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Chapter 2 Data Presentation
10/04/2015
OutlineOrganize data into a frequency distribution.Graphical presentation: Histogram, frequency polygon, cumulative frequency polygon. Graphical techniques: line chart, bar chart
andpie chart
Chapter 2 Data Presentation
10/04/2015Last Update: April 2007 Slide number 2
Recommended ReadingCustomised Text, Adapted from ‘Statistical Techniques in Business & Economics by Lind, Marchal 16th Edition’ McGraw Hill
Chapter 2
Page 17 - 49
Frequency Distribution
A grouping of data into mutually exclusive classes
It shows the number of observations in each class
10/04/2015 Slide number 3
Frequency Distribution - Terms
Class limits:Upper Limit: The highest possible value in a class
Lower limit: The lowest possible value in a class
10/04/2015 Slide number 4
Frequency Distribution - Terms
Class midpoint:A point that divides a class into two equal parts. Thisis 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.
10/04/2015 Slide number 5
Frequency Distribution – terms
Class Interval: 3.0-2.0= 1
EXAMPLE 1: Amount of rice sold (in ‘000 kg)
10/04/2015 Slide number 6
Class (amount sold
‘000 kg
Frequency (f) mid-point
2.0 - up to 3.0 1 (3+2)/2 = 2.5
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
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
10/04/2015 Slide number 7
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.
10/04/2015 Slide number 8
Step 1: Decide on the number of classes
2k> n
where k = number of classesn = 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 5classes, i.e., k=5.
Construction a Frequency Distribution
10/04/2015 Slide number 9
Step 2: Determine the class interval
i (Highest value - Lowest
value) Number of
classes
Construction a Frequency Distribution
i 33.8 - 10.3 4.7 5
5
.
10/04/2015 Slide number 10
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.
Construction a Frequency Distribution
10/04/2015 Slide number 11
EXAMPLE 10 continued
Interval= 5
Step 3: Set the individual class limits
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Hours studying Frequency, f
10 up to 15
15 up to 20
20 up to 25
25 up to 30
30 up to 35
EXAMPLE 10 continued
Step 4 & 5: Tally and Count the numbers in each class
10/04/2015 Slide number 13
Hours studying Frequency, f10 up to 15 715 up to 20 1220 up to 25 7
25 up to 30 3
30 up to 35 1
Constructing a Frequency Distribution
10/04/2015 Slide number 14
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 mustprovide 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
10/04/2015 Slide number 15
Relative Frequency Distribution
Relative Frequency = freq / freq
10/04/2015 Slide number 16
Hours f Relative Frequency
10 up to 15 7 7/30=.2333
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
Graphical Presentation of a FrequencyDistribution
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
10/04/2015 Slide number 17
Histogram for Hours Spent Studying
EXAMPLE 3
14
12
10
8
6
4
2
0
10/04/2015 Slide number 18
12.5 17.5 22.5 27.5
Hours spent studying
32.5
Fre
qu
ency
Graphical Presentation of a FrequencyDistribution
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.
10/04/2015 Slide number 19
Frequency Polygon for Hours Spent Studying
141210
86420
7.5
12.5
17.5
22.5
27.5
32.5
Hours spent studying
Fre
qu
en
cy
EXAMPLE 4
10/04/2015 Slide number 20
Cumulative Frequency
10/04/2015 Slide number 21
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
2926+
10/04/2015 Slide number 22
30
EXAMPLE 5
Constructing a Cumulative Frequency Polygon (Ogive)
• Scale the upper limit of the classes on theX-axis
• The cumulative frequency distribution is marked on the Y-axis
• The polygon cross the X-axis at the lower limitof the first class
10/04/2015 Slide number 23
10/04/2015Last Update: April 2007
Constructing a Cumulative Frequency Polygon
x-axis y-axisFirst limit with Cum Freq = 0
Slide number 24
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
35
30
25
2015
10
5
0
0
7
19
2629 30 30
10 15 20 25 3035
Hours Spent Studying
35
Cu
mu
late
d F
req
ue
ncy
10/04/2015 Slide number 25
Cumulative Frequency Polygon For HoursStudying
About 86.6% of the 30 students studied for less than 25 hours
86.6% x 30 = 26 students
35
30
25
2015
10
5
0
0
7
19
2629 30 30
10 15 20 25 3035
Hours Spent Studying
35
Cu
mu
late
d F
req
ue
nc
y
10/04/2015 Slide number 26
Slide number 27
10/04/2015
Cumulative Frequency Polygon For HoursStudying
About
15 20 25 30 35 35
Hours Spent Studying
students spent more than 25 hours studying.
4
Explanation: 26 students spent less than 25 hours,so we have 30 – 26 = 4 students spent more than25 hours
35
30
25
2015
10
5
0
0
7
19
2629 30 30
10
Cu
mu
late
d F
req
ue
ncy
Other Graphical Presentation of Data
10/04/2015 Slide number 28
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.
Line Graph – EXAMPLE 6
3736353433323130292827
Med
ian
Age
U.S. median age by gender
Males
Females
10/04/2015 Slide number 29
Year Males Females
1992 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
A bar chart for the number of unemployed per 100,000 population for selected cities during 2001
Bar Chart – EXAMPLE 7
7300
5400
6700
89008200
890010000
9000
8000
7000
6000
5000
4000
3000
2000
1000
01 2 3 4
Cities
5 6
# u
nem
plo
yed
/100
,000
Atlanta
Boston
Chicago Los
Angeles
New York
Washington
10/04/2015 Slide number 30
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
46%
24%
18%
7% 5%
# of runners
Nike Adidas Reebok AsicsOther
A sample of 200 runners were asked to indicate their favorite type of running shoe. Draw a pie chart based on the following information.
Pie Chart – EXAMPLE 8
10/04/2015 Slide number 31
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
Ethical Visual
10/04/2015Last Update: April 2007 Slide number 32
Charts Examples
10/04/2015Last Update: April 2007 Slide number 33
