Topic:Stem and Leaf Diagrams

From SharedExperienceProject

Topic Highlights

(What you will learn)

Representing data using stem-and-leaf diagrams
How to get the raw data from a stem-and-leaf diagram (and then do some basic analysis based on the data)
Why these are important and how to create and interpret them

Introduction and Motivation

(Why learn it)

In this topic, we are still firmly planted in the realm of descriptive statistics (see Introduction to Business Statistics if you need to refresh yourself about what this means).

A stem-and-leaf diagram is a way to represent 150 points or less, so they are more easily interpreted by the business manager. It lies somewhere between a frequency distribution and a histogram because it provides a graphical description, but it is still a table. It is different from the frequency distribution table and a histogram though, because the original raw data can be recovered.

Stem-and-leaf diagrams are a basic analysis tool, and everyone in business needs to know how to create and interpret them.

Learning Activities

(How the levels of understanding will be gained)

**Learning activities for this topic**
Type	Name	Direction
Reading	Section 2.1 of Bowerman et al. or: Section 2.2 of Kvanli et al.	Self-directed
In-class worksheet	Worksheet - Stem and Leaf Diagrams	Self-directed
In-class discussion	Stem and Leaf Diagrams (below on this page)	Instructor-directed
Practice problems	Practice Problems (below)	Self-directed
Personal activities	Reflection and review, e.g. review the learning objectives (below)	Self-directed

Learning Objectives

(Levels of understanding to be gained)

**Learning objectives for this topic**
Level of Understanding	Objective(s)
Very best	Can I think of ways to apply stem and leaf diagrams in my place of work? Can I solve Practice Problem 3?
Highly satisfactory	What is the advantage of using a stem-and-leaf diagram over a frequency distribution or histogram? Do I know the basic steps outlined under Basic steps for creating a stem-and-leaf? If necessary, are they on my equation sheet? Can I solve a simpler version of Practice Problem 3, like Example 1?
Satisfactory	Can I solve Practice Problem 1 and Practice Problem 2? Do I know the steps outlined under Basic steps for recovering the raw data? Am I familiar with the two simple equations we used in this topic? Are they on my formula sheet? Can I solve Example 3 and Example 4?
Maybe just enough to pass	Can I solve Example 2? Do I know the basic stem-and-leaf concept? Do I know what a stem-and-leaf diagram is?

Lecture Notes: Stem-and-Leaf Diagrams

The following notes are meant to facilitate a discussion about the concept of a Stem and Leaf Diagram.

Why use them?

Why would I want to bother with a stem-and-leaf diagram? Basically, the reason is same one we gave when studying frequency distributions and histograms: to allow us to represent data in a way that can be more easily interpreted by the manager. Check out that discussion if you don't recall it.

What are they?

The concept

A stem-and-leaf diagram is just a table that represents the raw data in groups given by the first few numbers, or stems, of the data points.

Let's start by considering just the basic concept. The figure below shows how three data points are represented using the stem-and-leaf concept.

Stem-and-leaf concept for single data points

You should note the following:

The left side of the figure is equivalent to the right side in each case. In other words:
- Writing the number 38 is exactly the same thing as writing:
- The data point (38) is given completely by knowing the stem (3), the leaf (8) and the so-called leaf unit (1)

The key relationship

Looking at the above figure, you should have also noticed that the only difference between the representations of 38, 3.8 and 380 is the leaf unit (which is 1, 0.1 and 10, respectively). It is critical that you understand the key relationship between the data, the stem and the leaf. It is given by the following equation, which is not in your textbook but which you may find helpful:

In the above examples, this applies as follows:

We'll make use of this as we go through some more examples.

A typical stem-and-leaf diagram

In practice, stem-and-leaf diagrams are used to represent up to 150 data points, unlike the case above which uses only one point to demonstrate the concept. The figure below shows a typical diagram for 12 raw data points. In the next section, we'll look at how to create one.

Typical stem-and-leaf diagram

How do I create a stem-and-leaf diagram?

Basic steps for creating a stem-and-leaf

Once you have collected some raw data, the following steps are used to create a stem-and-leaf diagram by hand.

1. Order the raw data by writing it out again from smallest to largest

2. Determine where to place the stem

(3. Re-write the data with no decimal place and draw a line between stem and leaf, if it helps you)

4. Set up the table, including the stem values

5. Place the leaf values (and count them to make sure you haven't missed any)

6. Find and write the leaf unit

Example 1

Let's work through an example using the data given below.

In this example, your goal is to create a stem-and-leaf diagram for the data.

Raw data

Look at the basic steps given above. The first step is to order the raw data from smallest to largest. Do it now on a sheet of paper:

We do this to make the following steps easier
It is useful to count and make sure you have the same number of points as the raw data (15 in this case)
You should end up with the following

Ordered data

The second step is to determine where to place the stem. We want to choose it so we have 5-10 stem values. You should see that this means placing it between the first and second digits, e.g. [ stem | leaf ] = [ 1 | 1 ] for the first data point, which gives stem values of 1, 2, 3 and 4

The third step is to rewrite the data with no decimal place. This data already has no decimal place, so you can draw lines between stem and leaf and move on.

The fourth step is to set up the diagram by setting up the table and writing the stem values. Do this now to obtain the following.

Empty stem-and-leaf diagram

The fifth step is to go back to the ordered data and place the leaf values. Do this now and put the data in your diagram. Be sure to place the leaf values in order from smallest to largest. You should end up with the following. You should also count to make sure you have the right number of data points (15 in this case).

Diagram with leaf component

Finally, before you are finished you need to compute the leaf unit. You may be able to do this in your head. If not, then use the key relationship. Do this now and add it to your table. The computation is shown in the figure below using data point 23.

Computing the leaf unit

If you have some examples on your equation sheet, then you can probably also infer the unit from them. For example, we know the following:

For stem = 3, leaf = 8:
- data = 38 if unit = 1
- data = 3.8 if unit = 0.1
- data = 380 if unit = 10

How do I get the raw data from a stem-and-leaf?

It's possible and often required to analyze a stem-and-leaf diagram. For example, you could be given one and asked to compute the mean of the data. Or pull out the minimum and maximum. There are many such examples.

Basic steps for recovering the raw data

This is easier than creating the diagram in the first place:

1. Figure out the first data point

2. Do the rest of the data points in the same way

3. Check that you have the right number of points

4. Do any analysis that might have been required

Let's give it a go...

Example 2

Imagine you are in a midterm. The first question gives you the stem-and-leaf diagram below and asks you to pull out the original raw data.

Given stem-and-leaf diagram

The first step is to figure out what the first data point is. Do this now using the key relationship. The solution is given in the figure below.

Getting the first data point

Now do the rest of the data points, which should be very quick. Be sure to write them out so you don't make a mistake later.

Next, count to make sure you have the same number of data points. How many are there this time?

You should have the solution given below, from which you can do any required calculations.

The raw data corresponding to the stem-and-leaf diagram

What else do I need to know?

Something tricky you might not have thought about is the fact that the same data point can be represented in several ways using the stem and leaf concept.

For example:

The number 562 can be represented as 5 | 62 and as 56 | 2
How do you know which is right?

In our earlier examples, we went with what gave us the right number of stems. This is basically the rule, but it's not always obvious first time through.

Let's do some examples to make sure you have it.

Example 3

Where would you place the stem for the following data?

The answer is given by the example below.

Stem placement
56 \| 2

Example 4

Where would you place it for the following data?

The answer is given by the example below.

Stem placement
5 \| 62

Practice Problems

We've covered the basics, now build your skills with the following problems. Don't look at the solutions until you've worked the problem through.

Practice Problem 1

a) What is the leaf unit for the data in Example 4?

b) Complete the stem-and-leaf diagram for the data in Example 4.

Solution

Practice Problem 2

Compute the mean, min and max of the data represented below. (Hint: You will need recover the raw data from the stem-and-leaf diagram first.)

Solution

Practice Problem 3

Create the stem-and-leaf diagram for the data given below:

Solution

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