# 2.3: Stem and Leaf Displays

- Page ID
- 2084

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Learning Objectives

- Create and interpret basic stem and leaf displays
- Create and interpret back-to-back stem and leaf displays
- Judge whether a stem and leaf display is appropriate for a given data set

A stem and leaf display is a graphical method of displaying data. It is particularly useful when your data are not too numerous. In this section, we will explain how to construct and interpret this kind of graph.

As usual, an example will get us started. Consider Table \(\PageIndex{1}\) that shows the number of touchdown passes (TD passes) thrown by each of the \(31\) teams in the National Football League in the \(2000\) season.

**Table \(\PageIndex{1}\): **Number of touchdown passes

\[\begin{matrix} 37 & 33 & 33 & 32 & 29 & 28 & 28 & 23 & 22\\ 22 & 22 & 21 & 21 & 21 & 20 & 20 & 19 & 19\\ 18 & 18 & 18 & 18 & 16 & 15 & 14 & 14 & 14\\ 12 & 12 & 9 & 6 & & & & & \end{matrix}\]

A stem and leaf display of the data is shown in Figure \(\PageIndex{1}\). The left portion of Figure \(\PageIndex{1}\) contains the stems. They are the numbers \(3, 2, 1,\; and\; 0\), arranged as a column to the left of the bars. Think of these numbers as \(10’s\) digits. A stem of \(3\), for example, can be used to represent the \(10’s\) digit in any of the numbers from \(30\) to \(39\). The numbers to the right of the bar are leaves, and they represent the \(1’s\) digits. Every leaf in the graph therefore stands for the result of adding the leaf to \(10\) times its stem.

\[\begin{array}{c|c c c c c c c c c c c c c} 3 &2 &3 &3 &7\\ 2 &0 &0 &1 &1 &1 &2 &2 &2 &3 &8 &8 &9\\ 1 &2 &2 &4 &4 &4 &5 &6 &8 &8 &8 &8 &9 &9\\ 0 &6 &9\\ \end{array}\]

**Figure \(\PageIndex{1}\): **Stem and leaf display of the number of touchdown passes

To make this clear, let us examine Figure \(\PageIndex{1}\) more closely. In the top row, the four leaves to the right of stem \(3\) are \(2, 3, 3,\; and\; 7\). Combined with the stem, these leaves represent the numbers \(32, 33, 33,\; and\; 37\), which are the numbers of TD passes for the first four teams in Table \(\PageIndex{1}\). The next row has a stem of \(2\) and \(12\) leaves. Together, they represent \(12\) data points, namely, two occurrences of \(20\) TD passes, three occurrences of \(21\) TD passes, three occurrences of \(22\) TD passes, one occurrence of \(23\) TD passes, two occurrences of \(28\) TD passes, and one occurrence of \(29\) TD passes. We leave it to you to figure out what the third row represents. The fourth row has a stem of \(0\) and two leaves. It stands for the last two entries in Table \(\PageIndex{1}\), namely \(9\) TD passes and \(6\) TD passes. (The latter two numbers may be thought of as \(09\) and \(06\).)

One purpose of a stem and leaf display is to clarify the shape of the distribution. You can see many facts about TD passes more easily in Figure \(\PageIndex{1}\) than in Table \(\PageIndex{1}\). For example, by looking at the stems and the shape of the plot, you can tell that most of the teams had between \(10\) and \(29\) passing TDs, with a few having more and a few having less. The precise numbers of TD passes can be determined by examining the leaves.

We can make our figure even more revealing by splitting each stem into two parts. Figure \(\PageIndex{2}\) shows how to do this. The top row is reserved for numbers from \(35\) to \(39\) and holds only the \(37\) TD passes made by the first team in Table \(\PageIndex{1}\). The second row is reserved for the numbers from \(30\) to \(34\) and holds the \(32, 33,\; and\; 33\) TD passes made by the next three teams in the table. You can see for yourself what the other rows represent.

\[\begin{array}{c|c c c c c c c c c c c c c} 3 &7\\ 3 &2 &3 &3 \\ 2 &8 &8 &9 \\ 2 &0 &0 &1 &1 &1 &2 &2 &2 &3 \\ 1 &5 &6 &8 &8 &8 &8 &9 &9\\ 1 &2 &2 &4 &4 &4 \\ 0 &6 &9 \end{array}\]

**Figure \(\PageIndex{2}\): **Stem and leaf display with the stems split in two

Figure \(\PageIndex{2}\) is more revealing than Figure \(1PageIndex{2}\) because the latter figure lumps too many values into a single row. Whether you should split stems in a display depends on the exact form of your data. If rows get too long with single stems, you might try 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 placed back to back along a common column of stems. The result is a “back-to-back stem and leaf graph.” Figure \(\PageIndex{3}\) shows such a graph. It compares the numbers of TD passes in the \(1998\) and \(2000\) seasons. The stems are in the middle, the leaves to the left are for the \(1998\) data, and the leaves to the right are for the \(2000\) data. For example, the second-to-last row shows that in 1998 there were teams with \(11, 12,\; and\; 13\) TD passes, and in \(2000\) there were two teams with \(12\) and three teams with \(14\) TD passes.

\[\begin{array}{c|c|c c c c c c c c c c} \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; 1\; 1 & 4 \\ &3 &7\\ \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; 3\; 3\; 2 &3 &2 &3 &3\\ \; \; \; \; \; \; \; \; \; \; \; \; \; 8\; 8\; 6\; 5 &2 &8 &8 &9\\ \; \; 4\; 4\; 3\; 3\; 1\; 1\; 1\; 0 &2 &0 &0 &1 &1 &1 &2 &2 &2 &3\\ 9\; 8\; 7\; 7\; 7\; 6\; 6\; 6\; 5 &1 &5 &6 &8 &8 &8 &8 &9 &9\\ \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; 3\; 2\; 1 &1 &2 &2 &4 &4 &4 &4\\ \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; 7 &0 &6 &9 \end{array}\]

* Figure \(\PageIndex{3}\): Back-to-back stem and leaf display*.

The left side shows the \(1998\) TD data and the right side shows the \(2000\) TD data. Figure \(\PageIndex{3}\) helps us see that the two seasons were similar, but that only in \(1998\) did any 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 to whole numbers that can be represented with a one-digit stem and a one-digit leaf. Second, all the numbers are positive. If the data include numbers with three or more digits, or contain decimals, they can be rounded to two-digit accuracy. Negative values are also easily handled. Let us look at another example.

Table \(\PageIndex{2}\) shows data from the case study Weapons and Aggression. Each value is the mean difference over a series of trials between the times it took an experimental subject to name aggressive words (like “punch”) under two conditions. In one condition, the words were preceded by a non-weapon word such as "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 of the aggressive word by the weapon word. Negative differences imply that the priming by the weapon word was less than for a neutral word.

**Table \(\PageIndex{2}\): **The effects of priming (thousandths of a second)

\[\begin{matrix} 43.2 & 42.9 & 35.6 & 25.6 & 25.4 & 23.6 & & \\ 20.5 & 19.9 & 14.4 & 12.7 & 11.3 & 10.2 & & \\ 10.0 & 9.1 & 7.5 & 5.4 & 4.7 & 3.8 & 2.1 & 1.2\\ -0.2 & -6.3 & -6.7 & -8.8 & -10.4 & -10.5 & & \\ -14.9 & -14.9 & -15.0 & -18.5 & -27.4 \end{matrix}\]

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

The data are displayed with stems and leaves in Figure \(\PageIndex{4}\). Since stem and leaf displays can only portray two whole digits (one for the stem 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 \(4\) and a leaf of \(3\). Similarly, \(42.9\) is rounded to \(43\). To represent negative numbers, we simply use negative stems. For example, 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 Table \(\PageIndex{2}\).

\[\begin{array}{c|c c c c c c c} 4 & 3 & 3 \\ 3 &6 \\ 2 &0 &0 &4 &5 &6\\ 1 &0 &0 &1 &3 &4\\ 0 &1 &2 &4 &5 &5 &8 &9\\ -0 &0 &6 &7 &9\\ -1 &0 &0 &5 &5 &5 &9\\ -2 &7 \end{array}\]

**Figure \(\PageIndex{4}\): **Stem and leaf display with negative numbers and rounding

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 \(9\), whereas 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 as possible between the "\(0\)" and "\(-0\)" rows. In Table \(\PageIndex{2}\), none of the values are \(0\) before rounding. The "\(0\)" that appears in the "\(-0\)" row comes from the original value of \(-0.2\) in the table.

Although stem and leaf displays are unwieldy for large data sets, they are often useful for data sets with up to \(200\) observations. Figure \(\PageIndex{5}\) portrays the distribution of populations of \(185\) US cities in \(1998\). To be included, a city had to have between \(100,000\) and \(500,000\) residents.

\[\begin{array}{c|ccccccccccccccccccccccccccccccccccccccccccc} 4 &8 &9 &9 \\ 4 &6 \\ 4 &4 &4 &5 &5\\ 4 &3 &3 &3\\ 4 &0 &1\\ 3 &9 &9\\ 3 &6 &7 &7 &7 &7 &7\\ 3 &5 &5\\ 3 &2 &2 &3\\ 3 &1 &1 &1\\ 2 &8 &8 &9 &9\\ 2 &6 &6 &6 &6 &6 &7\\ 2 &4 &4 &4 &4 &5 &5\\ 2 &2 &2 &3 &3 &3\\ 2 &0 &0 &0 &0 &0 &0\\ 1 &8 &8 &8 &8 &8 &8 &8 &8 &8 &8 &8 &8 &9 &9 &9 &9 &9 &9 &9 &9 &9 &9 &9\\ 1 &6 &6 &6 &6 &6 &6 &7 &7 &7 &7 &7 &7\\ 1 &4 &4 &4 &4 &4 &4 &4 &4 &4 &4 &4 &4 &5 &5 &5 &5 &5 &5 &5 &5 &5 &5 &5 &5\\ 1 &2 &2 &2 &2 &2 &2 &2 &2 &2 &2 &2 &2 &2 &2 &2 &2 &2 &2 &2 &3 &3 &3 &3 &3 &3 &3 &3 &3\\ 1 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 \end{array}\]

**Figure \(\PageIndex{5}\):** Stem and leaf display of populations of \(185\) US cities with populations between \(100,000\) and \(500,000\) in \(1998\).

Since a stem and leaf plot shows only two-place accuracy, we had to round the numbers to the nearest \(10,000\). For example, the largest 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 \(100,000\) and the 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 of important information. Also, their extreme values must fit into two successive digits, as the data in Figure \(\PageIndex{5}\) fit into the \(10,000\) and \(100,000\) places (for leaves and stems, respectively). Deciding what kind of graph is best suited to displaying your data thus requires good judgment. Statistics is not just recipes!