chapter 2 250110 083240

Chapter 2Chapter 2

Frequency DistributionsFrequency Distributions

and Graphsand Graphs

Reference: Allan G. Bluman (2007) Elementary Statistics: A Step-by Step Approach. New York : McGraw Hill

ObjectivesObjectives 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.

Organizing DataOrganizing 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.

Organizing DataOrganizing Data A frequency distributionfrequency distribution is the organizing of raw data in

table form, using classes and frequencies

3 types: Categorical frequency distribution Grouped frequency distribution Ungrouped frequency distribution

No. of data values contained in a specific class

Types of Frequency 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.

Categorical Frequency Distribution - Categorical Frequency Distribution - Example

Step 1:Step 1: Construct a frequency distribution table.

Step 2:Step 2: Tally the data and place the results in the 2nd column.

Step 3:Step 3: Count the tallies & place the results in the 3rd column.

A

Class

B

Tally

C

Frequency

D

Percent

A

B

O

AB

Categorical Frequency Distribution - Categorical Frequency Distribution - Example

Step 4:Step 4: Find the percentage of values in each class by using the formula:% = f/n x 100% where f = frequency of the class AND

n = total number of values

Step 5:Step 5: Find the totals for the 3rd and 4th column.

Blood Type Frequency Distribution - Blood Type Frequency Distribution - Example

Class Tally Frequency Percent

A 5 20

B 7 28

O 9 36

AB 4 16

Total 25 100

Grouped Frequency DistributionsGrouped Frequency Distributions

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.

Lifetimes of Boat BatteriesLifetimes of Boat Batteries - Example

Class

limits

Class boundaries Tally 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

52 – 58 51.5 – 58.5 |||| | 6 24

59 – 65 58.5 – 65.5 | 1 25

25

Terms Associated with a 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.

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

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.

Terms Associated with a Grouped Terms Associated with a Grouped Frequency DistributionFrequency Distribution

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.

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

Guidelines for Constructing Guidelines for Constructing a Frequency Distributiona Frequency Distribution

There should be between 5 and 20 classes.

The class width should be an odd number. To ensure the midpoint of each class has the

same place value as the data The classes must be mutually exclusive.

Non-overlapping class limits

Guidelines for Constructing Guidelines for Constructing a Frequency Distributiona Frequency Distribution

The classes must be continuous. No gaps in a frequency distribution. A class must be included even though there’s

no value in it. The classes must be exhaustive.

Enough classes to accommodate all the data.

The class must be equal in width. To avoid distorted view of the data Exception occurs when a distribution has a

class that’s open- ended.

Open-Ended Distribution

Age Frequency

10 – 20 3

21 – 31 6

32 – 42 4

43 – 53 10

54 and above 8

Minutes Frequency

Below 100 16

110 – 114 24

115 – 119 38

120 – 124 14

125 – 129 5

Example 1Example 2

Procedure for Constructing 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.

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.

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

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.)

Grouped Frequency Distribution - Grouped Frequency Distribution - Example

10 8 6 14

22 13 17 19

11 9 18 14

13 12 15 15

5 11 16 11


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.


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.


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.


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.


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.


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.


Class Limits Class Boundaries Frequency Cumulative Frequency

05 to 07 4.5 - 7.5 2 2

08 to 10 7.5 - 10.5 3 5

11 to 13 10.5 - 13.5 6 11

14 to 16 13.5 - 16.5 5 16

17 to 19 16.5 - 19.5 3 1920 to 22 19.5 - 22.5 1 20

Note: The dash “-” represents “to”.

Ungrouped Frequency 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.

Number of Miles TraveledNumber of Miles Traveled - Example

Class Frequency

5 24

10 16

15 10

To organize the data in a meaningful, intelligible way.

To enable the reader to determine the nature or shape of distribution.

To facilitate computational procedures for measures of average and spread.

To enable the researcher to draw charts and graphs for the presentation of data.

To enable the reader to make comparisons among different data sets.

Reasons for constructing a frequency Reasons for constructing a frequency distributiondistribution

Histograms, Frequency 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).

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


Example of a HistogramExample of a Histogram

2017141185

6

5

4

3

2

1

0

Number of Cigarettes Smoked per Day

Freq

uenc

y

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.


Example of a Frequency PolygonExample of a Frequency Polygon

Frequency Polygon

262320171411852

6

5

4

3

2

1

0


Fre

quen

cy

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


Example of an OgiveExample of an Ogive

262320171411852

20

10

0


Cum

ulat

ive

Fre

quen

cyOgive

Other Types of GraphsOther Types of Graphs

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

Other Types of GraphsOther Types of Graphs- Pareto Chart- Pareto Chart

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.

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


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

Other Types of Graphs Other Types of Graphs - Time Series Graph- Time Series Graph

19941993199219911990

89

87

85

83

81

79

77

75

Year

Rid

ersh

ip (

in m

illio

ns)PORT AUTHORITY TRANSIT RIDERSHIP


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.

Other Types of Graphs Other Types of Graphs - Pie Graph- Pie Graph

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

chapter 2 250110 083240

Technology

frequency distribution

frequency polygons

frequency age

frequency minutes

class width

specific class

previous class

upper class limits