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Lecture 1b: Describing Data: Frequency Distributions and

Graphic Presentation

North South University School of BusinessSlide 1 of 39

Outline

Organize data into a frequency distribution Portray a frequency distribution in a histogram,

frequency polygon, and cumulative frequency polygon

Present data using such graphic techniques as linecharts bar charts and pie charts


charts, bar charts, and pie charts

Frequency Distribution

A Frequency Distribution is a grouping of data into mutually exclusive

categories showing the number of


categories showing the number of observations in each class/category

Constructing a frequency distribution

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


Presenting data (graph) Drawing conclusions

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Frequency distribution : Construction Procedure

1. Collect at least 30 consecutive data points from aprocess

2. Arrange the data in increasing order3. Decide on the number of intervals or classes. No. of


intervals, k = 1 + 3.3 log10 n. Alternatively, use 2k n.4. Decide the class interval or width,

kLHi

5. Set the individual class limits

6. Tally the data into the classes

7. Count the number of items in each class

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

Cl FCl F ThClass intervalClass interval: The class

Definitions


Class FrequencyClass Frequency: The number of observations in

each class

interval is obtained by subtracting the lower limit of

a class from the upper limit of the same class. The class intervals should be equal

Example 1Dr. Tillman is Dean of the School of Business,Socastee University. He wishes to prepare areport showing the number of hours per weekstudents spend studying. He selects a randomsample of 30 students and determines thenumber of hours each student studied last week


15.0, 23.7, 19.7, 15.4, 18.3, 23.0, 14.2, 20.8, 13.5, 20.7, 17.4, 18.6, 12.9, 20.3, 13.7, 21.4, 18.3, 29.8, 17.1, 18.9, 10.3, 26.1, 15.7, 14.0, 17.8, 33.8, 23.2,

12.9, 27.1, 16.6

Organize the data into a frequency distribution

number of hours each student studied last week

Example 1 (contd)

Step OneStep One:: 30 Data points are available

Step TwoStep Two:: Arrange the data in increasing order


10.3, 12.9, 12.9, 13.5, 13.7, 14.0, 14.2, 15.0, 15.4, 15.7, 16.6, 17.1, 17.4, 17.8, 18.3, 18.3, 18.6, 18.9, 19.7, 20.3, 20.7, 20.8, 21.4, 23.0, 23.2, 23.7, 26.1,

27.1, 29.8, 33.8

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Example 1 (contd)Step ThreeStep Three:: Decide on the number of classes using the

formula 2k n

where k = number of classes n = 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

Step FourStep Four: Determine the class interval or width using the formula

33.8 33.8 10.310.355 = 4.7= 4.7

H H LLkk

i i >> ==

Example 1 (contd)


where H=highest value, L=lowest value

Round up for an interval of 5 hours.

Set the lower limit of the first class at 7.5 hours, giving a total of 6 classes

Hours studying Frequency, f

7.5 up to 12.5 1

Step FiveStep Five: Set the individual class limits andSteps Six and SevenSteps Six and Seven: Tally and count the number of items in

each class

Example 1 (contd)


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

Class Midpoint: find the midpoint of each interval, use the following formula: Upper limit + lower limit

2Hours studying Midpoint f

7.5 up to 12.5 (12.5+7.5)/2 =10.0 1

12 5 up to 17 5 (17 5+12 5)/2=15 0 12

Example 1 (contd)


12.5 up to 17.5 (17.5+12.5)/2=15.0 12

17.5 up to 22.5 (22.5+17.5)/2=20.0 10

22.5 up to 27.5 (27.5+22.5)/2=25.0 5

27.5 up to 32.5 (32.5+27.5)/2=30.0 1

32.5 up to 37.5 (37.5+32.5)/2=35.0 1

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Hours f Relative Frequency

7.5 up to 12.5 1 1/30=.0333

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

Example 1 (contd)


12.5 up to 17.5 12 12/30=.400

17.5 up to 22.5 10 10/30=.333

22.5 up to 27.5 5 5/30=.1667

27.5 up to 32.5 1 1/30=.0333

32.5 up to 37.5 1 1/30=.0333

TOTAL 30 30/30=1

Graphic Presentation of a Frequency Distribution

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

Cumulative FrequencyCumulative Frequency distribution


A Histogram is a graph in 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

Histogram for Hours Spent Studying


midpoint

Practice Problem 1 The commissions earned for the first quarter of last year by the 11

members of the sales staff at Master Chemical Company are:

$1,650, $1,475, $1,510, $1,670, $1,595, $1,760, $1,540, $1,495,$1,590, $1,625, $1,510

(a) What are the values such as $1,650 and $1,475 called?

(b) Using $1,400 up to $1,500 as the first class, $1,500 up to $1,600


as the second class, and so forth, organize the quarterlycommissions into a frequency distribution.

(c) What are the numbers in the right column of your frequencydistribution called?

(d) Describe the distribution of quarterly commissions, based on thefrequency distribution. What is the largest amount of commissionearned? What is the smallest? What is the typical amount earned?

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Solution: Practice Problem 1

a. The raw data or ungrouped data.

b.


c. Class frequencies.

d. The largest concentration of commissions is $1,500 up to $1,600. The smallest commission is about $1,400 and the largest is about $1,800.

Practice Problem 2

Barry Bonds of the San Francisco Giants established anew single season home run record by hitting 73 homeruns during the 2001 season. The longest of the homeruns travelled 488 feet and the shortest 320 feet. Youneed to construct a frequency distribution of these home


q yrun lengths.

(a) How many classes would you use?(b) What class interval would you suggest?(c) What actual classes would you suggest?


a. 26 = 64 < 73 < 128 = 27. So 7 classes arerecommended.

b. The interval width should be at least (488 - 320)/7 =24. Class intervals of 25 or 30 feet are both reasonable.


c. If we use a class interval of 25 feet and begin with alower limit of 300 feet, eight classes would be necessary.A class interval of 30 feet beginning with 300 feet is alsoreasonable. This alternative requires only seven classes.

Practice Problem 3 Table below shows the relative frequency distribution for the vehicles sold

last month at Whitner Autoplex.

(a) How many vehicles sold for $18,000 up to $21,000?

(b) What percent of the vehicles sold for a price between $18,000 and$21,000?

(c) What percent of the vehicles sold for $30,000 or more?


Selling Price ($ thousands)

Frequency Relative Frequency

15 up to 18 8 0.1000

18 up to 21 23 0.2875

21 up to 24 17 0.2125

24 up to 27 18 0.2250

27 up to 30 8 0.1000

30 up to 33 4 0.0500

33 up to 36 2 0.0250

Total 80 1.0000

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a. 23

b. 28.75%, found by (23/80) 100

c. 7.5%, found by (6/80) 100


Graphic Presentation of a Frequency Distribution

A Frequency PolygonFrequency Polygon consists of line segments connecting the


points formed by the class midpoint and the class frequency

810

1214

ency

Frequency Polygon for Hours Spent Studying


02

468

10 15 20 25 30 35Hours spent studying

Freq

ue

A Cumulative Cumulative Frequency Frequency

DistributionDistribution is used to determine how

many or what

To create a cumulative frequency

polygon, scale the upper limit of each class along the X-

Cumulative Frequency distribution


many or what proportion of the data values are

below or above a certain value

class along the X-axis and the

corresponding cumulative

frequencies along the Y-axis

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Cumulative Frequency Table for Hours Spent Studying

Hours Studying

Upper Limit

f Cumulative Frequency

7.5 up to 12.5 12.5 1 1 12 5 up to 17 5 17 5 12 13 (1+12)


12.5 up to 17.5 17.5 12 13 (1+12)17.5 up to 22.5 22.5 10 23 (13+10) 22.5 up to 27.5 27.5 5 28 (23+5) 27.5 up to 32.5 32.5 1 29 (28+1)

32.5 up to 37.5 37.5 1 30 (29+1)

20253035

Frequency

Cumulative Frequency Distributionfor Hours Spent Studying


05

1015

10 15 20 25 30 35Hours Spent Studying

Practice Problem 4

The annual imports of a selected group of electronic suppliers are shown inthe following frequency distribution.

(a) Portray the imports as a histogram.

(b) Portray the imports as a relative frequency polygon.

(c) Summarize the important facets of the distribution (such as classes with


the highest and lowest frequencies).

Imports ($ millions) Number of Suppliers2 up to 5 65 up to 8 138 up to 11 20

11 up to 14 1014 up to 17 1



c. The smallest annual sales volume of imports by asupplier is about $2 million, the largest about $17 million.The highest frequency is between $8 million and $11million.

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Practice Problem 5 A sample of the hourly wages of 15 employees at the Home Depot in

Brunswick, Georgia, was organized into the table below.

(a) What is the table called?

(b) Develop a cumulative frequency distribution and portray the distributionin a cumulative frequency polygon.

(c) On the basis of the cumulative frequency polygon, how many employeesearn $11 an hour or less? Half of the employees earn an hourly wage of


p y y ghow much or more? Four employees earn how much or less?

Hourly Wages Number of Employees$8 up to $10 310 up to 12 712 up to 14 414 up to 16 1

Solution: Practice Problem 5a. A frequency distribution.

b. Hourly Wages

Number of Employees

Less than $8 0Less than 10 3L th 12 10


c. About seven employees earn $11.00 or less. Abouthalf the employees earn $11.25 or more. About fouremployees earn $10.25 or less.

Less than 12 10Less than 14 14Less than 16 15

Line graphs are typically used to show the change or trend in a variable over time

Year Males Females1992 30.5 32.91993 30.8 33.21994 31 1 33 5

Line Graphs


1994 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

Example 3 (contd)


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Construct a bar chart for the number of unemployed per 100,000 population for selected cities during 2001

C ity N u m b e r o f u n e m p lo y e d

A BBar Chartar Chart can be used to depict any of the levels of measurement (nominal, ordinal, interval, or ratio)

Bar Chart


y p yp e r 1 0 0 ,0 0 0 p o p u la tio n

A tla n ta , G A 7 3 0 0 B o s to n , M A 5 4 0 0 C h ic a g o , IL 6 7 0 0

L o s A n g e le s , C A 8 9 0 0 N e w Y o rk , N Y 8 2 0 0

W a sh in g to n , D .C . 8 9 0 0

Bar Chart for the Unemployment Data

7300

5400

6700

89008200

8900

6000700080009000

10000

ed/1

00,0

00

Atlanta


010002000300040005000

1 2 3 4 5 6

Cities

# un

empl

oye

BostonChicagoLos AngelesNew YorkWashington

Pie Chart

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

A Pie ChartPie 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


ype o s oe # o u e s % o o a

Nike 92 46.0

Adidas 49 24.5

Reebok 37 18.5

Asics 13 6.5

Other 9 4.5

Pie Chart for Running Shoes

18.50%6.50%

4.50%NikeAdidasReebok


46%

24.50%ReebokAsicsOther

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Practice Problem 6

The Clayton County Commissioners want to showtaxpayers attending the forthcoming meeting whathappens to their tax dollars. The total amount of taxescollected is $2 million. Expenditures are: $440,000 forschools, $1,160,000 for roads, $320,000 for


, $ , , , $ ,administration, and $80,000 for supplies. A pie chartseems ideal to show the portion of each tax dollar goingfor schools, roads, administration, and supplies. Convertthe dollar amounts to percents of the total and portraythe percents in the form of a pie chart.



Assignment-1

Problems 5, 6 (pages 30-31) (page 27) Problem 11 (page 37) (page 33)

Problem 16 (page 41) (page 39)


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