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QQS1013ELEMENTARY STATISTIC
CHAPTER 2 DESCRIPTIVE 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
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.
2
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.
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 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
Example 2
Quantitative raw data
3
Qualitative raw data
These data also called ungrouped data
2.2 Organizing and Graphing Qualitative Data
2.2.1 Frequency Distributions/ Table2.2.2 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
the frequency of that category.
4
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
Percentage = (Relative Frequency)* 100
Example 3
5
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 WjIs W W Wj Is W W Is W WjWj Is Wj Sv W W W Wj St WWj Sv W Is P Sv Wj Wj W WSt 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 16Perdana 4 0.08 8Waja 10 0.20 20Satria 5 0.10 10Savvy 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 a gap is inserted between each of the bars.
It has=> simple/ vertical bar chart=> horizontal bar chart
6
=> 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 2.3
Wira
Iswara
Perdana
Waja
Satria
Savvy
0 2 4 6 8 10 12 14 16 18 20
UUM Staff-owned Vehicles Produced By Pro-ton
Frequency
Typ
es o
f V
ehic
le
7
Another example of horizontal bar chart: Figure 2.4
Figure 2.4: Number of students at Diversity College who are immigrants, 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 representative
frequencies.
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 100Sailing 36 36 40
Total 142 167 191
Solution:
8
2004 2005 20060
20406080
100120140160180200
Activities Breakdown (Jun)
Sailing
Walking
Caving
Climbing
Year
Nu
mb
er
of
pa
rtic
ipa
nts
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
Another example of horizontal bar chart: Figure 2.7
2004 2005 20060
20
40
60
80
100
120
Activities Breakdown (Jun)
Climbing
Caving
Walking
Sailing
Year
Nu
mb
er
of
pa
rtic
ipa
nts
9
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.
Example 7 (Table 2.6 and Figure 2.8):
10
Table 2.6 Figure 2.8
Example 8 (Table 2.7 and Figure 2.9):
Movie Genres
Frequency Relative Frequency
Angle Size
ComedyActionRomanceDramaHorrorForeignScience Fiction
54362828221616
0.270.180.140.140.110.080.08
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
200 1.00 360o
Figure 2.9Figure 2.9
11
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 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
Example 9
12
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.
YearRidership
(in millions)19901991199219931994
88.085.075.776.675.4
Solution:
1990 1991 1992 1993 199475
77
79
81
83
85
87
89
Year
Rid
ersh
ip (
in m
illi
on
s)
The graph shows a decline in ridership through 1992 and then leveling off for the years 1993 and 1994.
13
Exercise 1
1. The following data show the method of payment by 16 customers in a supermarket checkout line. Here, C = cash, CK = check, CC = credit card, D = debit and O = other.
C CK CK C CC D O CCK 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
ABCDE
13125911
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 1996No. 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 TR N T M RM M N R NT R M N MT 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.
14
Month Export ImportSeptember
OctoberNovemberDecember
28303224
20281714
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)
PerlisKedahPulau PinangPerakSelangorWilayah Persekutuan Kuala LumpurNegeri SembilanMelakaJohorPahangTerengganuKelantanSarawakSabah
435512163721664
100339022387610501255986878456
2.3 Organizing and Graphing Quantitative Data
2.3.1 Stem and Leaf Display2.3.2 Frequency Distribution2.3.3 Relative Frequency and Percentage
Distributions.
15
2.3.4 Graphing Grouped Data2.3.5 Shapes of Histogram2.3.6 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 37 614 41 38 44 13 22 18 19
Solution:
0 9 5 7 61 2 0 2 3 1 2 4 3 8 92 5 3 8 23 6 1 7 84 1 4
2.3.2 Frequency Distributions
A frequency distribution for quantitative data lists all the classes and
the number of values that belong to each class.
16
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.:
Class Width (class size)
Class width = Upper boundary – Lower boundary
class boundary
17
e.g. : Width of the first class = 600.5 – 400.5 = 200
Class Midpoint or Mark
e.g:
Constructing Frequency Distribution Tables
1. To decide the number of classes, we used Sturge’s formula, which is
18
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,
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
19
Solution:
i) Number of classes, c = 1 + 3.3 log 30 = 1 + 3.3(1.48)
= 5.89 6 class
ii) Class width,
iii) Starting Point = 135
Table 2.10 Frequency Distribution for Data of Table 2.9
Total Home Runs Tally f135 – 152153 – 170171 – 188189 – 206207 – 224225 – 242
|||| |||||||||| |||| ||||||||
1025634
20
2.3.3 Relative Frequency and Percentage Distributions
Example 12 (Refer example 11)
Table 2.11: Relative Frequency and Percentage Distributions
Total Home Runs
Class Boundaries Relative Frequency
%
135 – 152153 – 170171 – 188189 – 206207 – 224225 – 242
134.5 less than 152.5152.5 less than 170.5170.5 less than 188.5188.5 less than 206.5206.5 less than 224.5224.5 less than 242.5
0.33330.06670.1667
0.20.1
0.1333
33.336.6716.67
2010
13.33Sum 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.
21
134.5 152.5 170.5 188.5 206.5 224.5 242.5
Example 13 (Refer example 11)
1
0
2
4
6
8
10
12
Total home runs
Fre
qu
en
cy
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 a polygon.
Example 13
Figure 2.11: Frequency polygon for Table 2.10
1
0
2
4
6
8
10
12
Total home runs
Fre
qu
ency
134.5 152.5 170.5 188.5 206.5 224.5 242.5
22
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.
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 curve or simply a frequency curve.
The most common of shapes are:
(i) Symmetric
Figure 2.13 & 2.14: Symmetric histograms
23
(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
A cumulative 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 Home
RunsClass Boundaries Cumulative Frequency
135 – 152153 – 170171 – 188189 – 206207 – 224225 – 242
134.5 less than 152.5152.5 less than 170.5170.5 less than 188.5188.5 less than 206.5206.5 less than 224.5224.5 less than 242.5
1010+2=1210+2+5=1710+2+5+6=2310+2+5+6+3=2610+2+5+6+3+4=30
Ogive
24
566337
30 – 3940 – 4950 – 5960 - 6970 – 7980 - 89
30
Number of students (f)
Total
Earnings (RM)
0
5
10
15
20
25
30
35
29.5 39.5 49.5 59.5 69.5 79.5 89.5
Earnings
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) Cumulative Frequency
(F)
Less than 29.5Less than 39.5Less than 49.5Less than 59.5Less than 69.5Less than 79.5Less than 89.5
051117202330
Figure 2.17
Example 16 (Ogive Greater Than)
Cu
mu
lati
ve
Fre
qu
ency
25
566337
30 – 3940 – 4950 – 5960 - 6970 – 7980 - 89
30
Number of students (f)
Total
Earnings (RM)
0
5
10
15
20
25
30
35
29.5 39.5 49.5 59.5 69.5 79.5 89.5
Earnings
Cumulative Frequency
Figure 2.18
Figure 2.18
302519131070
More than 29.5More than 39.5More than 49.5More than 59.5More than 69.5More than 79.5More than 89.5
Cumulative Frequency (F)
Earnings (RM)
26
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
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
27
2.4.3 Relationship among mean, median & mode
2.4.1 Ungrouped Data
1. Mean
Mean for population data:
Mean for sample data:
where: = the sum af all values N = the population size
n = the sample size, µ = the population mean
= 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:
Thus, these five homes were sold for an average price of RM186 thousand @ RM186 000.
28
The mean has the advantage that its calculation includes each value of the data set.
2. Weighted Mean
Used when have different needs.
Weight mean :
where w is a weight.
Example 18
Consider the data of electricity components purchasing from a factory in the table below:
Type Number of component (w) Cost/unit (x)
12345
120050025001000800
RM3.00RM3.40RM2.80RM2.90RM3.25
Total 6000
Solution:
Mean cost of a unit of the component is RM2.97
29
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.
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 order
3 5 8 10 19
(2) Determine the depth of the 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
Example 20
30
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
(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.
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.
4. Mode
31
Mode is the value that occurs with the highest frequency in a data 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:
Mean for sample data:
Where the midpoint and f is the frequency of a class.
32
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 is calculated in the following table:
Numberof order
f x fx
10 – 1213 – 1516 – 1819 – 21
4122014
11141720
44168340280
n = 50 = 832
The value of mean sample is:
Thus, this mail-order company received an average of 16.64 orders per day during these 50 days.
33
Numberof order
f
10 – 1213 – 1516 – 1819 – 21
4122014
n = 50
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
1 – 1011 – 2021 – 3031 – 4041 – 50
8141297
Solution:
34
1st Step: Construct the cumulative frequency distribution
Time to travel to work
Frequency Cumulative Frequency
1 – 1011 – 2021 – 3031 – 4041 – 50
8141297
822344350
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.
3. Mode
35
n2=50
2=25
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
1 – 1011 – 2021 – 3031 – 4041 – 50
8141297
36
Solution:
Based on the table,
= 10.5, = (14 – 8) = 6, = (14 – 12) = 2 and i = 10
We can also obtain the mode by using the histogram;
Figure 2.19
37
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, and mode for a symmetric histogram and
Figure 2.21: 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.
Figure 2.22: Mean, median, and mode for a histogram and frequency distribution curve skewed to the left
38
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
(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 value – Smallest value
Example 25:
Find the range of production for this data set,Solution:
39
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.
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
Sample
Example 26
40
Let x denote the total production (in unit) of company
Company Production
ABCDE
62931267534
Find the variance and standard deviation,
Solution:
Company Production (x) x2
ABCDE
62931267534
38448649
15 87656251156
1156
Since s2 = 1182.50;
Therefore,
The properties of variance and standard deviation:
41
(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 never negative. Also, larger values of variance or standard deviation indicate greater amounts of variation.
(3) The value of s can increase dramatically with the inclusion of one or more outliers.
(4) 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.
2.5.2 Grouped Data
1. Range
Class Frequency
41 – 5051 – 6061 – 7071 – 8081 – 9091 - 100
13713106
Total 40
Upper bound of last class = 100.5Lower bound of first class = 40.5Range = 100.5 – 40.5 = 60
Range = Upper bound of last class – Lower bound of first class
42
2. Variance and Standard Deviation
Variance Standard Deviation
Population
Sample
Example 27
Find the variance and standard deviation for the following data:
Solution:
No. of order f x fx fx2
10 – 1213 – 1516 – 1819 – 21
4122014
11141720
44168340280
484235257805600
Total n = 50 857 14216
Variance, Standard Deviation,
43
No. of order f
10 – 1213 – 1516 – 1819 – 21
4122014
Total n = 50
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
populationx
CV
samplex
sCV
Example 28
75.25820.72 ss
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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:
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.
2.6.1 Ungrouped Data
1. Quartiles
2. Interquatile Range
2.6.2 Grouped Data
1. Quartile
2. Interquartile Range
1. Quartiles
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Quartiles are three summary measures that divide ranked data set into four equal parts.
The 1st quartiles – denoted as Q1
The 2nd quartiles – median of a data set or Q2
The 3rd quartiles – denoted as Q3
Example 29
1. 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
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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
2. Table below lists 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
Step 2: Determine the depth for Q1 and Q3
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
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Q1 = 79.4 + 0.25 (79.9 – 79.4) = 79.525
Q3 = 98.0 + 0.75 (103.5 – 98.0) = 102.125
2. Interquartile Range
The difference between the third quartile and the first quartile for a data set.
IQR = Q3 – Q1
Example 30
By referring to example 29, calculate the IQR.
Solution:
IQR = Q3 – Q1 = 102.125 – 79.525 = 22.6
2.6.2 Grouped Data
1. Quartiles
From Median, we can get Q1 and Q3 equation as follows:
;
Example 31
Refer to example 23, find Q1 and Q3
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Solution:
1st Step: Construct the cumulative frequency distribution
Time to travel to work
Frequency Cumulative Frequency
1 – 1011 – 2021 – 3031 – 4041 – 50
8141297
822344350
2nd Step: Determine the Q1 and Q3
Class Q1 is the 2nd class
Therefore,
Class Q3 is the 4th class
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Therefore,
2. Interquartile Range
IQR = Q3 – Q1
Example 32:
Refer to example 31, 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)
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Also called Skewness Coefficient or Pearson Coefficient of
Skewness
s
ModeMeanS
ors
ModeMeanS
k
k
)(3
If Sk +ve right skewed
If Sk -ve left skewed
If Sk = 0 symmetry
If Sk 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
So, from the Sk value this distribution is right skewed.
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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 – 4 165 – 9 25
10 – 14 4815 – 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 2100 – 199 2200 – 249 3250 – 299 3300 – 349 6350 – 399 3400 – 499 4500 – 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 7689 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 2.83.2 2.9 4.0 3.0 3.8 2.5 2.3 3.52.1 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 3.02.7 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
studentsNumber of text
book owned
1291115108
553210
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 2130 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.
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Price (RM) Frequency
12 – 1415 – 1718 – 2021 – 2324 – 2627 - 29
51425763
Find the mean, median and mode for this data.
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