Welcome to Chapter 2 MBA 541

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Welcome to Chapter 2 MBA 541

BENEDICTINE UNIVERSITY

• Statistics and Frequency Distributions

• Describing Data: Frequency Distributions and Graphic Presentation

• Chapter 2

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Chapter Two

Please, Read Chapter Two

in Lind before viewing this presentation.

StatisticalTechniques in

Business &Economics

Lind

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GoalsWhen you have completed this chapter, you

will be able to:

• ONE– Organize data into a frequency distribution.

• TWO– Portray a frequency distribution in a histogram,

frequency polygon, and cumulative frequency polygon.

• THREE– Present data using such graphic techniques as

line charts, bar charts, and pie charts.

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Frequency Distribution

A Frequency Distribution is a grouping of data into

mutually exclusive categories showing the number of

observations in each class.

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Constructing a Frequency Distribution

Constructing a frequency distribution involves:

• Determining the question to be addressed

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Constructing a Frequency Distribution

Constructing a frequency distribution involves:

• Determining the question to be addressed• Collecting raw data

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Constructing a Frequency Distribution

Constructing a frequency distribution involves:

• Determining the question to be addressed• Collecting raw data• Organizing data (frequency distribution)

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Constructing a Frequency Distribution

Constructing a frequency distribution involves:

• Determining the question to be addressed• Collecting raw data• Organizing data (frequency distribution)• Presenting data (graph)

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Constructing a Frequency Distribution

Constructing a frequency distribution involves:

• Determining the question to be addressed• Collecting raw data• Organizing data (frequency distribution)• Presenting data (graph)• Drawing conclusions

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A Simple Example of Creating a Frequency Distribution

• The next few slides illustrate the steps in constructing a frequency distribution.

• As Example 1, this illustrates, clarifies, and demonstrates the conceptual steps that were just discussed.

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A Simple Example of Creating a Frequency Distribution

Background Information:

• This company employs 17 individuals.

• This company maintains employee records.

• The company wants to understand employee length of service data.

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A Simple Example of Creating a Frequency Distribution

Step 1: Determine the question to be addressed

• An employer wants to answer the question, “How many people have worked for how many years at this business?”

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A Simple Example of Creating a Frequency Distribution

Step 2: Collect the raw data.

Length of Service(in years)

4 3 2 10 6 6

5 8 4 8 4

6 2 3 3 7 5

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A Simple Example of Creating a Frequency Distribution

Step 3: Determine the class interval or width.

Here we will choose five classes with an interval of two years.

Length of Service(in years)

4 3 2 10 6 6

5 8 4 8 4

6 2 3 3 7 5

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A Simple Example of Creating a Frequency Distribution

Step 4: Tally the results into classes.

Length of Service(in years)

4 3 2 10 6 6

5 8 4 8 4

6 2 3 3 7 5

Frequency Distribution

1-2 years 2

3-4 years 6

5-6 years 57-8 years 39-10 years 1Total 17

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A Simple Example of Creating a Frequency Distribution

Step 5: Present data (graph).

Frequency Distribution of Employee Service

0

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3

4

5

6

7

1-2 3-4 5-6 7-8 9-10

Years of Service

Fre

qu

en

cy

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A Simple Example of Creating a Frequency Distribution

Step 6: Draw conclusions.

Frequency Distribution

1-2 years 2

3-4 years 6

5-6 years 57-8 years 39-10 years 1Total 17

• The majority of the employees have between 3 and 6 years of service.

• Only one employee has more than 8 years of service.

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Definitions

Class Interval: The class interval is obtained by subtracting the lower limit of a class from the lower limit of the next class. The class intervals should be equal.

Class Midpoint: A point that divides a class into two equal parts. This is the average of the upper and lower class limits.

Class Frequency: The number of observations in each class.

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Example 2

Background: Dr. Tillman is Dean of the School of Business at Socastee University. He wishes to prepare a report showing the number of hours per week that students spend studying. He selects a random sample of 30 students and determines the number of hours that each student studied last week.

Hours Spent Studying Last Week

15.0 17.4 10.3 18.3 14.0 18.3

20.7 18.9 16.6 20.3 14.2 33.8

17.1 27.1 15.4 15.7 21.4 13.5

12.9 19.7 12.9 23.0 17.8 29.8

23.7 18.6 26.1 13.7 20.8 23.2

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Example 2

Step One: Decide on the number of class using the formula:

Where: k = number of classes, andn = number of observations.

•There are 30 observations so n = 30.•Two raised to the fifth power is 32.•Therefore, we should have at least 5 classes;

i.e., k = 5.

nk2

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Example 2

Step Two: Determine the class interval or width by using the formula:

Where: H = highest value, andL = lowest value.

• Round up for an interval of 5 hours.• Set the lower limit of the first class at 7.5 hours.• Generate a total of 6 classes.

H- L 33.8-10.3

i = =4.7k 5

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Example 2

Step Three: Set the individual class limits and

Steps Four and Five: Tally and count the number of items in each class.

Hours Studying Frequency, f

7.5 up to 12.5 1

12.5 up to 17.5 12

17.5 up to 22.5 10

22.5 up to 27.5 5

27.5 up to 32.5 1

32.5 up to 37.5 1

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Example 2

Hours Studying Midpoint Frequency, f

7.5 up to 12.5 10.0 1

12.5 up to 17.5 15.0 12

17.5 up to 22.5 20.0 10

22.5 up to 27.5 25.0 5

27.5 up to 32.5 30.0 1

32.5 up to 37.5 35.0 1

Class Midpoint: find the midpoint of each interval by using the following formula:

Upper Limit Lower LimitClass Midpoint

2

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Example 2 – Chart 2-11

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32333435363738

A B C D E F G H I J KHours Studying

15.0 Bin Frequency Bins Bin Frequency Midpoints23.7 10.3 1 12.5 12.5 1 10.019.7 15.0 7 17.5 17.5 12 15.015.4 19.7 11 22.5 22.5 10 20.018.3 24.4 7 27.5 27.5 5 25.023.0 29.1 2 32.5 32.5 1 30.014.2 More 2 37.5 37.5 1 35.020.8 More 013.520.717.418.612.920.313.721.418.329.817.118.910.326.115.714.017.833.823.212.927.116.6

Histogram for Hours Spent Studying

0

5

10

15

10.0 15.0 20.0 25.0 30.0 35.0

Hours Spent Studying

Frequency

Line Graph for Hours Spent Studying

0

5

10

15

10.0 15.0 20.0 25.0 30.0 35.0

Hours Spent Studying

Frequency

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Example 2

Hours Studying Midpoint Frequency, f RelativeFrequency

7.5 up to 12.5 10.0 1 1/30 = 0.0333

12.5 up to 17.5 15.0 12 12/30 = 0.4000

17.5 up to 22.5 20.0 10 10/30 = 0.3333

22.5 up to 27.5 25.0 5 5/60 = 0.1667

27.5 up to 32.5 30.0 1 1/30 = 0.0333

32.5 up to 37.5 35.0 1 1/30 = 0.0333

Total 35.0 30 30/30 = 1.0000

A Relative Frequency Distribution shows the percent of observations in each class.

Class FrequencyRelative Frequency

Total Number of Observations

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Graphic Presentations of a Frequency Distribution

A Histogram is a graph on which the class midpoints or limits are marked on the horizontal axis and the class frequencies on the vertical axis.

The class frequencies are represented by the heights of the bars and the bars are drawn adjacent to each other.

The three commonly used graphic forms are Histograms, Frequency Polygons, and

Cumulative Frequency Polygons.

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Histogram For Hours Spent Studying

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4

6

8

10

12

14

10 15 20 25 30 35

Hours spent studying

Fre

qu

ency

midpoint

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Graphic Presentations of a Frequency Distribution

A Frequency Polygon consists of line segments connecting the points formed by the class midpoints and the class frequency.

The three commonly used graphic forms are Histograms, Frequency Polygons, and

Cumulative Frequency Polygons.

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Frequency Polygon For Hours Spent Studying

Line Graph for Hours Spent Studying

0

5

10

15

10.0 15.0 20.0 25.0 30.0 35.0

Hours Spent Studying

Freq

uenc

y

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Graphic Presentations of a Frequency Distribution

A Cumulative Frequency Polygon is used to determine how many or what proportion of the data values are below or above a certain value.

To create a cumulative frequency polygon, scale the upper limit of each class along the X-axis and the corresponding cumulative frequencies along the Y-axis.

The three commonly used graphic forms are Histograms, Frequency Polygons, and Cumulative Frequency Polygons.

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Cumulative Frequency TableFor Hours Spent Studying

Hours Studying Upper Limit

Frequency CumulativeFrequency

7.5 up to 12.5 12.5 1 1

12.5 up to 17.5 17.5 12 1+12 = 13

17.5 up to 22.5 22.5 10 13+10 = 23

22.5 up to 27.5 27.5 5 23+5 = 28

27.5 up to 32.5 32.5 1 28+1 = 29

32.5 up to 37.5 37.5 1 29+1 = 30

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Cumulative Frequency Polygon

For Hours Spent Studying

0

5

10

15

20

25

30

35

10 15 20 25 30 35

Hours Spent Studying

Frequency

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Line GraphsLine graphs are typically used to show the change

or trend in a variable over time.Year Males Females

1992 30.5 32.9

1993 30.8 33.2

1994 31.1 33.5

1995 31.4 33.8

1996 31.6 34.0

1997 31.9 34.3

1998 32.2 34.6

1999 32.5 34.9

2000 32.8 35.2

2001 33.2 35.5

2002 33.5 35.8

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Example 3 - Line Graphs

US Median Age by Gender

2728293031323334353637

Year

Ag

e Males

Female

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Example 3 – Chart 2-2

1234567891011121314151617181920

A B C D E F G H I J KYear Males Females1992 30.5 32.91993 30.8 33.21994 31.1 33.51995 31.4 33.81996 31.6 34.01997 31.9 34.31998 32.2 34.61999 32.5 34.92000 32.8 35.22001 33.2 35.52002 33.5 35.8

US Median Age by Gender

2728293031323334353637

Year

Ag

e Males

Female

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Bar Chart

A Bar Chart can be used to depict any of the levels of measurement: nominal, ordinal, interval or ratio.

City Number of unemployedper 100,000 population

Atlanta, GA 7300

Boston, MA 5400

Chicago, IL 6700

Los Angeles, CA 8900

New York, NY 8200

Washington, DC 8900

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Bar Chart For the Unemployment Data

Number of unemployed per 100,000 population

01,0002,0003,0004,0005,0006,0007,0008,0009,000

10,000

Cities

Num

ber

of

unem

plo

yed p

er

100,0

00 p

opula

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Example 4 – Chart 2-3

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A B C D E F G H I J

City Number of unemployed per 100,000 population

Atlanta, GA 7300

Boston, MA 5400

Chicago, IL 6700

Los Angeles, CA

8900

New York, NY 8200

Washington, D.C.

8900

Number of unemployed per 100,000 population

0100020003000400050006000700080009000

10000

Atlant

a, G

A

Bosto

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A

Chica

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L

Los Ang

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, CA

New Y

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NY

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Pie ChartA Pie Chart is useful for displaying a relative frequency distribution. A circle is divided proportionally to the relative frequency and portions of the circle are allocated for the different groups.A sample of 200 runners were asked to indicate their favorite type of running shoe.Draw a pie chart based on the following information.

Type of shoe # of runners % of total

Nike 92 46.0

Adidas 49 24.5

Reebok 37 18.5

Asics 13 6.5

Other 9 4.5

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Pie Chart For Running Shoes

Pie Chart for running shoes

46.0%

24.5%

18.5%

6.5%

4.5%

Nike

AdidasReebok

AsicsOther

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Example 5 – Chart 2-4

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A B C D E F G H I

Type of shoe

# of runners

Nike 92Adidas 49Reebok 37Asics 13Other 9

Pie Chart for running shoes

46.0%

24.5%

18.5%

6.5%

4.5%

Nike

Adidas

Reebok

Asics

Other