Lec 02_ Organising Data(2)

10/04/2015

Chapter 2 Data Presentation

Outline

Organize data into a frequency distribution.

Graphical presentation: Histogram, frequency

polygon, cumulative frequency polygon.

Graphical techniques: line chart, bar chart and

pie chart

Slide number 2

Chapter 2 Data Presentation

Recommended Reading

Customised Text, Adapted from ‘Statistical Techniques in

Business & Economics by Lind, Marchal 16th Edition’

McGraw Hill

Chapter 2

Page 17 - 49

10/04/2015Last Update: April 2007

Slide number 310/04/2015

Frequency Distribution

A grouping of data

into mutually

exclusive classes

It shows the number

of observations in

each class


Frequency Distribution - Terms

Class limits:

Upper Limit: The highest possible value in a class

Lower limit: The lowest possible value in a class


Frequency Distribution - Terms

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.

Class interval: (Class Width)

The class interval is obtained by subtracting the

lower limit of a class from the lower limit of the

next class.


Frequency Distribution – terms

Class

(amount sold

‘000 kg

Frequency (f) mid-point

2.0 - up to 3.0 1

3.0 - up to 4.0 0 3.5

4.0 - up to 5.0 2 4.5

5.0 - up to 6.0 8 5.5

6.0 - up to 7.0 5 6.5

7.0 - up to 8.0 4 7.5

Total 20

Class

Interval:

3.0-2.0

= 1

(3+2)/2 = 2.5

EXAMPLE 1: Amount of rice sold (in ‘000 kg)


Steps

Decide on the number of classes

Determine the class interval

Set the individual class limits

Tally the observations into the classes

Count the number of items in each class

Constructing a Frequency Distribution


EXAMPLE 2

Dr. Tillman is Dean of the School of Business Socastee

University. He wishes to prepare a report showing the number of hours per week students spend on studying. He selects a random sample of 30 students and determines the number 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 these data into a frequency distribution.


Step 1: Decide on the number of classes 2k > n

where k = number of classes

n = number of observations

oThere are 30 observations so n=30.

o2 raised to the 5th power is 32.

i.e. 25 = 32

oTherefore, we should have at least 5

classes, i.e., k=5.

Construction a Frequency Distribution


Step 2: Determine the class interval


classes ofNumber

ue)Lowest val - lueHighest va(i

.

57.45

10.3 - 3.83i


Step 3: Set the individual class limits


Ensure that the lower limit of the first class is

smaller or equal than the smallest value and

the upper limit of the last class is larger or

equal to the largest value

Set the lower limit of the first class at

10 hours, giving a total of 5 classes.


EXAMPLE 10 continued

Hours studying Frequency, f

10 up to

up to

up to 25

25 up to 30

30 up to 35

15

20

15

20

Interval

= 5

Step 3: Set the individual class limits


EXAMPLE 10 continued

Hours studying Frequency, f

10 up to 15 7

15 up to 20 12

20 up to 25 7

25 up to 30 3

30 up to 35 1

Step 4 & 5: Tally and Count the numbers in each class


Constructing a Frequency Distribution

Preferably between 5 – 15 classes

If possible, the classes interval should be the same for all classes

The classes must be mutually exclusive, i.e. avoid overlapping classes. Each data point must fall in only one class.

The classes must be all inclusive, i.e. the classes must provide a place to record every value in the data set.

Preferably no open-ended classes.

open-ended classes: classes without lower or upper limit example: below 7.5 ; above 37.5


A relative frequency distribution

shows the percent of observations in

each class.

Relative Frequency


Hours f Relative

Frequency

10 up to 15 7

15 up to 20 12 12/30=.400

20 up to 25 7 7/30=.2333

25 up to 30 3 3/30=.1000

30 up to 35 1 1/30=.0333

TOTAL 30 30/30=1

Relative Frequency Distribution

Relative Frequency = freq / freq

7/30=.2333


Graphical Presentation of a Frequency

Distribution

Histograms

Classes marked on the horizontal axis

Frequency marked on the vertical axis

Frequencies of each class are

represented by the height of the bars

The bars are adjacent to each other


Histogram for Hours Spent Studying

0

2

4

6

8

10

12

14

12.5 17.5 22.5 27.5 32.5

Fre

qu

en

cy

Hours spent studying

EXAMPLE 3


Graphical Presentation of a Frequency

Distribution

Frequency Polygon

mid-point of the classes are marked on

the horizontal axis

Frequency marked on the vertical axis

Line segments connect the points that

represent the frequencies of their

respective classes.


Frequency Polygon for Hours Spent Studying

0

2

4

6

8

10

12

14

7.5 12.5 17.5 22.5 27.5 32.5

Fre

qu

en

cy

Hours spent studying

EXAMPLE 4


Cumulative Frequency

A cumulative frequency distribution is

used to determine how many or what

proportion of the data values are below

or above a certain value.

The cumulative frequency of a particular

class is found by adding the frequency of

that class to the cumulative frequency of

the previous class.


Hours f Cumulative

Frequency

(cf)

10 up to 15 7

15 up to 20 12

20 up to 25 7

25 up to 30 3

30 up to 35 1

TOTAL 30

Cumulative Frequency Distribution

7

19

29

26+

30

EXAMPLE 5


Constructing a Cumulative Frequency

Polygon (Ogive)

• Scale the upper limit of the classes on the

X-axis

• The cumulative frequency distribution is

marked on the Y-axis

• The polygon cross the X-axis at the lower limit

of the first class

Slide number 2410/04/2015Last Update: April 2007

Constructing a Cumulative Frequency

Polygon

x-axis y-axisFirst limit with

Cum Freq = 0

Hours f Cumulative

Frequency

(cf)

10 up to 15 7 7

15 up to 20 12 19

20 up to 25 7 26

25 up to 30 3 29

30 up to 35 1 30

TOTAL 30


Cumulative Frequency Polygon (OGIVE) For Hours Studying

About students spent less than 20 hours studying.19

0

7

19

2629 30 30

0

5

10

15

20

25

30

35

10 15 20 25 30 35 35

Hours Spent Studying

Cu

mu

late

d F

req

ue

nc

y


Cumulative Frequency Polygon For Hours Studying

About 86.6% of the 30 students studied for less than _______ hours25

86.6% x 30 = 26 students

0

7

19

2629 30 30

0

5

10

15

20

25

30

35

10 15 20 25 30 35 35


Cu

mu

late

d F

req

uen

cy


Cumulative Frequency Polygon For Hours Studying

About students spent more than 25 hours studying.4

so we have 30 – 26 = 4 students spent more than

25 hours

Explanation: 26 students spent less than 25 hours,

0

7

19

2629 30 30

0

5

10

15

20

25

30

35

10 15 20 25 30 35 35


Cu

mu

late

d F

req

ue

nc

y


Other Graphical Presentation of Data

Line Graph

used to show the change or trend in a variable over time

Bar Chart

depicts both the qualitative and quantitative data

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.


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

Line Graph – EXAMPLE 6

27

28

29

30

31

32

33

34

35

36

37

Med

ian A

ge

U.S. median age by gender

Males

Females


A bar chart for the number of unemployed per

100,000 population for selected cities during 2001

City No. 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

Bar Chart – EXAMPLE 7

7300

5400

6700

89008200

8900

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

1 2 3 4 5 6

# u

ne

mp

loye

d/1

00

,00

0

Cities

Atlanta

Boston

Chicago

Los Angeles

New York

Washington


46%

24%

18%

7% 5%

# of runners

Nike Adidas Reebok Asics Other

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

Pie Chart – EXAMPLE 8

Slide number 32

Ethical Visual


Slide number 33

Charts Examples


Documents

Lec 02_ Organising Data(2)