QQS1013 ELEMENTARY 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
Grouped Data 2.7 Measure of Skewness
Standard
Deviation
for
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 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
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Qualitative raw data
These data also called ungrouped data
2.2 Organizing and Graphing Qualitative Data 2.2.1 2.2.2 2.2.3
Frequency Distributions/ Table Relative Frequency and Percentage Distribution 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.
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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 category Sum of all frequencies
Percentage = (Relative Frequency)* 100
5
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 Is
W W
P W
Is Wj
Is Is
P W
Is W
W Is
St W
Wj Wj
Wj Wj St
Is Sv W
Wj W W
Sv Is W
W P W
W Sv St
W Wj St
Wj Wj P
St W Wj
W W Sv
Construct a frequency distribution table for these data with their relative frequency and percentage.
Solution: Frequency
Relative Frequency
Wira
19
19/50 = 0.38
Iswara Perdana Waja Satria Savvy
8 4 10 5 4
0.16 0.08 0.20 0.10 0.08
0.38*100 = 38 16 8 20 10 8
50
1.00
100
Category
Total
Percentage (%)
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 6
=> 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, UUM Staff-owned Vehicles Produced By Proton
e l
Savvy
Satria ic h e Waja V f o Perdana s e p y Iswara T Wira 0
5
10 Frequency
15
20
Figure 2.3
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.
Climbing Caving Walking Sailing Total
2004 21 10 75 36
2005 34 12 85 36 142
2006 36 21 100 40 167
191
8
Solution: Activities Breakdown (Jun) 200
ts 180 n a 160 ip c it 140 r a 120 p f 100 o r 80 e b 60 m 40 u N 20
Sailing Walking Caving Climbing
0 2004
2005
2006
Year
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, Activities Breakdown (Jun) 120
s t n a 100 p i ic tr 80 a p f 60 o r e b 40 m u 20 N
Climbing Caving Walking Sailing
0 2004
2005
2006
Year
Figure 2.6
9
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.
o
To construct a pie chart, we multiply 360 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): Movie Genres
Frequency
Relative Frequency
Angle Size
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=64.8 o 360*0.14=50.4 360*0.14=50.4o o 360*0.11=39.6 o 360*0.08=28.8 o 360*0.08=28.8
o
Comedy Action Romance Drama Horror Foreign Science Fiction
200
1.00
360o
Figure 2.9 Figure 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 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
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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 the findings.
Ridership
Year
(in millions) 88.0
1990 1991 1992 1993 1994
85.0 75.7 76.6 75.4
Solution:
89 ) s 87 n io ll 85 i m 83 n (i p i 81 h s r e 79 d i R
77 75 1990
1991
1992
1993
1994
Year
The graph shows a decline in ridership through 1992 and then leveling off for the years 1993 and 1994.
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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 CC
CK D
C CC
CC C
D CK
O CK
C 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
A B C
13 12 5
D E
9 11
Relative Frequency
Percentage
Angle Size
3. Draw a time series graph to represent the data for the number of worldwide airline fatalities for the given years.
Year No. of fatalities
1990
1991
1992
1993
1994
1995
440
510
990
801
732
557
1996 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 R M T T
N N M R R
R T N M R
T M R N N
T R N M 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 four months of sales in certain year. Using the provided information, present this data in component bar graph.
Month September October November December 6.
Export 28 30 32 24
Import 20 28 17 14
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 Perlis Kedah Pulau Pinang Perak Selangor Wilayah Persekutuan Kuala Lumpur Negeri Sembilan Melaka Johor Pahang Terengganu Kelantan Sarawak Sabah
Quantity (mm) 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 2.3.2 2.3.3
2.3.1
2.3.4
Stem and Leaf Display Frequency Distribution Relative Frequency Distributions. Graphing Grouped Data
2.3.5 2.3.6
Shapes of Histogram Cumulative Frequency Distributions.
and
Percentage
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 7 36 13 11 12 31 28 37 6 14 41 38 44 13 22 18 19
Solution: 0 1 2 3 4
9 2 5 6 1
5 0 3 1 4
7 2 8 7
6 3 1 2 4 3 8 9 2 8
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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.
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 boundary
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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 midpoint or mark =
Lower limit + Upper limit 2
e.g:
Midpoint of the 1st class =
401 600 500.5 2
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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 c
i
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
i
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iii) Starting Point = 135 Table 2.10 Frequency Distribution for Data of Table 2.9 Total Home Runs 135 – 152 153 – 170 171 – 188 189 – 206 207 – 224 225 – 242
Tally |||| |||| || |||| |||| | ||| ||||
f 10 2 5 6 3 4
f 30 20
Relative Frequency and Percentage Distributions
2.3.3
Frequency of that class Sum of all frequencies f = f
Relative frequency of a class =
Percentage = (Relative frequency) 100
Example 12 (Refer example 11) 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
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
0.3333 0.0667 0.1667 0.2 0.1
33.33 6.67 16.67 20 10
225 – 242
224.5 less than 242.5 Sum
0.1333 1.0
13.33 100%
Graphing Grouped Data
2.3.4 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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Example 13 (Refer example 11) 12 10 8
y c n e u q
6
rF e
4 2 0
134.5
152.5
170.5
1188.5
206.5
224.5
242.5
Total home runs
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 12 10 y c n e u q e r F
8 6 4 2 0
134.5
152.5
170.5
1
188.5
206.5
224.5
242.5
Total home runs
Figure 2.11: Frequency polygon for Table 2.10
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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 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
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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 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 Runs
Class Boundaries
Cumulative Frequency
135 – 152
134.5 less than 152.5
10
153 – 170 171 – 188 189 – 206 207 – 224 225 – 242
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=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) Earnings (RM)
Number of students (f)
30 – 39 40 – 49 50 – 59 60 - 69 70 – 79 80 - 89
5 6 6 3 3 7
Total
30
Cumulative Frequency (F)
Less than 29.5 Less than 39.5
0 5
Less than 49.5 Less than 59.5 Less than 69.5 Less than 79.5 Less than 89.5
11 17 20 23 30
35
y c30 n e25 u q 20 e r15 F e10 v it 5 la u0 m u C
29.5
39.5
49.5
59.5
69.5
79.5
89.5
Earnings Figure 2.17
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Example 16 (Ogive Greater Than)
Earnings (RM)
Number of students (f)
30 – 39 40 – 49 50 – 59 60 - 69 70 – 79 80 - 89
5 6 6 3 3 7
Total y 35 c n 30 e u 25 q 20 e r F 15 e 10 v i tl 50 a u m u Figure C 2.18
Earnings RM
Cumulative Fre uenc F
More than 29.5 More than 39.5
30 25
More than 49.5 More than 59.5 More than 69.5 More than 79.5 More than 89.5
19 13 10 7 0
30
29.5
39.5
49.5
59.5
69.5
79.5
89.5
Earnings Figure 2.18
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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.
For symmetry data
Smallest value
K1
2.4.1
K1
Median
For right skewed data Smallest K1 value
2.4
K3
Largest value
For left skewed data Smallest value
Median
Median
K3
K3
Largest value
Largest value
Measures of Central Tendency Ungrouped Data (1) Mean (2) Weighted mean (3) Median (4) Mode
2.4.2
Grouped Data (1) Mean (2) Median (3) Mode
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2.4.3
Relationship among mean, median & mode
2.4.1 Ungrouped Data 1.
Mean
Mean for population data:
Mean for sample data:
x
x N
x n
x = the sum af all values
where:
N = the population size n = the sample size, µ = 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:
x
x
n
265 127 191 158 189 5 930 5
186 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 :
xw
wx w
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)
1 2 3 4 5
1200 500 2500 1000 800
RM3.00 RM3.40 RM2.80 RM2.90 RM3.25
Total
6000
Solution: xw =
wx w 1200(3) 500(3.4)
2500(2.8)
1200 500 =
1000(2.9)
2500 1000
800(3.25)
800
17800
6000 = 2.967 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. n
Depth of Median =
1 2
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 (2)
5
8
10
19
Determine the depth of the Median Depth of Median = =
n 1 2 5 1 2
=3
(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
(2)
5
8
10
15
19
Determine the depth of the Median Depth of Median = = =
(3)
1
n
2
1
6
2 3.5
Determine the value of the Median Therefore the median is located in the middle of 3 rd position and 4th position of the data set.
Median
8 10 2
9
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 a data set.
Example 21 1.
What is the mode for given data? 77
2.
69
74
81
71
68
74
73
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 μ=
N
Mean for sample data: x =
Where
fx n
x 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 mailorder company. Calculate the mean.
Number of order
f
10 – 12 13 – 15 16 – 18 19 – 21
4 12 20 14 n = 50
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
11 14 17 20
44 168 340 280
n = 50
fx = 832
The value of mean sample is: x=
fx 832 = = 16.64 n 50
Thus, this mail-order company received an average of 16.64 orders per day during these 50 days.
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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: n
Median = L+ m
Where: n = F = i = Lm = = f
2
- F
i fm
the total frequency the total frequency before class median the class width the lower boundary of the class median the frequency of the class median
m
Example 23 Based on the grouped data below, find the median:
Time to travel to work
Frequency
1 – 10 11 – 20 21 – 30 31 – 40 41 – 50
8 14 12 9 7
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Solution: 1st Step: Construct the cumulative frequency distribution Time to travel to work
Frequency
Cumulative Frequency
1 – 10
8
8
11 – – 30 20 21 31 – 40 41 – 50
14 12 9 7
22 34 43 50
n 2 So,
F = 22,
50 2
rd
Class median is the 3 class
25
fm = 12, Lm= 21.5 and i = 10
Therefore,
n
Median = Lm 2 - F i fm
25 - 22 10 12
= 21.5 = 24
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.
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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:
Mode = Lmo +
Δ1
i Δ1 + Δ2
Where:
Lmois the lower boundary of class mode 1 is the difference between the frequency of class mode and the frequency of the class before the class mode
2 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 – 10 11 – 20
8 14
21 31 – – 30 40 41 – 50
12 9 7
36
Solution: Based on the table, Lmo= 10.5,
1= (14 – 8) = 6, 2 = (14 – 12) = 2
and i = 10
6 5. Mode= 10.5 10 17 6 2 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 frequency distribution curve the right
(3)
Figure 2.21: Mean, median, and mode for a histogram and frequency distribution curve skewed to
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.
38
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 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) (2)
Range 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,
40
Solution: Range = Largest value – Smallest value = 267 277 – 49 651 = 217 626
Disadvantages: being influenced by outliers. o 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
x
2
Population
2
x
2
N
2
2
N
2
Sample s
2
x
2
x
n
n
s
2
s
2
1
41
Example 26 Let x denote the total production (in unit) of company
Company
Production
A B C
62 93 126
D E
75 34
Find the variance and standard deviation,
Solution: Company
Production (x)
x2
A B C
62 93 126
3844 8649 15 876
D E
75 34
5625 1156
x
1156
s2 =
x
2
x -
2
35150
2
n n -1 2 390 35150-
=
5
5 1
= 1182.50 2
Since s = 1182.50; Therefore,
s 1182.50 34.3875
42
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 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 srcinal data while the units of standard deviation are the same as the units of the srcinal data values.
2.5.2 Grouped Data 1. Range
Range = Upper bound of last class – Lower bound of first class
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
43
2.
Variance and Standard Deviation Variance Population
2
fx
2
s2
fx
N
2
2
N 2
Sample
Standard Deviation 2
fx
fx
2
s2
n n 1
s2
Example 27 Find the variance and standard deviation for the following data:
No. of order
f
10 – 12 13 – 15
4 12
16 19 – – 18 21
20 14
Total
n = 50
Solution: 2
No. of order
f
x
fx
fx
10 – 12 13 – 15 16 – 18 19 – 21
4 12 20 14
11 14 17 20
44 168 340 280
484 2352 5780 5600
857
14216
Total
n = 50
44
Variance,
s2
Standard Deviation,
fx
fx
2
s
s
2
7.5820 2.75
n
n 1 14216
2
832
2
50 50 1
7.5820 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.
s 100% ( sample ) x CV x 100 % ( population ) CV
45
Example 28 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: CV1
CV2
20 700 20
100% 2 86 . %
1070
100% 1 87 . %
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
2.6.2 Grouped Data
1. Quartiles
1. Quartile
2. Interquatile Range
2. Interquartile Range
46
1. Quartiles
Quartiles are three summary measures that divide ranked data set into four equal parts.
st
The 1 quartiles – denoted as Q1
Depth of Q 1 =
n 1 4
nd
The 2 quartiles – median of a data set or Q 2
The 3 quartiles – denoted as Q3
rd
Depth of Q 3 =
3( n 1) 4
Example 29 1. Table below lists the total revenue for the 11 top tourism company in Malaysia 109.7 86.8
79.9
21.2
76.4
80.2
82.1
79.4
89.3
98.0
103.5
103.5
109.7
Solution: Step 1: Arrange the data in increasing order 76.4 79.4 121.2
79.9
80.2
82.1
86.8
89.3
98.0
Step 2: Determine the depth for Q1 and Q3 47
n 1
Depth of Q 1 =
4
=
3( n 1)
Depth of Q 3 =
4
11 1
= 3
4
=
3 11 1 4
= 9
Step 3: Determine the Q1 and Q3 76.4 79.4 121.2
79.9
80.2
82.1
86.8
89.3
98.0 103.5
109.7
Q1 = 79.9 Q3 = 103.5
2. Table below lists the total revenue for the 12 top tourism company in Malaysia 109.7 98.0
79.9 103.5
74.1 86.8
121.2
76.4
80.2
82.1
79.4
89.3
Solution: Step 1: Arrange the data in increasing order 74.1 76.4 79.4 109.7 121.2
79.9
80.2
82.1
86.8
89.3
98.0
103.5
Step 2: Determine the depth for Q1 and Q3 Depth of Q 1 =
Depth of Q 3 =
n 1 4
=
12 1
3( n 1) 4
4
=
= 3.25 3 12 1 4
= 9.75
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Step 3: Determine the Q1 and Q3 74.1 76.4 79.4 109.7 121.2
79.9
80.2
82.1
86.8
89.3
98.0 103.5
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:
Q1 L +
Q1
n 4-F i fQ 1
;
Q3 L+
Q3
3n - F 4 i fQ 3
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Example 31 Refer to example 23, find Q 1 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 n ClassQ
1
50
125 . 4 4 nd
Class Q1 is the 2 class Therefore,
n -F Q1 LQ 4 i f Q . -8 125 10 10.5 14 13.7143 1
1
50
ClassQ
3
3n 4
3 50 4
37 5 .
th
Class Q3 is the 4 class
Therefore, n -F Q3 LQ 4 i fQ . - 34 375 30.5 10 9 34.3889 3
3
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
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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
S k Mean Mode s or Sk
3( Mean Mode)
s
If Sk +ve
right skewed
If Sk -ve
If Sk = 0
If Sk takes a value in between (-0.9999, -0.0001) or (0.0001,
left skewed
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 Sk
Sk
Mean - Mode 28 23 . 11905 s 4.2 OR 3 Mean - Median 32825
s
4.2
. 21429
So, from the Sk value this distribution is right skewed. 52
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 5 9 10 – – 14 15 – 19
16 25 48 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 – 99 100 – 199 200 – 249 250 – 299 300 – 349 350 – 399 400 – 499 500 – 999
2 2 3 3 6 3 4 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 89
90 86
89 71
70 76
79 77
80 89
83 70
83 65
75 90
76 86
Calculate the a. mean and interpret the value. b. median and interpret the value. c. mode and interpret the value. d. standard deviation.
53
4. The followings data shows the information of serving time (in minutes) for 40 customers in a post office: 2.0 3.2 2.1 4.6 2.7
4.5 2.9 3.1 2.8 3.9
2.5 4.0 3.6 5.1 2.9
2.9 3.0 4.3 2.7 2.9
4.2 3.8 4.7 2.6 2.5
2.9 2.5 2.6 4.4 3.7
3.5 2.3 4.1 3.5 3.3
a.
Construct a frequency distribution table with 0.5 of class width.
b. c. d. e. f. g.
Construct a histogram. Calculate the mode and median of the data. Find the mean of serving time. Determine the skewness of the data. Find the first and third quartile value of the data. Determine the value of interquartile range.
2.8 3.5 3.1 3.0 2.4
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 12 9 11
Number of text book owned 5 5 3
15 10 8
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 30
25 53
26 28
17 40
38 20
16 35
60 31
21
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?
54
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
5 14 25 7
24 26 27 –- 29
6 3
Find the mean, median and mode for this data.
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