon words than when preceded by neutral words. The value -27.4indicates that another subject was 27.4 milliseconds slower pronouncing aggressive words when they were preceded by weaponwords.

The data are displayed with stems and leaves in the . Since stem and leaf displays can only portray two whole digits (one for thestem and one for the leaf) the numbers are first rounded. Thus, the value 43.2 is rounded to 43 and represented with a stem of 4and a leaf of 3. Similarly, 42.9 is rounded to 43. To represent negative numbers, we simply use negative stems. Forexample, the bottom row of the figure represents the number -27. The second-to-last row represents the numbers -10, -10, -15,etc. Once again, we have rounded the original values from .

Observe that the figure contains a row headed by "0" and another headed by"-0". The stem of 0 is for numbers between 0 and 9whereas the stem of -0 is for numbers between 0 and -9. For example, the fifth row of the table holds the numbers 1, 2, 4,5, 5, 8, 9 and the sixth row holds 0, -6, -7, and -9. Values that are exactly 0 before rounding should be split as evenly aspossible between the "0" and "-0" rows. In , none of the values are 0 before rounding. The "0" that appears in the "-0" row comes from theoriginal value of -0.2 in the figure.

Although stem and leaf displays are unwieldy for large datasets, they are often useful for datasets with up to 200 observations. portrays the distribution of populations of 185 US cities in 1998. To be included, a cityhad to have between 100,000 and 500,000 residents.

Since a stem and leaf plot shows only two-place accuracy, we had to round the numbers to the nearest 10,000. For example thelargest number (493,559) was rounded to 490,000 and then plotted with a stem of 4 and a leaf of 9. The fourth highest number(463,201) was rounded to 460,000 and plotted with a stem of 4 and a leaf of 6. Thus, the stems represent units of 100,000 andthe leaves represent units of 10,000. Notice that each stem value is split into five parts: 0-1, 2-3, 4-5, 6-7, and 8-9.

Whether your data can be suitably represented by a stem and leaf graph depends on whether they can be rounded without loss ofimportant information. Also, their extreme values must fit into two successive digits, as the data in fit into the 10,000 and 100,000 places (for leaves and stems, respectively). Deciding what kind of graph isbest suited to displaying your data thus requires good judgment. Statistics is not just recipes!