## Introduction

If a given data is ungrouped, we can use various methods of showing the frequency with which certain classes of values appear. For example, let’s say we have the following list of numbers: 14, 15, 25, 27, 32, 37, 38, 41, 42, 42, 45.

In order to represent the frequencies, we can draw a frequency distribution table:

Furthermore, we can also draw a histogram:

But, disadvantage of the frequency table and histogram is that we only have informations about frequencies, while we don’t see the given data points. This is why we use **stem and leaf plot**.

## Stem and leaf plot (S – L plot)

A stem and leaf plot is a special table with two columns. The main idea is to divide each data point into a ‘stem’ and ‘leaf’. The left – side column is the ‘stem’ and the right – side column is the ‘leaf’. The ‘stems’ are the digits that given numbers have in common, while the ‘leaves’ are what differs them. For instance, the S – L plot for the previous example would be:

In the ‘stem’ column are tens digits; $1, 2, 3$ and $4$. In the ‘leaf’ column are the ones digits.

## Examples

In the first example we practice reading stem and leaf plots.

* Example 1: *The following stem and leaf plot shows the number of sunglasses in each of the stores of some chain of stores.

How many stores have fewer than $26$ sunglasses?

**Solution:**

For instance, $2 \vert 0$ means $20$ sunglasses. Therefore, $5$ stores have fewer than $26$ sunglasses. Those stores have $8, 11, 13, 16$ and $25$ sunglasses.

In the following examples we practice drawing stem and leaf plots.

* Example 2: *Draw a stem and leaf plot for the following list of scores on some test:

$$43, 67, 72, 85, 48, 61, 62, 58, 89, 93.$$

**Solution:**

The ‘stem’ values are the tens digits and the ‘leaves’ values are the ones digits. We can place the numbers in order from smallest to largest, but it is not necessary.

* Example 3: *The height of the students was measured and the data (in

*cm*) is:

$$182 \ 153 \ 164 \ 173 \ 184 \ 175 \ 180 \ 155$$

$$201 \ 177 \ 180 \ 183 \ 186 \ 188 \ 182 \ 178$$

$$169 \ 168 \ 173 \ 159 \ 152 \ 162 \ 163 \ 185.$$

Draw the stem and leaf plot.

**Solution:**

* Example 4: *Draw a stem and leaf plot for the following list of numbers:

$$21.2, 23.4, 20.7, 32.4, 33.8, 21.3, 32.3, 32.6, 23.5, 34.9, 20.5.$$

**Solution:**

For instance, $33 \vert 8$ means $33.8$.

* Example 5: *Draw a stem and leaf plot for the following list of numbers:

$$-14.9, -15.0, -26.3, 32.8, 23.6, 10.1, 11.2, 33.3, 10.0, 24.1, 10.9, 22.9.$$

**Solution:**

Obviously, the numbers range from $-26.3$ to $33.3$. First, we should round the numbers. For example, $33.3$ is rounded to $33$ and shown with the stem of $3$ and leaf of $3$. Similarly, $-14.9$ is rounded to $-15$ and so on.

The stem and leaf plot is:

As we can see, stem and leaf plots are useful because they give us a quick overview of a distribution. Furthermore, by looking at stem and leaf plot, we notice where the majority of data points lie. On the other hand, stem and leaf plots are used only for small data sets.