Topic:Frequency Polygons and Cumulative Frequencies (Ogives)

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Contents 1 Topic Highlights 2 Introduction and Motivation 3 Learning Activities 4 Learning Objectives 5 Lecture Notes: Frequency Polygons 5.1 What is a frequency polygon? 5.1.1 Example 1 5.2 Other Similar Plots 6 Lecture Notes: Cumulative Frequencies - Ogives 6.1 What is a cumulative frequency (Ogive)? 6.1.1 Example 2 6.2 The Relative Ogive 6.2.1 Example 3 6.3 Putting it all Together 6.3.1 Example 4 7 Practice Problems 7.1 Practice Problem 1 7.2 Practice Problem 2 7.3 Practice Problem 3

Topic Highlights

(What you will learn)

• The concept of a frequency polygon, which is nothing more than a slight twist on the frequency histogram
  
• The concept of a cumulative frequency, or Ogive
  
• Why these are important and how to create and interpret them

Introduction and Motivation

(Why learn it)

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

We will build on what we learned in the topic Frequency Distributions and Histograms in two ways:

• A frequency polygon is a very small twist on the frequency histogram, and as you will see, it provides a slightly different representation of the data
• A cumulative frequency plot (or Ogive) is a new concept that builds on the frequency distribution table we saw earlier

Both are basic analysis tools, and you should 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	Read before class: Section 2.1 of Bowerman et al. or: Section 2.3 and 2.4 of Kvanli et al.	Self-directed
In-class worksheet	Worksheet - Freq Polygons and Cumulative Frequencies
In-class discussion	Frequency Polygons (below on this page) Cumulative Frequencies - Ogives (also below)	Instructor-directed
Practice problems	Practice Problems (below)	Self-directed
Personal activities	e.g. reflection and review Review the learning objectives in preparation for the quiz	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 frequency polygons and cumulative frequencies in my place of work?
Highly satisfactory	What is the advantage of using a cumulative frequency?
Satisfactory	Am I very clear on the differences between frequency, relative frequency and cumulative frequency, as discussed in Example 3? Can I solve Practice Problem 1 (under Practice Problems below)? Can I solve Practice Problem 2?
Maybe just enough to pass	On my own, can I solve Example 1, Example 2 and Example 3 (below)?  Do I know what a frequency polygon is? Do I know what a cumulative frequency is? And how it is calculated?

Lecture Notes: Frequency Polygons

The following notes are meant to facilitate a discussion about frequency polygons.

What is a frequency polygon?

A frequency polygon is just a different way of plotting the data in a frequency distribution table.

Recall in the topic Frequency Distributions and Histograms, where we took the numbers in a frequency distribution table and plotted them to create a frequency histogram. A frequency polygon is the same thing, only we plot the data points in a scatter plot instead of using a bar chart.

Example 1

Let's start with an example. The figure below shows the table we came up with in Example 1 of the topic Frequency Distributions and Histograms. The figure below that shows the histogram we came up with.

Frequency distribution table

Corresponding histogram

The frequency polygon is like the histogram, but instead it is a plot of the midpoints of the classes, connected by straight lines. This is shown below.

Frequency polygon

Make sure you know how to compute the midpoint values. At first, it might help you to rewrite the frequency distribution table (as in the figure below).

Re-written frequency distribution table (only for helping you plot the freqency polygon)

Other Similar Plots

You will see in the Practice Problems (below) that there are other plots like the frequency polygon, i.e. for relative frequency and relative Ogive.

Lecture Notes: Cumulative Frequencies - Ogives

The following notes are meant to facilitate a discussion about the concept of a cumulative frequency.

What is a cumulative frequency (Ogive)?

The cumulative frequency, also known as an Ogive, is another way to analyze the frequency distribution table. Unlike a frequency distribution which tells you how many data points are within each class, a cumulative frequency tells you how many are less than or within each of the class limits.

It is useful for analyses that require quick results about the proportion of data that lies below a certain level.

Let's look at an example.

Example 2

The frequency distribution table from Example 1 is repeated below

.

From this, let's compute the cumulative frequency for that data as follows:

• Cumulative frequency for class 1 = frequency for class 1
  
• Cumulative frequency for class 2 = frequency for class 1 + frequency for class 2
  
• Cumulative frequency for class 3 = frequency for class 2 + frequency for class 3
• And so on, to result in the new column of data shown below

The result is shown below:

Frequency distribution table with cumulative frequency (Ogive)

That's pretty much an Ogive. Just make sure you can do this if asked for one.

The Relative Ogive

Just like we did in the topic Frequency Distributions and Histograms when we introduced the concept of relative frequency, we can also have a relative version of the cumulative frequency (also known as a relative Ogive).

Example 3

Do this for this data in Example 2 above.

This should be pretty straightforward for you. Just create a new column called Relative Cumulative Frequency (or Relative Ogive) and compute the relative values as you have done before.

The result is given below.

Frequency distribution table with relative cumulative frequency (relative Ogive)

Putting it all Together

You've seen quite a few new terms over this and the last topic. Now let's put them all together and make sure you know the difference.

Example 4

In the topic Frequency Distributions and Histograms, we studied the concepts of:

• Frequency data
• Relative frequency data

Here in this topic, we've learned about:

• Cumulative frequency data (also known as an Ogive)
• Relative cumulative frequency data (also known as a relative Ogive)

Be sure you know the differences between these very well, as shown below.

Putting it all together

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

Refer to the data in Example 4 to determine what proportion of students represented by the table in are shorter than 182 cm.

Solution
0.74, or 74%

Practice Problem 2

Create a relative frequency polygon using paper and pencil based on the table in Example 4.

Solution
Notice that unlike the frequency polygon case, lines are drawn here on both sides of the lowest and highest midpoints, at equal distance to the left and right. In other words, the relative frequency polygon MUST start and end at zero. This is described in Section 2.3 of Kvanli et al.

Practice Problem 3

The following is a relative Ogive plot based on data in Example 4. See Section 2.4 of Kvanli et al for a description of how this is obtained:

a) Based on the above plot, approximately what percentage of students is above 179 cm tall?

Solution
40%

b) If you are 170 cm tall, approximately what percentage of students is taller than you?

Solution
70%

c) To be in the top 20% of the class in terms of height, approximately how tall would you have to be?

Solution
184 cm