A stem-and-leaf display is a device for presenting quantitative data in a graphical format in order to assist in visualizing the shape of a distribution. This graphical technique evolved from Arthur Bowley's work in the early 1900s, and it is a useful tool in exploratory data analysis. A stem-and-leaf display is often called a *stemplot** *(although, the latter term more specifically refers to another chart type).

Stem-and-leaf displays became more commonly used in the 1980s after the publication of John Tukey 's book on exploratory data analysis in 1977. The popularity during those years is attributable to the use of monospaced (typewriter) typestyles that allowed computer technology of the time to easily produce the graphics. However, the superior graphic capabilities of modern computers have lead to the decline of stem-and-leaf displays.

While similar to histograms, stem-and-leaf displays differ in that they retain the original data to at least two significant digits and put the data in order, thereby easing the move to order-based inference and non-parametric statistics.

## Construction of Stem-and-Leaf Displays

A basic stem-and-leaf display contains two columns separated by a vertical line. The left column contains the *stems* and the right column contains the *leaves*. To construct a stem-and-leaf display, the observations must first be sorted in ascending order. This can be done most easily, if working by hand, by constructing a draft of the stem-and-leaf display with the leaves unsorted, then sorting the leaves to produce the final stem-and-leaf display. Consider the following set of data values:

It must be determined what the stems will represent and what the leaves will represent. Typically, the leaf contains the last digit of the number and the stem contains all of the other digits. In the case of very large numbers, the data values may be rounded to a particular place value (such as the hundreds place) that will be used for the leaves. The remaining digits to the left of the rounded place value are used as the stem. In this example, the leaf represents the ones place and the stem will represent the rest of the number (tens place and higher).

The stem-and-leaf display is drawn with two columns separated by a vertical line. The stems are listed to the left of the vertical line. It is important that each stem is listed only once and that no numbers are skipped, even if it means that some stems have no leaves. The leaves are listed in increasing order in a row to the right of each stem. Note that when there is a repeated number in the data (such as two values of

Now, let's consider a data set with both negative numbers and numbers that need to be rounded:

For negative numbers, a negative is placed in front of the stem unit, which is still the value

## Applications of Stem-and-Leaf Displays

Stem-and-leaf displays are useful for displaying the relative density and shape of data, giving the reader a quick overview of distribution. They retain (most of) the raw numerical data, often with perfect integrity. They are also useful for highlighting outliers and finding the mode.

However, stem-and-leaf displays are only useful for moderately sized data sets (around 15 to 150 data points). With very small data sets, stem-and-leaf displays can be of little use, as a reasonable number of data points are required to establish definitive distribution properties. With very large data sets, a stem-and-leaf display will become very cluttered, since each data point must be represented numerically. A box plot or histogram may become more appropriate as the data size increases.

## Stem-and-Leaf Display

This is an example of a stem-and-leaf display for EPA data on miles per gallon of gasoline.