Data needs to be organized so that we can get useful information. You may be interested in NBA and wish to know how often players score during a match. You may also be interested in the range of successful shots. In order to obtain this information, data has to be ordered and grouped. This is the purpose of stem and leaf diagrams.

Introduction Stem-and-Leaf Graphs Definition

Stem-and-leaf graphs are a way of organizing data, where we split it into its stem (first digit) and the leaf (last digit).

We use stem-and-leaf graphs to give us a view of the skewness of the data and compare it to other distributions.

Distribution is the shape of the graph when all the data is plotted.

Data can be either positively skewed, negatively skewed, or have no skewness. Positively skewed data is when the extreme values are on the right-hand side. Negatively skewed data is when the extreme values are on the left-hand side. No skewness is when the extreme values are evenly distributed on both sides.

Construct a Stem-and-Leaf Graph

Step 1-Identify your stems and draw out the graph. The stem is the digit that appears on the left of the graph. The stem values are a way of categorising the data. For example, if we have the data values 63, 68, 45, 48, 52 and 57, we could have 4, 5 and 6 as stems to represent the values that are in the 40s, 50s or 60s.
Step 2 -Add in your leaves. Once your graph has been drawn, you can simply fill the leaves in to display the data. It is important however to note that the leaves must appear in order of size. So, if you have 45, 48 and 43, the leaves should be displayed 3, 5 and then 8.
Step 3- Write your key. What does 4|3 mean? Does it mean 43, 4.3 or something else? If 4|3 means 43, we write “4|3 means 43” for the key.

In this section we will explore different types of stem and leaf diagrams.

Single Stem-and-Leaf Graphs

A single stem-and-leaf graphs divides the data into two parts: the stem and the leaf. The first digit is the stem and the second digit it the leaf.

Here we are analyzing the distribution of one data set.

Express 12, 15, 17, 22, 25, 27, 29, 34, 35, 38 and 39 as a stem-and-leaf graph.

Solution:

\[ \begin{array}{c|c} \text{stem} & \text{leaf}\\ \hline 1 & 2,5,7\\ \hline 2 & 5,7,9 \\ \hline 3 & 4,5,6,9 \\ \hline \end{array} \]

Back-to-Back Stem-and-Leaf Graphs

We use back-to-back stem and leaf graphs to compare two distributions. Both distributions share the same stem.

We can compare two distributions.

Draw a back-to-back stem-and-leaf graph for the following distributions:

Distribution A: 11,13,14,24,25,25,31,32,32,34,35.

Distribution B: 12,15,17,18,19,25,27,29,34,35,38.

Solution

\[ \begin{array}{c|c|c} \text{Stem (A)} & \text{leaf} & \text{Stem (B)}\\ \hline 1,3,4 & 1 & 2,5,7,8,9\\ \hline 4,5,5 &2& 5,7,9 \\ \hline 1,2,2,4 & 3 & 4,5,8,9 \\ \hline \end{array} \]

We can use a key for example 1|2|3 which means 21 in distribution A and 23 in distribution B

The back-to-back stem and leaf diagram is a useful way to compare skewness.

We can see that stem B has most data on the right side of the mean (positively skewed) whereas stem A has most data on the left side of the mean (negatively skewed).

Stem-and-Leaf Plot Statistics

Stem-and-leaf diagrams can help you to quickly find statistics.

From the stem-and-leaf graph, we can find the median, range, upper quartile, lower quartiles, and interquartile range. The median is the midway point of the data set. The range is the difference between the maximum value and the minimum value. The lower quartile is the value under which a quarter of the data is found when they are arranged in increasing order. The upper quartile is the value under which three-quarters of the data is found when they are arranged in increasing order. The interquartile range is the difference between the upper quartile and the lower quartile.

\begin{array}{c|c} \text{stem} & \text{leaf}\\ \hline 1 & 2,5\\ \hline 2 & 5,7,9 \\ \hline 3 & 4,5,6,8 \\ \hline \end{array} Key 1|2 means 12

The range is the difference between the maximum and the minimum.

Range= 38 - 12 = 26

Median is the the middle value. There are 9 data points. The middle value will be the 5th data point which is 29.

The lower quartile is the value where 25% of the data is equal or below. This is the third data point which is 25.

The upper quartile is the value where 75% of the data is equal or below. This is the 7th data point which is 35.

The interquartile range is the upper quartile - lower quartile = 35-25 = 10.

Example of Stem-and-Leaf Graphs

In this section we will look at examples of stem-and-leaf diagrams.

Below is the stem and leaf diagram for the number of pair of shoes 11 households have

\[ \begin{array}{c|c} \text{stem} & \text{leaf}\\ \hline 0 & 1,7,9\\ \hline 1 & 5,7,9,10 \\ \hline 2 & 2 \\ \hline 3 & 4,5,8,9 \\ \hline \end{array} \]

key 1|5 means 15 pairs of shoes

Stem and leaf diagrams can be used for non integer numbers e.g 1.2 ,3.4, 3.5,3.6, 4.2, 4.5

The stem and leaf diagram would be the following: \begin{array}{c|c} \text{stem} & \text{leaf}\\ \hline 1 & 2\\ \hline 3 & 4,5,6 \\ \hline 4 & 2,5 \\ \hline \end{array} Key 1|2 means 1.2

Stem-and-Leaf Graphs - Key takeaways

Stem and leaf diagrams can be used for one or two data sets
We can find the median, range, upper quartile, lower quartiles and the interquartile range.
We can compare the skewness between two data sets.