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2-12-1

Chapter 2Chapter 2

Frequency Distributions Frequency Distributions and Graphsand Graphs

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2-22-2 OutlineOutline

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

Polygons, and Ogives 2-4 Other Types of Graphs

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2-32-3 ObjectivesObjectives

Organize data using frequency distributions.

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

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2-42-4 ObjectivesObjectives

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

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2-52-5 2-2 Organizing Data2-2 Organizing Data

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

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

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2-62-6 2-2 Organizing Data2-2 Organizing Data

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

The following slide shows an example of a frequency distribution.

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2-72-72-2 Three Types of Frequency 2-2 Three Types of Frequency

DistributionsDistributions

Categorical frequency distributionsCategorical frequency distributions -- can be used for data that can be placed in specific categories, such as nominal- or ordinal-level data.

Examples -Examples - political affiliation, religious affiliation, blood type etc.

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2-82-82-2 Blood Type Frequency 2-2 Blood Type Frequency

Distribution - Distribution - Example

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2-92-92-2 Ungrouped Frequency 2-2 Ungrouped Frequency

DistributionsDistributions

Ungrouped frequency distributions -Ungrouped frequency distributions - can be used for data that can be enumerated and when the range of values in the data set is not large.

Examples -Examples - number of miles your instructors have to travel from home to campus, number of girls in a 4-child family etc.

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2-102-102-2 Number of Miles Traveled2-2 Number of Miles Traveled -

Example

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2-112-112-2 Grouped Frequency 2-2 Grouped Frequency

DistributionsDistributions

Grouped frequency distributions -Grouped frequency distributions - can be used when 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.

Examples -Examples - the life of boat batteries in hours.

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2-122-122-2 Lifetimes of Boat Batteries2-2 Lifetimes of Boat Batteries -

Example

Classlimits

ClassBoundaries

Frequency Cumulativefrequency

24 - 30 23.5 - 37.5 4 4

38 - 51 37.5 - 51.5 14 18

52 - 65 51.5 - 65.5 7 25

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2-132-132-2 Terms Associated with a 2-2 Terms Associated with a

Grouped Frequency DistributionGrouped Frequency Distribution

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

In the lifetimes of boat batteries example, the values 24 and 30 of the first class are the class class limitslimits.

The lower classlower class limit is 24 and the upper classupper class limit is 30.

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

2-2 Terms Associated with a 2-2 Terms Associated with a Grouped Frequency DistributionGrouped Frequency Distribution

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The class widthclass width for a class in a frequency distribution is found by subtracting the lower (or upper) class limit of one class minus the lower (or upper) class limit of the previous class.

2-2 Terms Associated with a 2-2 Terms Associated with a Grouped Frequency DistributionGrouped Frequency Distribution

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2-162-162-2 Guidelines for Constructing 2-2 Guidelines for Constructing

a Frequency Distributiona Frequency Distribution

There should be between 5 and 20 classes.

The class width should be an odd number.

The classes must be mutually exclusive.

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2-172-172-2 Guidelines for Constructing 2-2 Guidelines for Constructing

a Frequency Distributiona Frequency Distribution

The classes must be continuous. The classes must be exhaustive. The class must be equal in width.

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2-182-182-2 Procedure for Constructing 2-2 Procedure for Constructing

a Grouped Frequency Distributiona Grouped Frequency Distribution

Find the highest and lowest value. Find the range. Select the number of classes

desired. Find the width by dividing the range

by the number of classes and rounding up.

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2-192-19

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, find the frequencies, and

find the cumulative frequency.

2-2 Procedure for Constructing 2-2 Procedure for Constructing a Grouped Frequency Distributiona Grouped Frequency Distribution

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In a survey of 20 patients who smoked, the following data were obtained. Each value represents the number of cigarettes the patient smoked per day. Construct a frequency distribution using six classes. (The data is given on the next slide.)

2-2 Grouped Frequency Distribution - 2-2 Grouped Frequency Distribution - Example

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2-212-212-2 Grouped Frequency Distribution - 2-2 Grouped Frequency Distribution -

Example

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2-222-22

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

Step 2:Step 2: Find the range: R = H – L = 22 – 5 = 17.

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

2-2 Grouped Frequency Distribution - 2-2 Grouped Frequency Distribution - Example

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Step 4: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.

2-2 Grouped Frequency Distribution - 2-2 Grouped Frequency Distribution - Example

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Step 5:Step 5: Select a starting point for the lowest class limit. For convenience, this value is chosen to be 5, the smallest data value. The lower class limits will be 5, 8, 11, 14, 17, and 20.

2-2 Grouped Frequency Distribution - 2-2 Grouped Frequency Distribution - Example

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Step 6: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 computed as 8 - 1, etc.

2-2 Grouped Frequency Distribution - 2-2 Grouped Frequency Distribution - Example

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Step 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.

2-2 Grouped Frequency Distribution - 2-2 Grouped Frequency Distribution - Example

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Step 8:Step 8: Tally the data, write the numerical values for the tallies in the frequency column, and find the cumulative frequencies.

The grouped frequency distribution is shown on the next slide.

2-2 Grouped Frequency Distribution - 2-2 Grouped Frequency Distribution - Example

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Class Limits Class Boundaries Frequency Cumulative Frequency

05 to 07 4.5 - 7.5 2 208 to 10 7.5 - 10.5 3 511 to 13 10.5 - 13.5 6 1114 to 16 13.5 - 16.5 5 1617 to 19 16.5 - 19.5 3 1920 to 22 19.5 - 22.5 1 20

Note: The dash “-” represents “to”.

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2-292-292-3 Histograms, Frequency 2-3 Histograms, Frequency

Polygons, and OgivesPolygons, and Ogives

The three most commonly used The three most commonly used graphs in research are:graphs in research are:

The histogram. The frequency polygon. The cumulative frequency graph,

or ogive (pronounced o-jive).

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The histogramhistogram is a graph that displays the data by using vertical bars of various heights to represent the frequencies.

2-3 Histograms, Frequency 2-3 Histograms, Frequency Polygons, and OgivesPolygons, and Ogives

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2-312-31 Example of a HistogramExample of a Histogram

2017141185

6

5

4

3

2

1

0

Number of Cigarettes Smoked per Day

Freq

uenc

y

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A frequency polygonfrequency polygon is a graph that displays the data by using lines that connect points plotted for frequencies at the midpoint of classes. The frequencies represent the heights of the midpoints.

2-3 Histograms, Frequency 2-3 Histograms, Frequency Polygons, and OgivesPolygons, and Ogives

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2-332-33 Example of a Frequency PolygonExample of a Frequency Polygon

Frequency Polygon

262320171411852

6

5

4

3

2

1

0

Number of Cigarettes Smoked per Day

Fre

quen

cy

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A cumulative frequency graphcumulative frequency graph or ogiveogive is a graph that represents the cumulative frequencies for the classes in a frequency distribution.

2-3 Histograms, Frequency 2-3 Histograms, Frequency Polygons, and OgivesPolygons, and Ogives

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2-352-35 Example of an OgiveExample of an Ogive

Ogive

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2-362-36 2-4 Other Types of Graphs2-4 Other Types of Graphs

Pareto charts -Pareto charts - a Pareto chart is used to represent a frequency distribution for a categorical variable.

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2-372-372-4 Other Types of Graphs-Pareto 2-4 Other Types of Graphs-Pareto

ChartChart

When constructing a Pareto chart - Make the bars the same width. Arrange the data from largest to

smallest according to frequencies. Make the units that are used for the

frequency equal in size.

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2-382-38 Example of a Pareto ChartExample of a Pareto Chart

Homicide

RobberyRape

Assault

13 29 34164 5.412.114.268.3

100.0 94.6 82.5 68.3

250

200

150

100

50

0

100

80

60

40

20

0

Defect

CountPercentCum %

Pe r

cent

Co u

nt

Enforcement Officers in U.S. National Parks During 1995. Pareto Chart for the number of Crimes Investigated by Law

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2-392-39 2-4 Other Types of Graphs2-4 Other Types of Graphs

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

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2-402-402-4 Other Types of Graphs - 2-4 Other Types of Graphs -

Time Series GraphTime Series Graph

19941993199219911990

89

87

85

83

81

79

77

75

Year

Rid

ersh

ip (

in m

illio

ns)PORT AUTHORITY TRANSIT RIDERSHIP

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2-412-41 2-4 Other Types of Graphs2-4 Other Types of Graphs

Pie graph -Pie graph - A pie graph is a circle that is divided into sections or wedges according to the percentage of frequencies in each category of the distribution.

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2-422-422-4 Other Types of Graphs - 2-4 Other Types of Graphs - Pie Pie GraphGraph

Robbery (29, 12.1%)

Rape (34, 14.2%)

Assaults (164, 68.3%)

Homicide (13, 5.4%)

Pie Chart of the Number of Crimes Investigated byLaw Enforcement Officers In U.S. National Parks During 1995