Sqqs1013 ch2-a122

SQQS1013 Elementary Statistics
DESCRIPTIVE STATISTICSDESCRIPTIVE STATISTICS
2.1 INTRODUCTION
Raw data - Data recorded in the sequence in
which there are
collected and before they are processed or ranked
Array data - Raw data that is arranged in ascending or
descending order.
Here is a list of question asked in a large statistics class and the “raw data” given by one
of the students:
1. What is your sex (m=male, f=female)?
Answer : m
2. How many hours did you sleep last night?
Answer: 5 hours
3. Randomly pick a letter – S or Q.
Answer: S
4. What is your height in inches?
Answer: 67 inches
5. What’s the fastest you’ve ever driven a car (mph)?
Answer: 110 mph
• Quantitative raw data • Qualitative raw data
 These data also called ungrouped data.
Chapter 2: Descriptive Statistics 1
Example 2
Example 1

2.2 ORGANIZING AND GRAPHING QUALITATIVE DATA
2.2.1 Frequency Distributions Table
• A frequency distribution for qualitative data lists all categories and the
number of elements that belong to each of the categories.
• It exhibits the frequencies are distributed over various categories
• Also called as a frequency distribution table or simply a frequency
table.
e.g. : The number of students who belong to a certain category is called
the frequency of that category.
2.2.2 Relative Frequency and Percentage Distribution
• A relative frequency distribution is a listing of all categories along with their
relative frequencies (given as proportions or percentages).
• It is commonplace to give the frequency and relative frequency distribution
together.
• Calculating relative frequency and percentage of a category
Relative Frequency of a category
= Frequency of that category
Sum of all frequencies
FORMULA
ξ∆Σ λϖ
β

Percentage (%) = (Relative Frequency)* 100
A sample of UUM staff-owned vehicles produced by Proton was identified and the
make of each noted. The resulting sample follows (W = Wira, Is = Iswara, Wj =
Waja, St = Satria, P = Perdana, Sv = Savvy):
Construct a frequency distribution table for these data with their relative frequency
and percentage.
W W P Is Is P Is W St Wj
Is W W Wj Is W W Is W Wj
Wj Is Wj Sv W W W Wj St W
Wj Sv W Is P Sv Wj Wj W W
St W W W W St St P Wj Sv
Solution:
Category Frequency
Relative
Frequency
Percentage (%)
Wira 19
Iswara 8
Perdana 4
Waja 10
Satria 5
Savvy 4
Total
2.2.3 Graphical Presentation of Qualitative Data
a) Bar Graphs
Example 3

• A graph made of bars whose heights represent the frequencies of
respective categories.
• Such a graph is most helpful when you have many categories to
represent.
• Notice that a gap is inserted between each of the bars.
• It has
o simple/ vertical bar chart
o horizontal bar chart
o component bar chart
o multiple bar chart
• Simple/ Vertical Bar Chart
To construct a vertical bar chart, mark the various categories on the horizontal
axis and mark the frequencies on the vertical axis
• Horizontal Bar Chart
To construct a horizontal bar chart, mark the various categories on the vertical
axis and mark the frequencies on the horizontal axis.
• • Component Bar Chart
UUM Staff-owned Vehicles Produced
By Proton
0 5 10 15 20
Wira
Perdana
Satria
TypesofVehicle
Frequency

 To construct a component bar chart, all categories is in one bar and every
bar is divided into components.
 The height of components should be tally with representative frequencies.
Suppose we want to illustrate the information below, representing the number of
people participating in the activities offered by an outdoor pursuits centre during
Jun of three consecutive years.
2004 2005 2006
Climbing 21 34 36
Caving 10 12 21
Walking 75 85 100
Sailing 36 36 40
Total 142 167 191
Solution:
• Multiple Bar Chart
 To construct a multiple bar chart, each bars that representative any
categories are gathered in groups.
 The height of the bar represented the frequencies of categories.
 Useful for making comparisons (two or more values).
 The bar graphs for relative frequency and percentage distributions can be
drawn simply by marking the relative frequencies or percentages, instead of the
class frequencies.
Example 4
Activities Breakdown (Jun)
0
50
100
150
200
2004 2005 2006
Year
Numberofparticipants
Sailing
Walking
Caving
Climbing

b) Pie Chart
• A circle divided into portions that represent the relative frequencies or
percentages of a population or a sample belonging to different
categories.
• An alternative to the bar chart and useful for summarizing a single
categorical variable if there are not too many categories.
• The chart makes it easy to compare relative sizes of each
class/category.
• The whole pie represents the total sample or population. The pie is
divided into different portions that represent the different categories.
• To construct a pie chart, we multiply 360o
by the relative frequency for
each category to obtain the degree measure or size of the angle for the
corresponding categories.
Movie
Genres
Frequency Relative Frequency Angle Size
Comedy
Action
Romance
Drama
Horror
Foreign
Science
Fiction
54
36
28
28
22
16
16
0.27
0.18
0.14
0.14
0.11
0.08
0.08
360*0.27=97.2o
360*0.18=64.8o
360*0.14=50.4o
360*0.14=50.4o
360*0.11=39.6o
360*0.08=28.8o
360*0.08=28.8o
Total 200 1.00 360o
Activities Breakdown (Jun)
0
20
40
60
80
100
120
2004 2005 2006
Year
Numberofparticipants
Climbing
Caving
Walking
Sailing
Example 5

c) Line Graph/Time Series Graph
• A graph represents data that occur over a specific period time of time.
• Line graphs are more popular than all other graphs combined because
their visual characteristics reveal data trends clearly and these graphs
are easy to create.
• When analyzing the graph, look for a trend or pattern that occurs over
the time period.
• Example is the line ascending (indicating an increase over time) or
descending (indicating a decrease over time).
• Another thing to look for is the slope, or steepness, of the line. A line
that is steep over a specific time period indicates a rapid increase or
decrease over that period.
• Two data sets can be compared on the same graph (called a
compound time series graph) if two lines are used.
• Data collected on the same element for the same variable at different
points in time or for different periods of time are called time series data.
• A line graph is a visual comparison of how two variables—shown on the
x- and y-axes—are related or vary with each other. It shows related
information by drawing a continuous line between all the points on a
grid.
• Line graphs compare two variables: one is plotted along the x-axis
(horizontal) and the other along the y-axis (vertical).
• The y-axis in a line graph usually indicates quantity (e.g., RM, numbers
of sales litres) or percentage, while the horizontal x-axis often measures
units of time. As a result, the line graph is often viewed as a time series
graph

A transit manager wishes to use the following data for a presentation showing
how Port Authority Transit ridership has changed over the years. Draw a time
series graph for the data and summarize the findings.
Year
Ridership
(in millions)
1990
1991
1992
1993
1994
88.0
85.0
75.7
76.6
75.4
Solution:
The graph shows a decline in ridership through 1992 and then leveling off for the years
1993 and 1994.
EXERCISE 1
Example 6
75
77
79
81
83
85
87
89
1990 1991 1992 1993 1994
Year
Ridership(inmillions)

1. The following data show the method of payment by 16 customers in a supermarket
checkout line. ( C = cash, CK = check, CC = credit card, D = debit and O =
other ).
C CK CK C CC D O C
CK CC D CC C CK CK CC
a. Construct a frequency distribution table.
b. Calculate the relative frequencies and percentages for all categories.
c. Draw a pie chart for the percentage distribution.
2. The frequency distribution table represents the sale of certain product in ZeeZee
Company. Each of the products was given the frequency of the sales in certain
period. Find the relative frequency and the percentage of each product. Then,
construct a pie chart using the obtained information.
Type of
Product
Frequency Relative Frequency Percentage Angle Size
A
B
C
D
E
13
12
5
9
11
3. Draw a time series graph to represent the data for the number of worldwide airline
fatalities for the given years.
Year 1990 1991 1992 1993 1994 1995 1996
No. of
fatalities
440 510 990 801 732 557 1132
4. A questionnaire about how people get news resulted in the following information
from 25 respondents (N = newspaper, T = television, R = radio, M = magazine).
N N R T T
R N T M R
M M N R N
T R M N M
T R R N N
a. Construct a frequency distribution for the data.
b. Construct a bar graph for the data.
5. The given information shows the export and import trade in million RM for four
months of sales in certain year. Using the provided information, present this data
in component bar graph.
Month Export Import
September
October
November
December
28
30
32
24
20
28
17
14
6. The following information represents the maximum rain fall in
millimeter (mm) in each state in Malaysia. You are supposed to help a
meteorologist in your place to make an analysis. Based on your knowledge,

present this information using the most appropriate chart and give your
comment.
State Quantity (mm)
Perlis
Kedah
Pulau Pinang
Perak
Selangor
Wilayah Persekutuan
Kuala Lumpur
Negeri Sembilan
Melaka
Johor
Pahang
Terengganu
Kelantan
Sarawak
Sabah
435
512
163
721
664
1003
390
223
876
1050
1255
986
878
456
2.3 ORGANIZING AND GRAPHING QUANTITATIVE DATA
2.3.1 Stem-and-Leaf Display
• In stem and leaf display of quantitative data, each value is divided into two
portions – a stem and a leaf. Then the leaves for each stem are shown
separately in a display.
• Gives the information of data pattern.
• Can detect which value frequently repeated.
25 12 9 10 5 12 23 7
36 13 11 12 31 28 37 6
14 41 38 44 13 22 18 19
Solution:
2.3.2 Frequency Distributions
Example 7

• A frequency distribution for quantitative data lists all the classes and the
number of values that belong to each class.
• Data presented in form of frequency distribution are called grouped data.
• The class boundary is given by the midpoint of the upper limit of one
class and the lower limit of the next class. Also called real class limit.
• To find the midpoint of the upper limit of the first class and the lower limit
of the second class, we divide the sum of these two limits by 2.
e.g.:
+
=
400 401
400.5
2
• Class Width (class size)
Class width = Upper boundary – Lower boundary
e.g. :
Width of the first class = 600.5 – 400.5 = 200
• Class Midpoint or Mark
class
boundary
FORMULA
ξ∆Σ λϖ
β
FORMULA
ξ∆Σ λϖ
β

Lower limit + Upper limit
class midpoint or mark =
2
e.g:
+
=
401 600
Midpoint of the 1st class = 500.5
2
• Constructing Frequency Distribution Tables
1. To decide the number of classes, we used Sturge’s formula, which is
c = 1 + 3.3 log n
where c is the no. of classes
n is the no. of observations in the data set.
2. Class width,
>
>
Largest value - Smallest value
Number of classes
Range
i
i
c
This class width is rounded up to a convenient number.
3. Lower Limit of the First Class or the Starting Point
Use the smallest value in the data set.
Example 8
FORMULA
ξ∆Σ λϖ
β
FORMULA
ξ∆Σ λϖ
β

The following data give the total home runs hit by all players of each of the 30 Major
League Baseball teams during 2004 season.
i) Number of classes, c = 1 + 3.3 log 30
= 1 + 3.3(1.48)
= 5.89 ≈ 6 class
ii) Class width,
−
>
>
≈
242 135
6
17.8
18
i
iii) Starting Point = 135
Table 2.10 : Frequency Distribution for Data of Table 2.9
Total Home Runs Tally f
135 – 152
153 – 170
171 – 188
189 – 206
207 – 224
225 – 242
|||| ||||
||
||||
|||| |
|||
||||
10
2
5
6
3
4
=∑ 30f
2.3.3 Relative Frequency and Percentage Distributions
FORMULA
ξ∆Σ λϖ
β

0
2
4
6
8
10
12
1 Total home runs
•
∑
Frequency of that class
Relative frequency of a class =
Sum of all frequencies
=
Percentage = (Relative frequency) 100
f
f
(Refer example 8)
Table 2.11: Relative Frequency and Percentage Distributions
Total Home Runs Class Boundaries
Relative
Frequency
%
135 – 152
153 – 170
171 – 188
189 – 206
207 – 224
225 – 242
134.5 less than 152.5
152.5 less than 170.5
170.5 less than 188.5
188.5 less than 206.5
206.5 less than 224.5
224.5 less than 242.5
0.3333
0.0667
0.1667
0.2000
0.1000
0.1333
33.33
6.67
16.67
20.00
10.00
13.33
Total 1.0 100%
2.3.4 Graphing Grouped Data
a) Histograms
 A histogram is a graph in which the class boundaries are marked on the
horizontal axis and either the frequencies, relative frequencies, or percentages
are marked on the vertical axis. The frequencies, relative frequencies or
percentages are represented by the heights of the bars.
 In histogram, the bars are drawn adjacent to each other and there is a space
between y axis and the first bar.
(Refer example 8)
Frequency histogram for Table 2.9
b) Polygon
 A graph formed by joining the midpoints of the tops of successive bars in a
histogram with straight lines is called a polygon.
134.5 152.5 170.5 188.5 206.5 224.5 242.5
Example 9
Example 10

Frequency polygon for Table 2.11
 For a very large data set, as the number of classes is increased (and the width of
classes is decreased), the frequency polygon eventually becomes a smooth
curve called a frequency distribution curve or simply a frequency curve.
Frequency distribution curve
c) Shape of Histogram
 Same as polygon.
 For a very large data set, as the number of classes is increased (and the width
of classes is decreased), the frequency polygon eventually becomes a smooth
curve called a frequency distribution curve or simply a frequency curve.
 The most common of shapes are:
(i) Symmetric
(ii) Right skewed
(iii) Left skewed
Example 11
134.5 152.5 170.5 188.5 206.5 224.5 242.5

Symmetric histograms
Right skewed and Left skewed
 Describing data using graphs helps us insight into the main characteristics of the
data.
 When interpreting a graph, we should be very cautious. We should observe
carefully whether the frequency axis has been truncated or whether any axis has
been unnecessarily shortened or stretched.
2.3.5 Cumulative Frequency Distributions
• A cumulative frequency distribution gives the total number of values that
fall below the upper boundary of each class.
Example 12

Using the frequency distribution of table 2.11,
Total Home
Runs
Class Boundaries f
Cumulative
Frequency
135 – 152
153 – 170
171 – 188
189 – 206
207 – 224
225 – 242
134.5 less than 152.5
152.5 less than 170.5
170.5 less than 188.5
188.5 less than 206.5
206.5 less than 224.5
224.5 less than 242.5
10
2
5
6
3
4
• Ogive
 An ogive is a curve drawn for the cumulative frequency distribution by joining
with straight lines the dots marked above the upper boundaries of classes at
heights equal to the cumulative frequencies of respective classes.
 Two type of ogive:
(i) ogive less than
(ii) ogive greater than
 First, build a table of cumulative frequency.
(Ogive Less Than)
Earnings
(RM)
Number of
students (f) Earnings (RM)
Cumulative
Frequency (F)
30 – 39 5 Less than 29.5 0
40 – 49 6 Less than 39.5 5
50 – 59 6 Less than 49.5 11
60 - 69 3 Less than 59.5 17
70 – 79 3 Less than 69.5 20
80 - 89 7 Less than 79.5 23
Less than 89.5 30
Total 30
Graph Ogive Less Than
Example 13
0
5
10
15
20
25
30
35
29.5 39.5 49.5 59.5 69.5 79.5 89.5
CumulativeFrequency
Earnings

(Ogive More Than)
Earnings
(RM)
Number of
students (f) Earnings (RM)
Cumulative
Frequency (F)
30 – 39 5 More than 29.5 30
40 – 49 6 More than 39.5 25
50 – 59 6 More than 49.5 19
60 - 69 3 More than 59.5 13
70 – 79 3 More than 69.5 10
80 - 89 7 More than 79.5 7
More than 89.5 0
Total 30
Graph Ogive More Than
2.3.6 Box-Plot
• Describe the analyze data graphically using 5 measurement: smallest
value, first quartile (K1), second quartile (median or K2), third quartile
(K3) and largest value.
0
5
10
15
20
25
30
35
29.5 39.5 49.5 59.5 69.5 79.5 89.5
Earnings
CumulativeFrequency
Example 14

2.4 MEASURES OF CENTRAL TENDENCY
2.4.1 Ungrouped Data Measurement
• Mean
Mean for population data:
x
N
µ =
∑
Mean for sample data:
x
x
n
=
∑
where: x∑ = the sum of all values
N = the population size
n = the sample size,
µ = the population mean
x = the sample mean
The following data give the prices (rounded to thousand RM) of five homes sold
recently in Sekayang.
158 189 265 127 191
Find the mean sale price for these homes.
Chapter 2: Descriptive Statistics
Smallest
value
Largest
value
K1 Median K3
Largest
value
K1 Median K3
Largest
value
K1 Median K3
Smallest
value
Smallest
value
For symmetry data
For left skewed data
For right skewed data
20
Example 15
FORMULA
ξ∆Σ λϖ
β

Solution:
Thus, these five homes were sold for an average price of RM186 thousand @
RM186 000.
 The mean has the advantage that its calculation includes each value of
the data set.
• Weighted Mean
 Used when have different needs.
 Weight mean :
w
wx
x
w
=
∑
∑
where w is a weight.
Consider the data of electricity components purchasing from a factory in the table
below:
Type Number of component (w) Cost/unit (x)
Example 16
FORMULA
ξ∆Σ λϖ
β

1
2
3
4
5
1200
500
2500
1000
800
RM3.00
RM3.40
RM2.80
RM2.90
RM3.25
Total 6000
Solution:
1200(3) 500(3.4) 2500(2.8) 1000(2.9) 800(3.25)
1200 500 2500 1000 800
17800
6000
2.967
w
wx
x
w
=
+ + + +
+ + + +
∑
∑
=
=
=
Mean cost of a unit of the component is RM2.97
• Median
 Median is the value of the middle term in a data set that has been
ranked in increasing order.
 Procedure for finding the Median
Step 1: Rank the data set in increasing order.
Step 2: Determine the depth (position or location) of the median.
1
2
n +
Depth of Median =
Step 3: Determine the value of the Median.
Find the median for the following data:
10 5 19 8 3
Solution:
(1) Rank the data in increasing order
Example 17
FORMULA
ξ∆Σ λϖ
β

(2) Determine the depth of the Median
1
2
5 1
2
3
n +
+
Depth of Median =
=
=
(3) Determine the value of the median
Therefore the median is located in third position of the data set.
Hence, the Median for above data =
Find the median for the following data:
10 5 19 8 3 15
Solution:
(1) Rank the data in increasing order
(2) Determine the depth of the Median
1
2
6 1
2
3.5
n +
+
Depth of Median =
=
=
(3) Determine the value of the Median
Therefore the median is located in the middle of 3rd
position and 4th
position of the data set.
8 10
9
2
+
= =Median
Hence, the Median for the above data =
 The median gives the center of a histogram, with half of the data values
to the left of (or, less than) the median and half to the right of (or, more
than) the median.
 The advantage of using the median is that it is not influenced by outliers.
• Mode
 Mode is the value that occurs with the highest frequency in a data set.
Example 18

1. What is the mode for given data?
77 69 74 81 71 68 74 73
2. What is the mode for given data?
77 69 68 74 81 71 68 74 73
Solution:
1. Mode =
2. Mode =
 A major shortcoming of the mode is that a data set may have none or
may have more than one mode.
 One advantage of the mode is that it can be calculated for both kinds of
data, quantitative and qualitative.
2.4.2 Grouped Data Measurement
• Mean
Mean for population data:
fx
μ =
N
∑
Mean for sample data:
∑fx
x =
n
Where x the midpoint and f is the frequency of a class.
The following table gives the frequency distribution of the number of orders received
each day during the past 50 days at the office of a mail-order company. Calculate
the mean.
Chapter 2: Descriptive Statistics
Number of order f
10 – 12
13 – 15
16 – 18
19 – 21
4
12
20
14
n = 50
24
Example 19
Example 20
FORMULA
ξ∆Σ λϖ
β

Solution:
Because the data set includes only 50 days, it represents a sample. The value of
fx∑ is calculated in the following table:
Number of order f x fx
10 – 12
13 – 15
16 – 18
19 – 21
4
12
20
14
n = 50
The value of mean sample is:
Thus, this mail-order company received an average of 16.64 orders per day during
these 50 days.
• Median
 Step 1: Construct the cumulative frequency distribution.
 Step 2: Decide the class that contain the median.
Class Median is the first class with the value of cumulative frequency is
at least n/2.
 Step 3: Find the median by using the following formula:
Based on the grouped data below, find the median:
Time to travel to work Frequency
Median
 
 ÷
 ÷
 ÷
 
m
m
n
- F
2= L + i
f
Where:
n = the total frequency
F = the total frequency before class
median
i = the class width
= the lower boundary of the class
median
= the frequency of the class median
Example 21
FORMULA
ξ∆Σ λϖ
β

1 – 10
11 – 20
21 – 30
31 – 40
41 – 50
8
14
12
9
7
Solution:
1st
Step: Construct the cumulative frequency distribution
Time to travel to work Frequency Cumulative Frequency
1 – 10
11 – 20
21 – 30
31 – 40
41 – 50
8
14
12
9
7
Thus, 25 persons take less than 23 minutes to travel to work and another 25
persons take more than 23 minutes to travel to work.
• Mode
 Mode is the value that has the highest frequency in a data set.
 For grouped data, class mode (or, modal class) is the class with the
highest frequency.

 Formula of mode for grouped data:
Where:
is the lower boundary of class mode
is the difference between the frequency of class mode and the
frequency of the class before the class mode
is the difference between the frequency of class mode and the
frequency of the class after the class mode
i is the class width
Based on the grouped data below, find the mode
Time to travel to work Frequency
1 – 10
11 – 20
21 – 30
31 – 40
41 – 50
8
14
12
9
7
Solution:
Based on the table,
We can
also obtain the
mode by using
the histogram;
 
 ÷
 
Mode 1
mo
1 2
Δ
= L + i
Δ + Δ
moL
1∆
2∆
Example 22
FORMULA
ξ∆Σ λϖ
β

2.4.3 Relationship among Mean, Median & Mode
• As discussed in previous topic, histogram or a frequency distribution curve
can assume either skewed shape or symmetrical shape.
• Knowing the value of mean, median and mode can give us some idea
about the shape of frequency curve.
(1) For a symmetrical histogram and frequency curve with one peak, the
value of the mean, median and mode are identical and they lie at the
center of the distribution.
Mean, median, and mode for a symmetric histogram and frequency distribution curve
(2) For a histogram and a frequency curve skewed to the right, the value of
the mean is the largest that of the mode is the smallest and the value
of the median lies between these two.

Mean, median, and mode for a histogram and frequency distribution curve
skewed to the right
(3) For a histogram and a frequency curve skewed to the left, the value of the
mean is the smallest and that of the mode is the largest and the value
of the median lies between these two.
Mean, median, and mode for a histogram and frequency distribution curve
skewed to the left
2.5 DISPERSION MEASUREMENT
• The measures of central tendency such as mean, median and mode do not
reveal the whole picture of the distribution of a data set.

• Two data sets with the same mean may have a completely different spreads.
• The variation among the values of observations for one data set may be
much larger or smaller than for the other data set.
2.5.1 Ungrouped Data Measurement
• Range
RANGE = Largest value – Smallest value
Find the range of production for this data set,
Solution:
Range = Largest value – Smallest value
= 267 277 – 49 651
= 217 626
 Disadvantages:
o being influenced by outliers.
o based on two values only. All other values in a data set are ignored.
• Variance and Standard Deviation
 Standard deviation is the most used measure of dispersion.
 A Standard Deviation value tells how closely the values of a data set
clustered around the mean.
Example 23
FORMULA
ξ∆Σ λϖ
β

 Lower value of standard deviation indicates that the data set value are
spread over relatively smaller range around the mean.
 Larger value of data set indicates that the data set value are spread
over relatively larger around the mean (far from mean).
 Standard deviation is obtained the positive root of the variance:
Variance for population:
( )
N
N
x
x∑ ∑−
=
2
2
2
σ
Variance for sample:
( )
1
2
2
2
−
−
=
∑ ∑
n
n
x
x
s
Standard Deviation for population:
2
σσ =
Standard Deviation for sample:
2
ss =
Let x denote the total production (in unit) of company
Company Production
A
B
C
D
E
62
93
126
75
34
Find the variance and standard deviation,
Solution:
Company Production (x) x2
A
B
C
D
E
62
93
126
75
34
Example 24
FORMULA
ξ∆Σ λϖ
β
FORMULA
ξ∆Σ λϖ
β

390
 The properties of variance and standard deviation:
o The standard deviation is a measure of variation of all values from the
mean.
o The value of the variance and the standard deviation are never
negative. Also, larger values of variance or standard deviation indicate
greater amounts of variation.
o The value of s can increase dramatically with the inclusion of one or
more outliers.
o The measurement units of variance are always the square of the
measurement units of the original data while the units of standard
deviation are the same as the units of the original data values.
• Range
Range = Upper bound of last class – Lower bound of first class
FORMULA
ξ∆Σ λϖ
β

Class Frequency
41 – 50
51 – 60
61 – 70
71 – 80
81 – 90
91 - 100
1
3
7
13
10
6
Total 40
Upper bound of last class = 100.5
Lower bound of first class = 40.5
Range = 100.5 – 40.5 = 60
• Variance and Standard Deviation
Variance for population:
( )
2
2
2
−
σ =
∑
∑
fx
fx
N
N
Variance for sample:
( )
2
2
2
1
−
=
−
∑
∑
fx
fx
ns
n
Standard Deviation:
Population: 2
σσ =
Sample: 2
ss =
Find the variance and standard deviation for the following data:
No. of order f
10 – 12
13 – 15
16 – 18
19 – 21
4
12
20
14
Example 25
FORMULA
ξ∆Σ λϖ
β
FORMULA
ξ∆Σ λϖ
β

Total n = 50
Solution:
No. of order f x fx fx2
10 – 12
13 – 15
16 – 18
19 – 21
4
12
20
14
Total n = 50
Variance,
Standard Deviation,
Thus, the standard deviation of the number of orders received at the office of this mail-
order company during the past 50 days is 2.75.
2.5.3 Relative Dispersion Measurement
• To compare two or more distribution that has different unit based on their
dispersion OR
• To compare two or more distribution that has same unit but big different in
their value of mean.
• Also called modified coefficient or coefficient of variation, CV.

)(%100
)(%100
population
x
CV
sample
x
s
CV
−×





=
−×





=
σ
Given mean and standard deviation of monthly salary for two groups of worker who
are working in ABC company- Group 1: 700 & 20 and Group 2 :1070 & 20. Find the
CV for every group and determine which group is more dispersed.
Solution:
1
2
20
100 2 86
700
20
100 1 87
1070
= × =
= × =
CV % . %
CV % . %
The monthly salary for group 1 worker is more dispersed compared to group 2.
2.6 MEASURE OF POSITION
• Determines the position of a single value in relation to other values in a
sample or a population data set.
• Quartiles
 Quartiles are three summary measures that divide ranked data set into
four equal parts.
o The 1st
quartiles – denoted as Q1
1
4
+
1Depth of Q =
n
Example 26
FORMULA
ξ∆Σ λϖ
β
FORMULA
ξ∆Σ λϖ
β

o The 2nd
quartiles – median of a data set or Q2
o The 3rd
quartiles – denoted as Q3
3 1
4
+
3Depth of Q =
(n )
Table below lists the total revenue for the 11 top tourism company in Malaysia
109.7 79.9 21.2 76.4 80.2 82.1 79.4 89.3 98.0 103.5
86.8
Solution:
Step 1: Arrange the data in increasing order
76.4 79.4 79.9 80.2 82.1 86.8 89.3 98.0 103.5 109.7
121.2
Step 2: Determine the depth for Q1 and Q3
1 11 1
3
4 4
+ +
1Depth of Q = = =
n
( )3 11 13 1
9
4 4
++
3Depth of Q = = =
(n )
Step 3: Determine the Q1 and Q3
76.4 79.4 79.9 80.2 82.1 86.8 89.3 98.0 103.5 109.7
121.2
Q1 = 79.9 ; Q3 = 103.5
Table below list the total revenue for the 12 top tourism company in Malaysia
109.7 79.9 74.1 121.2 76.4 80.2 82.1 79.4 89.3
98.0 103.5 86.8
Solution:
Step 1: Arrange the data in increasing order
74.1 76.4 79.4 79.9 80.2 82.1 86.8 89.3 98.0 103.5 109.7
121.2
Example 27
Example 28
FORMULA
ξ∆Σ λϖ
β

Step 2: Determine the depth for Q1 and Q3
1 12 1
3 25
4 4
+ +
1Depth of Q = = =
n
.
( )3 12 13 1
9 75
4 4
++
3Depth of Q = = =
(n )
.
Step 3: Determine the Q1 and Q3
74.1 76.4 79.4 79.9 80.2 82.1 86.8 89.3 98.0 103.5 109.7
121.2
Q1 = 79.4 + 0.25 (79.9 – 79.4) = 79.525
Q3 = 98.0 + 0.75 (103.5 – 98.0) = 102.125
• Interquartile Range
 The difference between the third quartile and the first quartile for a data
set.
IQR = Q3 – Q1
By referring to example 28, calculate the IQR.
Solution:
IQR = Q3 – Q1 = 102.125 – 79.525 = 22.6
• Quartiles
 From Median, we can get Q1 and Q3 equation as follows:
1
1
1 Q
Q
n
- F
4Q L + i
f
 
 ÷
=  ÷
 ÷
 
Example 29
FORMULA
ξ∆Σ λϖ
β
FORMULA
ξ∆Σ λϖ
β

 
 ÷
=  ÷
 ÷
 
3
3
3 Q
Q
3n
- F
4Q L + i
f
Refer to example 22, find Q1 and Q3
Solution:
1st
Step: Construct the cumulative frequency distribution
Time to travel to work Frequency Cumulative Frequency
1 – 10
11 – 20
21 – 30
31 – 40
41 – 50
8
14
12
9
7
8
22
34
43
50
2nd
Step: Determine the Q1 and Q3
1
n 50
Class Q 12 5
4 4
.= = =
Class Q1 is the 2nd
class
Therefore,
1
1
1
4
12 5 8
10 5 10
14
13 7143
 
 ÷
= +  ÷
 ÷
 
 
= +  ÷
 
=
Q
Q
n
- F
Q L i
f
. -
.
.
( )
3
3 503n
Class Q 37 5
4 4
.= = =
Class Q3 is the 4th
class
Example 30

Therefore,
3
3
3
4
37 5 34
30 5 10
9
34 3889
 
 ÷
= +  ÷
 ÷
 
 
= +  ÷
 
=
Q
Q
n
- F
Q L i
f
. -
.
.
• Interquartile Range
IQR = Q3 – Q1
Refer to example 30, calculate the IQR.
Solution:
IQR = Q3 – Q1 = 34.3889 – 13.7143 = 20.6746
2.7 MEASURE OF SKEWNESS
• To determine the skewness of data (symmetry, left skewed, right skewed)
• Also called Skewness Coefficient or Pearson Coefficient of Skewness
3( )
k k
mean mode mean median
s or s
s s
− −
= =
 If Sk +ve  right skewed
 If Sk -ve  left skewed
Example 31
FORMULA
ξ∆Σ λϖ
β

 If Sk = 0  symmetry
 If Sk takes a value in between (-0.9999, -0.0001) or (0.0001,
0.9999)  approximately symmetry.
The duration of cancer patient warded in Hospital Seberang Jaya recorded in a
frequency distribution. From the record, the mean is 28 days, median is 25 days
and mode is 23 days. Given the standard deviation is 4.2 days.
a. What is the type of distribution?
b. Find the skewness coefficient
Solution:
This distribution is right skewed because the mean is the largest value
( ) ( )
28 23
11905
4 2
3 3 28 25
21429
4 2
Mean - Mode
OR
Mean - Median
−
= = =
−
= = =
k
k
S .
s .
S .
s .
So, from the Sk value this distribution is right skewed.
ADDITIONAL INFORMATION
Use of Standard Deviation
1. Chebyshev’s Theorem
• According to Chebyshev’s Theorem, for any number k greater than 1, at least (1
– 1/k2
) of the data values lie within k standard deviations of the mean.
( )
%75@75.0
2
1
1
1
1
2
2
=
−=
−=
k
Example 32

• Thus; for example if k = 2, then
• Therefore, according to Chebyshev’s Theorem, at least 75% of the values of a
data set lie within two standard deviation of the mean
2. Empirical Rule
• For a bell-shaped distribution, approximately
 1.68%of the observations lie within one standard deviation of the mean.
 2.95% of the observations lie within two standard deviations of mean.
 3.99.7% of the observations lie within three standard deviations of the mean.
Measure of Position
1. Ungrouped Data - Quartile Deviation
• QD is a mean for Interquartile Range
• It used to compare the dissemination of two data set.
• If the QD value is high, it means that the data is more
disseminated.
Quartile Deviation = Interquartile Range / 2
= (Q3 - Q1) / 2

2. Ungrouped Data – Percentile
Pk = value of the (kn)th
term in a ranked set 100
Where: k = the number of percentile
n = the sample size
Percentile rank of xi = Number of values than xi X 100
Total number of values in the data set

EXERCISE 2
1. A survey research company asks 100 people how many times they have been to
the dentist in the last five years. Their grouped responses appear below.
Number of Visits Number of Responses
0 – 4 16
5 – 9 25
10 – 14 48
15 – 19 11
What are the mean and variance of the data?
2. A researcher asked 25 consumers: “How much would you pay for a television
adapter that provides Internet access?” Their grouped responses are as follows:
Amount ($) Number of Responses
0 – 99 2
100 – 199 2
200 – 249 3
250 – 299 3
300 – 349 6
350 – 399 3
400 – 499 4
500 – 999 2
Calculate the mean, variance, and standard deviation.
3. The following data give the pairs of shoes sold per day by a particular
shoe store in the last 20 days.
85 90 89 70 79 80 83 83 75 76
89 86 71 76 77 89 70 65 90 86
Calculate the
a. mean and interpret the value.
b. median and interpret the value.
c.mode and interpret the value.
d. standard deviation.
4. The followings data shows the information of serving time (in minutes) for 40
customers in a post office:
2.0 4.5 2.5 2.9 4.2 2.9 3.5 2.8
3.2 2.9 4.0 3.0 3.8 2.5 2.3 3.5
2.1 3.1 3.6 4.3 4.7 2.6 4.1 3.1
4.6 2.8 5.1 2.7 2.6 4.4 3.5 3.0
2.7 3.9 2.9 2.9 2.5 3.7 3.3 2.4
a.Construct a frequency distribution table with 0.5 of class width.

b.Construct a histogram.
c.Calculate the mode and median of the data.
d.Find the mean of serving time.
e.Determine the skewness of the data.
f.Find the first and third quartile value of the data.
g.Determine the value of interquartile range.
5. In a survey for a class of final semester student, a group of data was obtained for
the number of text books owned.
Number of students Number of text book owned
12
9
11
15
10
8
5
5
3
2
1
0
Find the average number of text book for the class. Use the weighted mean.
6.The following data represent the ages of 15 people buying lift tickets at a ski area.
15 25 26 17 38 16 60 21
30 53 28 40 20 35 31
Calculate the quartile and interquartile range.
7.A student scores 60 on a mathematics test that has a mean of 54 and a standard
deviation of 3, and she scores 80 on a history test with a mean of 75 and a
standard deviation of 2. On which test did she perform better?
8.The following table gives the distribution of the share’s price for ABC Company
which was listed in BSKL in 2005.
Price (RM) Frequency
12 – 14
15 – 17
18 – 20
21 – 23
24 – 26
27 - 29
5
14
25
7
6
3
Find the mean, median and mode for this data.

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