Three Popular Data Displays

This section elaborates on how to describe data. In particular, you will learn about the relative frequency histogram. Complete the exercises and check your answers.

Stem and Leaf Diagrams

Suppose 30 students in a statistics class took a test and made the following scores:

$\begin{array}{rrrrrrrrrr}86 & 80 & 25 & 77 & 73 & 76 & 100 & 90 & 69 & 93 \\ 90 & 83 & 70 & 73 & 73 & 70 & 90 & 83 & 71 & 95 \\ 40 & 58 & 68 & 69 & 100 & 78 & 87 & 97 & 92 & 74\end{array}$

How did the class do on the test? A quick glance at the set of 30 numbers does not immediately give a clear answer. However the data set may be reorganized and rewritten to make relevant information more visible. One way to do so is to construct a stem and leaf diagram as shown in Figure 2.1 "Stem and Leaf Diagram". The numbers in the tens place, from 2 through 9, and additionally the number 10, are the "stems," and are arranged in numerical order from top to bottom to the left of a vertical line. The number in the units place in each measurement is a "leaf," and is placed in a row to the right of the corresponding stem, the number in the tens place of that measurement. Thus the three leaves 9, 8, and 9 in the row headed with the stem 6 correspond to the three exam scores in the 60s, 69 (in the first row of data), 68 (in the third row), and 69 (also in the third row). The display is made even more useful for some purposes by rearranging the leaves in numerical order, as shown in Figure 2.2 "Ordered Stem and Leaf Diagram". Either way, with the data reorganized certain information of interest becomes apparent immediately. There are two perfect scores; three students made scores under 60; most students scored in the 70s, 80s, and 90s; and the overall average is probably in the high 70s or low 80s.

Figure 2.1 Stem and Leaf Diagram

Figure 2.2 Ordered Stem and Leaf Diagram

In this example the scores have a natural stem (the tens place) and leaf (the ones place). One could spread the diagram out by splitting each tens place number into lower and upper categories. For example, all the scores in the 80s may be represented on two separate stems, lower 80s, and upper 80s:

$\begin{array}{l|lll} 8 & 0 & 3 & 3 \\ 8 & 6 & 7 \end{array}$

The definitions of stems and leaves are flexible in practice. The general purpose of a stem and leaf diagram is to provide a quick display of how the data are distributed across the range of their values; some improvisation could be necessary to obtain a diagram that best meets that goal.

Note that all of the original data can be recovered from the stem and leaf diagram. This will not be true in the next two types of graphical displays.