Chapter 1: Frequency Distribution and Graphs

OUTLINE

• 1.1 Organizing Data
• 1.2 Histograms, Frequency Polygons, and Ogives
• 1.3 Other Types of Graphs Summary

Chapter 1: Frequency Distribution and Graphs

OBJECTIVES

After completing this chapter, you should be able to

1. Organize data using a frequency distribution.
2. Represent data in frequency distributions graphically, using histograms, frequency polygons, and ogives.
3. Represent data using bar graphs, Pareto charts, time series graphs, pie graphs, and dotplots.
4. Draw and interpret a stem and leaf plot.

Introduction

To describe situations, draw conclusions, or make inferences about events, the researcher must organize the data.

1. The most convenient method of organizing data is to construct a frequency distribution.

The researcher presents data so they can be understood by those who will benefit from reading the study.

2. The most useful method of presenting the data is by constructing statistical charts and graphs.

Useful R Libraries

• library(flextable)

• library(tidyverse)

• library(formattable)

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Organizing Data (OD) - raw data

1. When the data are in original form, they are called raw data and are listed next.

2. Since little information can be obtained from looking at raw data, the researcher organizes the data into what is called a frequency distribution.

OD - frequency distribution

• A frequency distribution is the organization of raw data in table form, using classes and frequencies.

• Each raw data value is placed into a quantitative or qualitative category called a class.

• The frequency of a class then is the number of data values contained in a specific class..

OD - Frequency Table

• Two types of frequency distributions that are most often used are the categorical frequency distribution and the grouped frequency distribution.

OD - Categorical Frequency Distributions

• The categorical frequency distribution is used for data that can be placed in specific categories, such as nominal- or ordinal-level data. Example: political affiliation, religious affiliation, or major field of study

OD - Categorical Frequency Distributions

EXAMPLE 1–1 Breakfast Beverages

1. Forty people were asked what beverage they drink for breakfast. Construct a categorical frequency distribution for the data and summarize the results. Use these classes: W = water, M = milk, J = juice, C = coffee, and T = tea.

OD - EXAMPLE 1 Solution: Step 1

• Step 1 Make a table as shown.

OD - EXAMPLE 1 Solution: Step 2 - 4

• Step 2 Count the tallies and place the results in the third column labeled Frequency.

• Step 3 Find the percentage of values in each class by using the formula:

\[ \% = \frac{f}{n}\] where \(f = \text{frequency}\) of the class and \(n = \text{total number of values}\). For example, in class W, the percentage is

\[ = \frac{14}{40}\times 100 = 35\% \]

• Step 4 Find the cumulative frequency by adding each frequency from a frequency distribution table to the sum of its predecessors.

\[ 35\%, 35+20=55\%, 35+20+15 = 70\% ... \]

OD - EXAMPLE 1 Solution: Step 5

• Step 5 Draw draw the Frequency Table

OD - Grouped Frequency Distributions

• When the range of the data is large, the data must be grouped into classes that are more than one unit in width, in what is called a grouped frequency distribution.

• For example, a distribution of the blood glucose levels in milligrams per deciliter (mg/dL) for 50 randomly selected college students is shown.

OD - Grouped Frequency Distributions2

• In this distribution, the values 58 and 64 of the first class are called class limits.

• The lower class limit is 58; it represents the smallest data value that can be included in the class.

• The upper class limit is 64; it represents the largest data value that can be included in the class.

• The numbers in the second column are called class boundaries.

• The basic rule of thumb is that the class limits should have the same decimal place value as the data, but the class boundaries should have one additional place value and end in a 5.

• The class width for a class in a frequency distribution is found by subtracting the lower (or upper) class limit of one class from the lower (or upper) class limit of the next class.

OD - Grouped Frequency Distributions3

The researcher must decide how many classes to use and the width of each class. To construct a frequency distribution, follow these rules:

• There should be between 5 and 20 classes.

• It is preferable but not absolutely necessary that the class width be an odd number. This ensures that the midpoint of each class has the same place value as the data. The class midpoint \(X_m\) is obtained by adding the lower boundary(lb) and upper boundary(ub) and dividing by 2, or adding the lower limit(ll) and upper limit(ul) and dividing by 2: \[ X_{m} = \frac{lb + up}{2}\]

or

\[X_{m} = \frac{ll + ul}{2}\]

OD - Grouped Frequency Distributions4

• The classes must be mutually exclusive.

• The classes must be continuous.

• The classes must be exhaustive.

• The classes must be equal in width. One exception occurs when a distribution has a class that is open-ended. A frequency distribution with an open-ended class is called an open-ended distribution.