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QQS1013
ELEMENTARY STATISTIC
CHAPTER 2DESCRIPTIVE STATISTICS
2.1 Introduction
2.2 Organizing and Graphing Qualitative Data
2.3 Organizing and Graphing Quantitative Data
2.4 Central Tendency Measurement
2.5 Dispersion Measurement
2.6 Mean, Variance and Standard Deviation for
Grouped Data
2.7 Measure of Skewness
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OBJECTIVES
After completing this chapter, students should be able to:
Create and interpret graphical displays involve qualitative
and quantitative data.
Describe the difference between grouped and ungrouped
frequency distribution, frequency and relative frequency,
relative frequency and cumulative relative frequency.
Identify and describe the parts of a frequency distribution:
class boundaries, class width, and class midpoint.
Identify the shapes of distributions.
Compute, describe, compare and interpret the three
measures of central tendency: mean, median, and mode for
ungrouped and grouped data. Compute, describe, compare and interpret the two measures
of dispersion: range, and standard deviation (variance) for
ungrouped and grouped data.
Compute, describe, and interpret the two measures of
position: quartiles and interquartile range for ungrouped and
grouped data.
Compute, describe and interpret the measures of skewness:
Pearson Coefficient of Skewness.
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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 descendingorder.
Example 1
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 (raw data): m
2. How many hours did you sleep last night?Answer: 5 hours
3. Randomly pick a letterS or Q.Answer: S
4. What is your height in inches?Answer: 67 inches
5. Whats the fastest youve ever driven a car (mph)?Answer: 110 mph
Example 2
Quantitative raw data
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Qualitative raw data
These data also called ungrouped data
2.2 Organizing and Graphing Qualitative Data
2.2.1 Frequency Distributions/ Table
2.2.2 Relative Frequency and Percentage Distribution
2.2.3 Graphical Presentation of 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.
The number of students who belong to a certain category is called
thefrequencyof that category.
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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 categorySum of all frequencies
Percentage = (Relative Frequency)* 100
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Example 3
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):
W W P Is Is P Is W St Wj
Is W W Wj Is W W Is W WjWj 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
Construct a frequency distribution table for these data with their relative
frequency and percentage.
Solution:
Category FrequencyRelative
FrequencyPercentage (%)
Wira 19 19/50 = 0.380.38*100
= 38
Iswara 8 0.16 16
Perdana 4 0.08 8
Waja 10 0.20 20
Satria 5 0.10 10
Savvy 4 0.08 8
Total 50 1.00 100
2.2.3 Graphical Presentation of Qualitative Data
1. Bar Graphs
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 agap is inserted between each of the bars.
It has=> simple/ vertical bar chart
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=> horizontal bar chart
=> component bar chart
=> 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
Refer to Figure 2.1 and Figure 2.2,
Figure 2.1 Figure 2.2
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.
Example 4: Refer Example 3,
Figure 2.30 5 10 15 20
Wira
Iswara
Perdana
Waja
Satria
Savvy
Frequency
TypesofVehicle
UUM Staff-owned Vehicles Produced By
Proton
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Another example of horizontal bar chart: Figure 2.4
Figure 2.4: Number of students at Diversity College who areimmigrants, by last country of permanent residence
Component Bar Chart
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 representativefrequencies.
Example 5
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 2006Climbing 21 34 36Caving 10 12 21Walking 75 85 100
Sailing 36 36 40Total 142 167 191
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Solution:
Figure 2.5
Mulztiple 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).
Example 6: Refer example 5,
Figure 2.6
0
20
40
60
80
100
120
140
160
180200
2004 2005 2006
Numberofparticipants
Year
Activities Breakdown (Jun)
Sailing
Walking
Caving
Climbing
0
20
40
60
80
100
120
2004 2005 2006
Numberofparticipants
Year
Activities Breakdown (Jun)
Climbing
Caving
Walking
Sailing
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Another example of horizontal bar chart: Figure 2.7
Figure 2.7: Preferred snack choices of students at UUM
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.
2. 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.
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Example 7 (Table 2.6 and Figure 2.8):
Table 2.6 Figure 2.8
Example 8 (Table 2.7 and Figure 2.9):
MovieGenres
Frequency RelativeFrequency
Angle Size
Comedy
ActionRomanceDrama
Horror
Foreign
ScienceFiction
54
362828
22
16
16
0.27
0.180.140.14
0.11
0.08
0.08
360*0.27=97.2o
360*0.18=64.8o
360*0.14=50.4
o
360*0.14=50.4o
360*0.11=39.6o
360*0.08=28.8o
360*0.08=28.8o
200 1.00 360o
Figure 2.9Figure 2.9
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3. 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 theslope, orsteepness, 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 variablesshown on
the x- and y-axesare 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
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Example 9
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 thefindings.
YearRidership
(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 offfor the years 1993 and 1994.
75
77
79
81
83
85
87
89
1990 1991 1992 1993 1994
Ridership(in
millions)
Year
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Exercise 1
1. The following data show the method of payment by 16 customers in asupermarket checkout line. Here, 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 ZeeZeeCompany. 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 ofProduct
Frequency RelativeFrequency
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 airlinefatalities for the given years.
Year 1990 1991 1992 1993 1994 1995 1996No. of
fatalities440 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.
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5. The given information shows the export and import trade in million RM for fourmonths of sales in certain year. Using the provided information, present this
data in component bar graph.
Month Export Import
SeptemberOctober
November
December
2830
32
24
2028
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
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2.3 Organizing and Graphing Quantitative Data
2.3.1 Stem and Leaf Display
2.3.2 Frequency Distribution
2.3.3 Relative Frequency and Percentage
Distributions.
2.3.4 Graphing Grouped Data
2.3.5 Shapes of Histogram
2.3.6 Cumulative Frequency Distributions.
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.
Example 10
25 12 9 10 5 12 23 736 13 11 12 31 28 37 61441 38 44 13 22 18 19
Solution:
0 9 5 7 6
1 2 0 2 3 1 2 4 3 8 9
2 5 3 8 2
3 6 1 7 8
4 1 4
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2.3.2 Frequency Distributions
Afrequency 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 calledreal 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 401400.5
2
class boundary
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Class Width (class size)
Class width = Upper boundaryLower boundary
e.g. :Width of the first class = 600.5400.5 = 200
Class Midpoint or Mark
Lower limit + Upper limitclass midpoint or mark =
2
e.g:
401 600Midpoint of the 1st class = 500.5
2
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Constructing Frequency Distribution Tables
1. To decide the number of classes, we used Sturges 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 valueNumber of classes
Range
i
ic
This class width is rounded to a convenient number.
3. Lower Limit of the First Class or the Starting Point
Use the smallest value in the data set.
Example 11
The following data give the total home runs hit by all players of each of
the 30 Major League Baseball teams during 2004 season
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Solution:
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 f135152
153170171188
189206
207224
225242
|||| ||||
||||||
|||| |
|||
||||
10
25
6
3
4
30f
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2.3.3 Relative Frequency and Percentage Distributions
Frequency of that classRelative frequency of a class =
Sum of all frequencies
=
Percentage = (Relative frequency) 100
f
f
Example 12 (Refer example 11)
Table 2.11: Relative Frequency and Percentage Distributions
Total HomeRuns
Class Boundaries RelativeFrequency
%
135152153170
171188
189206
207224
225242
134.5 less than 152.5152.5 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
0.33330.0667
0.1667
0.2
0.1
0.1333
33.336.67
16.67
20
10
13.33
Sum 1.0 100%
2.3.4 Graphing Grouped Data
1. 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.
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0
2
4
6
8
10
12
1
Frequency
Total home runs
Example 13 (Refer example 11)
Figure 2.10: Frequency histogram for Table 2.10
2. Polygon
A graph formed by joining the midpoints of the tops of
successive bars in a histogram with straight lines is called apolygon.
Example 13
Figure 2.11: Frequency polygon for Table 2.10
0
2
4
6
8
10
12
1
Frequency
Total home runs
134.5 152.5 170.5 188.5 206.5 224.5 242.5
134.5 152.5 170.5 188.5 206.5 224.5 242.5
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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 afrequency curve.
Figure 2.12: Frequency distribution curve
2.3.5 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 curveor simply afrequency curve.
The most common of shapes are:
(i) Symmetric
Figure 2.13 & 2.14: Symmetric histograms
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(ii) Right skewed and (iii) Left skewed
Figure 2.15 & 2.16: 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.
2.3.6 Cumulative Frequency Distributions
Acumulative frequency distribution gives the total number of
values that fall below the upper boundary of each class.
Example 14: Using the frequency distribution of table 2.11,
Total HomeRuns
Class Boundaries Cumulative Frequency
135152
153170
171188
189206
207224
225242
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
10+2=12
10+2+5=17
10+2+5+6=23
10+2+5+6+3=26
10+2+5+6+3+4=30
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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.
Example 15 (Ogive Less Than)
Earnings (RM) CumulativeFrequency
(F)
Less than 29.5Less than 39.5Less than 49.5
Less than 59.5Less than 69.5Less than 79.5Less than 89.5
05
11
17202330
Figure 2.17
5663
37
30 3940 4950 5960 - 69
70
7980 - 89
30
Number ofstudents (f)
Total
Earnings(RM)
CumulativeFrequency
0
5
10
1520
25
30
35
29.5 39.5 49.5 59.5 69.5 79.5 89.5
Earnings
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Example 16 (Ogive Greater Than)
Figure 2.18
Figure 2.18
566337
30 3940 4950 5960 - 6970 7980 - 89
30
Number of
students (f)
Total
Earnings
(RM)
302519131070
More than 29.5More than 39.5More than 49.5More than 59.5More than 69.5More than 79.5More than 89.5
CumulativeFre uenc F
Earnings
RM
0
5
10
15
20
25
30
35
29.5 39.5 49.5 59.5 69.5 79.5 89.5
EarningsCumulativeFrequency
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2.3.7 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.
2.4 Measures of Central Tendency
2.4.1 Ungrouped Data(1) Mean
(2) Weighted mean
(3) Median
(4) Mode
2.4.2 Grouped Data(1) Mean
(2) Median
(3) Mode
Smallest
value
Largest
value
K1 Median K3
Largestvalue
K1 Median K3
Largestvalue
K1 Median K3
Smallestvalue
Smallest
value
For symmetry data
For left skewed data
For right skewed data
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2.4.3 Relationship among mean, median & mode
2.4.1 Ungrouped Data
1. Mean
Mean for population data:x
N
Mean for sample data:x
xn
where: x = the sum af all valuesN = the population size
n = the sample size, = the population mean
x = the sample mean
Example 17
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.
Solution:
158 189 265 127 191
5
930
5
186
x
x
n
Thus, these five homes were sold for an average price of RM186thousand @ RM186 000.
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The mean has the advantage that its calculation includes each valueof the data set.
2. Weighted Mean
Used when have different needs.
Weight mean :
w
wxx
w
where w is a weight.
Example 18
Consider the data of electricity components purchasing from a factory inthe table below:
Type Number of component (w) Cost/unit (x)
1
23
4
5
1200
5002500
1000
800
RM3.00
RM3.40RM2.80
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
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3. 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.
Example 19
Find the median for the following data:
10 5 19 8 3
Solution:
(1) Rank the data in increasing order3 5 8 10 19
(2) Determine the depth of the Median1
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.
3 5 8 10 19
Hence, the Median for above data = 8
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Example 20
Find the median for the following data:
10 5 19 8 3 15
Solution:
(1) Rank the data in increasing order
3 5 8 10 15 19
(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.
8 109
2
Median
Hence, the Median for the above data = 9
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.
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4. Mode
Mode is the value that occurs with the highest frequency in adata set.
Example 21
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 = 74 (this number occurs twice): Unimodal
2. Mode = 68 and 74: Bimodal
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
1. Mean
Mean for population data:
fx =
N
Mean for sample data:
fxx =
n
Where x the midpoint andf is the frequency of a class.
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Example 22
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.
Solution:
Because the data set includes only 50 days, it represents a sample. The
value of fx is calculated in the following table:
Numberof order
f x fx
10121315
1618
1921
412
20
14
1114
17
20
44168
340
280
n = 50 fx= 832
The value of mean sample is:
fx 832x = = =16.64
n 50
Thus, this mail-order company received an average of 16.64 orders per
day during these 50 days.
Numberof order
f
1012
1315
1618
1921
4
12
20
14
n = 50
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2. 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:
Where:n = the total frequencyF = the total frequency before class mediani = the class width
= the lower boundary of the class median= the frequency of the class median
Example 23
Based on the grouped data below, find the median:
Time to travel to work Frequency
110
11202130
31404150
8
1412
97
Median mm
n- F
2= L + i f
mL
mf
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Solution:
1st Step: Construct the cumulative frequency distribution
Time to travelto work
Frequency CumulativeFrequency
110
11202130
3140
4150
8
1412
9
7
8
2234
43
50
Class median is the 3rd
class
So, F= 22, = 12, = 21.5 and i = 10
Therefore,
Thus, 25 persons take less than 24 minutes to travel to work and another
25 persons take more than 24 minutes to travel to work.
252
50
2
n
mf
mL
2
25 2221 5 10
12
24
Median
=
=
mm
n- F
= L if
-.
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3. 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.
To find mode for grouped data, use the following formula:
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
Example 24
Based on the grouped data below, find the mode
Time to travel to work Frequency
110
1120
21303140
4150
8
14
129
7
Mode 1mo
1 2
= L + i
+
moL
1
2
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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.(Figure 2.20)(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.
Figure 2.20: Mean, median, andmode for a symmetric histogram
and frequency distribution curve
Figure 2.21: Mean, median, and mode fora histogram and frequency distributioncurve skewed to
the right
(3) For a histogram and afrequency 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 thesetwo.
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Figure 2.22: Mean, median, and mode for a histogram and
frequency distribution curve skewed to the left
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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 adata 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
(1) Range
(2) Standard Deviation
2.5.2 Grouped data
(1) Range
(2) Standard deviation
2.5.3 Relative Dispersion Measurement
2.5.1 Ungrouped Data
1. Range
RANGE = Largest valueSmallest value
Example 25:
Find the range of production for this data set,
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Solution:
Range = Largest valueSmallest value
= 267 27749 651
= 217 626
Disadvantages:o being influenced by outliers.o Based on two values only. All other values in a data set are
ignored.
2. 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.
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 Standard Deviation
Population
N
N
xx
2
2
2
22
Sample
1
2
2
2
n
n
x
x
s
22ss
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Example 26
Let x denote the total production (in unit) of company
Company ProductionA
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
3844
8649
15 876
5625
1156
1156 351502 x
2
5
5 1
1182 50
39035150-=
=
2
2
2
xx -
ns =n -1
.
Since s2
= 1182.50;
Therefore,
1182 50
34 3875
s .
.
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The properties of variance and standard deviation:
(1) The standard deviation is a measure of variation of all values
from the mean.
(2) The value of the variance and the standard deviation are nevernegative. Also, larger values of variance or standard deviation
indicate greater amounts of variation.
(3) The value ofs can increase dramatically with the inclusion of
one or more outliers.
(4) The measurement units of variance are always the square ofthe measurement units of the original data while the units of
standard deviation are the same as the units of the original
data values.
2.5.2 Grouped Data
1. Range
Class Frequency
4150
5160
61707180
8190
91 - 100
1
3
713
10
6
Total 40
Upper bound of last class = 100.5
Lower bound of first class = 40.5Range = 100.540.5 = 60
Range = Upper bound of last classLower bound of first class
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2. Variance and Standard Deviation
Variance Standard Deviation
Population
2
2
2
fx
fx NN
22
Sample
2
2
2
1
fxfx
nsn
22ss
Example 27
Find the variance and standard deviation for the following data:
Solution:
No. of order f x fx fx2
101213151618
1921
41220
14
111417
20
44168340
280
48423525780
5600
Total n = 50 857 14216
No. of order f
1012
1315
1618
1921
4
12
20
14
Total n = 50
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1. Quartiles
Quartiles are three summary measures that divide ranked dataset into four equal parts.
The 1st quartilesdenoted as Q1
1
4
1Depth of Q =
n
The 2nd quartilesmedian of a data set or Q2
The 3rd quartilesdenoted as Q3
3 1
4
3Depth of Q =
(n )
Example 29
1. Table below lists the total revenue for the 11 top tourism company inMalaysia
109.7 79.9 21.2 76.4 80.2 82.1 79.4 89.3 98.0 103.586.8
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
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1 11 13
4 4
1Depth of Q = = =
n
3 11 13 1 94 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 98.0 103.5 109.7121.2
Q1 = 79.9
Q3 = 103.5
2. Table below lists the total revenue for the 12 top tourism company inMalaysia
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 98.0 103.5109.7 121.2
Step 2: Determine the depth for Q1
and Q3
1 12 13 25
4 4
1Depth of Q = = =
n.
3 12 13 19 75
4 4
3Depth of Q = = =
(n ).
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Step 3: Determine the Q1 and Q3
74.1 76.4 79.4 79.9 80.2 82.1 86.8 89.3 98.0 103.5109.7 121.2
Q1 = 79.4 + 0.25 (79.979.4) = 79.525
Q3 = 98.0 + 0.75 (103.598.0) = 102.125
2. Interquartile Range
The difference betweenthe third quartile and the first quartile
for a data set.
IQR = Q3Q1
Example 30
By referringto example 29, calculate the IQR.
Solution:
IQR = Q3Q1 = 102.12579.525 = 22.6
2.6.2 Grouped Data
1.Quartiles
From Median, we can get Q1 and Q3 equation as follows:
1
1
1 Q
Q
n- F
4Q L + if
;
3
3
3 Q
Q
3 n- F
4Q L + if
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Example 31
Refer to example 23, find Q1 and Q3
Solution:
1st Step: Construct the cumulative frequency distribution
Time to travelto work
Frequency Cumulative Frequency
110
11
202130
3140
4150
8
1412
9
7
8
2234
43
50
2nd Step: Determine the Q1 and Q3
1
n 50Class Q 12 5
4 4
.
Class Q1 is the 2nd
class
Therefore,
1
1
1
4
12 5 810 5 10
14
13 7143
Q
Q
n- F
Q L if
. -.
.
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3
3 503nClass Q 37 5
4 4.
Class Q3 is the 4th
class
Therefore,
3
3
3
4
37 5 3430 5 10
9
34 3889
Q
Q
n- F
Q L if
. -.
.
2.Interquartile Range
IQR = Q3Q1
Example 32:
Referto example 31, calculate the IQR.
Solution:
IQR = Q3Q1 = 34.388913.7143 = 20.6746
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2.7 Measure of Skewness
To determine the skewness of data (symmetry, left skewed,
right skewed) Also called Skewness Coefficient orPearson Coefficient of
Skewness
IfSk+ve right skewed
IfSk-ve left skewed
IfSk= 0
IfSk takes a value in between (-0.9999, -0.0001) or (0.0001,
0.9999)
approximately symmetry.
Example 33
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 2311905
4 2
3 3 28 2521429
4 2
Mean - Mode
OR
Mean - Median
k
k
S .s .
S .s .
So, from the Skvalue this distribution is right skewed.
s
ModeMeanS
or
s
ModeMeanS
k
k
)(3
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Exercise 2:
1. A survey research company asks 100 people how many times they have been tothe dentist in the last five years. Their grouped responses appear below.
Number of Visits Number of Responses
04 16
59 25
1014 48
1519 11
What are the mean and variance of the data?
2. A researcher asked 25 consumers: How much would you pay for a televisionadapter that provides Internet access? Their grouped responses are as follows:
Amount ($) Number of Responses
099 2
100199 2
200249 3
250299 3
300349 6
350399 3
400499 4
500999 2
Calculate the mean, variance, and standard deviation.
3. The following data give the pairs of shoes sold per day by a particular shoe storein 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 thea. mean and interpret the value.b. median and interpret the value.c. mode and interpret the value.d. standard deviation.
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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 3.14.6 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 ofstudents
Number of textbook owned
12
9
11
15
108
5
5
3
2
10
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 skiarea.
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 standarddeviation 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?
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8. The following table gives the distribution of the shares price for ABC Companywhich was listed in BSKL in 2005.
Price (RM) Frequency
1214
1517
1820
2123
2426
27 - 29
5
14
25
7
6
3
Find the mean, median and mode for this data.