CHAPTER 2CHAPTER 2

Frequency Frequency Distributions and Distributions and

GraphsGraphs

• 2-1 Introduction• 2-2 Organizing Data• 2-3 Histograms, Frequency

Polygons, and Ogives• 2-4 Other Types of Graphs

Outline

• Organize data using frequency distributions.

• Represent data in frequency distributions graphically using histograms, frequency polygons, and ogives.

• Represent data using Pareto charts, time series graphs, and pie graphs.

Objectives

2-1 Introduction

• In a statistical study, data organization and presentation are very important for better understanding and interpretation.

2-2 Organizing Data

• When data are collected in original form, they are called raw data.

• When the raw data is organized into a frequency distribution, the frequency will be the number of values in a specific class of the distribution.

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

2-2 Types of Frequency Distribution

• There are three major type of frequency distribution:i) Categorical Frequency Distributionii) Ungrouped Frequency Distribution

iii) Grouped frequency Distribution

Categorical Frequency Distribution

• Categorical frequency distribution is used for data that can be categorized into specific group according to its attributes.

• Example: Gender, blood type, major field of study, religious affiliation, etc.

Class Frequency

A 5

B 7

AB 9

O 4

Total 25

Example of Categorical Frequency Distribution

Blood Type

Ungrouped Frequency Distribution

• Ungrouped frequency distribution is used for data that can be counted or measured, however the data range is not wide.

Class Frequency

0 5

1 10

2 7

3 4

Total 26

Example of Ungrouped Frequency Distribution

Number of children in a family

Grouped Frequency Distribution

• Grouped frequency distribution is used for data that can be counted or measured.

• The range of values in the data set is very large. The data must be grouped into classes that are more than one unit in width.

• Example: Traveling time in mins, body length in cm, life of boat batteries in hrs, etc.

Example of Grouped Frequency Distribution

Lifetimes of Boat Batteries

Class limits Class boundaries

Frequency Cumulative frequency

24-30 23.5-30.5 3 3

31-37 30.5-37.5 1 4

38-44 37.5-44.5 5 9

45-51 44.5-51.5 9 18

Data are classified in a range of values

2-2 Associated Terms for Grouped Frequency Distribution

Class Limits

Class Boundaries

Class Width

Class Limits

Class limits represent the smallest and largest data values that can be included in a class.

Upper class limits The largest value in a class

Lower class limits The smallest value in a class

Class Limits

26-3031-3536-40

Can you identify the upper and lower class limits for all the classes shown above?

Class Boundaries

Class boundaries used to separate the classes so that there are no gaps in the frequency distribution.

How to find class boundaries?

Lower class boundary = (Lower class limit of a class + Upper class limit of the previous class)/2

Upper class boundary = (Upper class limit of a class + Lower class limit of the subsequent class)/2

Class Limits

Class 1: 11-15Class 2: 16-20Class 3: 21-25

What is the lower class boundary for class 2?

What is the upper class boundary for class 2?

Class Width

Class width represents the size of a class.

Class width can be calculated by subtracting the upper class boundary from lower class boundary of the same class.

4.5 5 - 9 9.5

Lower class limit Upper class limit

Class

Lower class boundary

Upper class boundary

Class width

2-2 Guidelines for Constructing a Frequency Distribution

1. The classes must be mutually exclusive

Age

11-20

21-30

31-40

41-50

Mutually exclusive

- No overlapping in class limits

so that data can only be

placed in one class.

2. The classes must be continuous.

Age

10-20

21-31

32-42

43-53

Continuous

- There is no gap between

different classes.

3. There should be between 5 and 20 classes.

4. The classes must be equal in width.

2-2 Procedures for Constructing a Grouped Frequency Distribution

Find the highest and lowest value.

Find the range (Highest - Lowest).

Select the number of classes desired.

Find the width by dividing the range by the

number of classes and rounding up.

Step 1

Select a starting point (usually the lowest value); add the width to get the lower limits.

Find the upper class limits.

Find the boundaries.

Tally the data.

Step 2

Find the frequency.

Step 3

Find the cumulative frequency.

Step 4

2-2 Example for Grouped Frequency Distribution

A book store recorded the number of books sold on 20 consecutive Fridays. Construct a grouped frequency distribution for these

10 8 6 14

22 13 17 19

11 9 18 14

13 12 15 15

5 11 16 11

STEP 1: Find the highest and lowest values H = 22 and L = 5

STEP 2: Find the range Range = 22 – 5 = 17

STEP 3: Select the number of classes desired. In this case it is equal to 6.

STEP 4: Find the class width by dividing the range by the number of classes. Width = 17/6 = 2.83. This value is rounded up to 3.

STEP 5: Select a starting point for the lowest class limit. In this case, 5 was chosen as it is the smallest data value. The lower class limits will be 5, 8, 11, 14,

17, and 20.

STEP 6: The upper class limits will be 7, 10, 13, 16, 19, and 22. For example, the upper limit for the first class is 7.

STEP 7: Find the class boundaries by subtracting 0.5 from each lower class limit and adding 0.5 to the upper class limit.

STEP 8: Tally the data, write the numerical values for the tallies in the frequency column, and find

the cumulative frequencies.

Class Limits Class Boundaries

Frequency Cumulative Frequency

5 – 7 4.5 – 7.5 2 2

8 – 10 7.5 – 10.5 3 5

11 – 13 10.5 – 13.5 6 11

14 – 16 13.5 – 16.5 5 16

17 – 19 16.5 – 19.5 3 19

20 - 22 19.5 – 22.5 1 20

2-3 Histograms, Frequency Polygons, and Ogives

Three most commonly used graphs in research are:

Histogram. Frequency polygon. Cumulative frequency graph, or ogive.

Histogram

• Histogram is a graph that displays the data by using contiguous vertical bars of various heights to represent the frequencies of the classes.

Example of Histogram

Frequency

Class boundaries

Histogram VS Bar Chart

No gap between the bars

There is a gap between the bars

Frequency Polygon

• Frequency polygon is the graph that displays the data by using lines that connect points plotted for the frequencies at the midpoints of the classes. The frequency are represented by the heights of the points.

Example of Frequency Polygon

Frequency Polygon

How to plot a frequency polygon?

1. Plot a histogram.

2. Mark on the mid-point of each class interval.

3. Join all the mid-points with a straight line.

Mid-point of an interval = ½ (lower CB + upper CB)

Cumulative Graph / Ogive

• Cumulative graph or ogive is a graph

that represents the cumulative frequencies

for the classes in a frequency distribution.

Example of Frequency Polygon

Ogive

How to plot an ogive?

1. Cumulative frequencies are plotted

against the upper class boundaries.

2. The points are joined with a smooth curve.

Question

10 students were asked to solve a simple problem and the time taken by each was noted. Construct a histogram, frequency polygon, and ogive for this data.

Time (seconds) Frequency

10-19 1

20-29 3

30-39 4

40-49 2

2-3 Other Types of Graphs

Pareto chart is used to represent a

frequency distribution for a categorical

variable, and the frequencies are displayed

by the heights of vertical bars, which are

arranged in order from highest to lowest.

Example of Pareto Chart

Bars are arranged from highest to lowest frequency

• Time series graph represents data that occur over a specific period of time.

Example of Time Series Chart

Changes in number of

customers over a certain period

of time

• Pie chart is a circle divided into sections

according to the percentage of frequencies

in each category of the distribution.

• A pie chart should not consist of too many

segments (less than eight is suggested).

Example of Pie Chart