Stem and Leaf Plots

Stem and leaf plots are used to show the frequency with which certain classes of values occur. They are similar to frequency tables or histograms, and are extremely helpful in visualizing the shape of the distribution.

Here is an example of an empty plot:

For every number, the leaf is the ones digit, and is placed in the right column of the table. The stem would be the remaining digits (usually just the tens digit), and is placed in the left column of the table (in the row of its corresponding leaf).

Example

"There are $16$ students in a class. The scores of all the students on their final exams were $96, 71, 78, 83, 65, 89, 100, 87, 69, 90, 72, 85, 74, 63, 94, \text{and } 82$. Make a stem and leaf plot to show the distribution of the exam scores."

First, let's sort this array of numbers in increasing order: $63, 65, 69, 71, 72, 74, 78, 82, 83, 85, 87, 89, 90, 94, 96, 100$.

What are the stems of these numbers? The stems are the leftmost digits of the data values: $6, 7, 8, 9, \text{and }10$. (For two-digit numbers, the stems are the tens digits. Fo three digit numbers, such as $100,$ the stems contain the hundreds and tens digits.)

What are the leaves for each stem? When the stem is $6$, the leaves are $3, 5,$ and $9$ (because the scores in the range of $60-69$ are $63$, $65$, and $69$). When the stem is $7$, the leaves are $1, 2, 4$, and $8$ (because the scores in the range of $70-79$ are $71$, $72$, $74,$ and $78$). When the stem is $8$, the leaves are $2, 3, 5, 7$, and $9$ (because the scores in the range of $80-89$ are $82$, $83$, $85,$ $87,$ and $89$). When the stem is $9$, the leaves are $0, 4$, and $6$ (because the scores in the range of $90-99$ are $90$, $94$, and $96$). When the stem is $10$, the leaf is $0$ (because the score in the range of $100-109$ is $100$).

Solution

The leaves corresponding with a Stem of $1$ will be the values that have a $1$ as a tens digit, or the values that are in the range of $10-19.$ If we look at our data set, we can see that the three numbers fitting that description are $10, 14,$ and $17.$

Stem and Leaf Plots - Construction and Interpretation

A stem and leaf plot is one way to show how often classes of values appear in a data set.

Let's use the dataset below to build a stem-and-leaf plot together.

$$\text{Heights of Sunflowers in Inches: } \\ 13, 71, 21, 20, 24, 18, 40, 32, 39, 11, 14, 25, 35, 21, 21, 19, 28, 43, 52, 56, 41, 29, 27$$

Step One

Before you do ANYTHING, put your data in order from smallest to largest like you do when finding the median or the mode of a data set. For the love of stems and leaves, PUT YOUR VALUES IN ORDER.

$$\text{Heights of Sunflowers in Inches: } \\ 11, 13, 14, 18, 19, 20, 21, 21, 21, 24, 25, 27, 28, 29, 32, 35, 39, 40, 41, 43, 52, 56, 71$$

Step Two

Fill your plot. Here's how to do it.

The Stem column contains the non-ones digits in the data set, in ascending order. If the data set only contains two-digit numbers, the stems will be the tens digits of the data values. Also, you cannot repeat stems. Let's look at our data set, and figure out what will be in the Stems column. Our data ranges from $11 — 71.$ This means that the tens digits of our data range from $1 — 7.$ So, our stems will be $\boxed{1}, \boxed{2}, \boxed{3}, \boxed{4}, \boxed{5}, \boxed{6}, \boxed{7}.$

The Leaf column contains the ones digit for each data value, in same row as its corresponding tens digit, in ascending order. Remember how you can't repeat stems? Well, you should repeat leaves, if there are duplicate values in your data set. You should have the same number of leaves as you do values in your data set. Let's look at our data set, and figure out what will be in the Leaf column.

The values in our data set with a $1$ as a tens digit are $1\boxed{1}, 1\boxed{3}, 1\boxed{4}, 1\boxed{8}, 1\boxed{9}.$ So, in the same row as the $1$ stem, the leaves are $\boxed{1}, \boxed{3}, \boxed{4}, \boxed{8}, \boxed{9},$ in that order.

The values in our data set with a $2$ as a tens digit are $2\boxed{0}, 2\boxed{1}, 2\boxed{1}, 2\boxed{1}, 2\boxed{4}, 2\boxed{5}, 2\boxed{7}, 2\boxed{8}, 2\boxed{9}$ So, in the same row as the $2$ stem, the leaves are $\boxed{0}, \boxed{1}, \boxed{1}, \boxed{1}, \boxed{4}, \boxed{5}, \boxed{7}, \boxed{8}, \boxed{9},$ in that order.

The values in our data set with a $3$ as a tens digit are $3\boxed{2}, 3\boxed{5}, 3\boxed{9}.$ So, in the same row as the $3$ stem, the leaves are $\boxed{2}, \boxed{5}, \boxed{9},$ in that order.

The values in our data set with a $4$ as a tens digit are $4\boxed{0}, 4\boxed{1}, 4\boxed{3}.$ So, in the same row as the $4$ stem, the leaves are $\boxed{0}, \boxed{1}, \boxed{3},$ in that order.

The values in our data set with a $5$ as a tens digit are $5\boxed{2}, 5\boxed{6}.$ So, in the same row as the $5$ stem, the leaves are $\boxed{2}, \boxed{6},$ in that order.

In our data, there is no value with a $6$ as the tens digit. But, since our data values range from $11 —71,$ and $60—69$ is in that range, we still have to include a $6$ for a stem. It just won't have corresponding leaves. In general, you should include tens digits/stems that are in the range of your data, even if they don't have a corresponding leaf/ones digit.

The value in our data set with a $7$ as a tens digit is $7\boxed{1}.$ So, in the same row as the $7$ stem, the leaf is $\boxed{1}.$

Since the stem of $2$ has the most corresponding leaves, we can say that the class of values (or range of values) $20—29$ has the greatest frequency in our data set.

We can look at the plot to see that the $\boxed{three}$ leaves corresponding to Stem $3$ are $2, 5, 9,$ representing the values $32, 35, 39.$

There are $\boxed{zero}$ leaves corresponding to Stem $6,$ meaning there are no values in the range $60—69$ in our data set. You can't get lower than that!

Bonus

If you turn a stem-and-leaf plot on its side, you can see the basic distribution of the data.

(Video) How to Make a Stem and Leaf Plot

by MySecretMathTutor

My Secret Math Tutor has a good video that shows you how to construct and analyze stem and leaf plots from a set of data.

$\large{\text{Summary}}$

Stem and Leaf Plots are a way to organize a set of data. I find that they are the most useful with sets of data where most values are integers that have two digits.

The data set from the video is: $$23, 58, 62, 62, 63, 65, 67, 71, 71, 72, 80, 82, 82, 82$$

The stem column contains the tens place of all of the numbers in the set. The given set has numbers from $23$ to $82$, so the video lists $2$ through $8$ in the stem column. (Note: if some values in your data set have three or more digits, the stems are the non-ones digits.)

The leaf column has the ones place of each number in the set, in the row of its corresponding tens place. For example, the given set has $62, 62, 63, 67,$ and $67$. This means the leaves for the stem of $6$ are $2, 2, 3, 5,$ and $7$.