Download worksheet-5-mean-mode-of-grouped-data

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MATHEMATICS WORKSHEET
1
XI Grade (Semester 1)
Chapter 1
Worksheet 5th
Topic : Measures of Central Tendency of grouped data
TIME : 4 X 45 minutes
SMAK ST. ALBERTUS
(ST. ALBERT Senior High School)
Talang 1 Street Malang 65112, Indonesia
Phone (0341) 564556, 581037 Fax.(0341) 552017
Email: [email protected] homepage: http://www.dempo.org
1
Adapted from New Syllabu s Mathematics 3, Teh Keng Seng BSc, Dip Ed & Looi Chin Keong BSc. Dip Ed
STANDARD COMPETENCY :
1. To use the rules of statistics, the rules of counting, and the properties of probability in
problem solving.
BASIC COMPETENCY:
1.3. To calculate the centre of measurement, the location of measurement, and the dispersion
of measurement, altogether with their interpretations.
In this chapter, you will learn:

How to calculate the mean of a grouped frequency distribution.

How to calculate the mean of a grouped frequency distribution using an “assumed mean”
method.

How to calculate the mode of a grouped frequency distribution.

How to calculate the mode of a grouped frequency distribution using histogram.
F. Mean and Mode of grouped data
The Mean of grouped data

1. In order to calculate the mean of grouped data, you need to:





Find the mid-point of each interval ( xi )
Multiply the frequency of each interval by its mid-point ( f i .xi )
Find the sum of all the products f i .xi
Find the sum of all the frequencies
Divide the sum of the products f i .xi by the sum of the frequencies.
Mean = x 
 f .x
f
i
i
i
Example 29
The following set of raw data shows the lengths, in millimeters, measured to the nearest
mm, of 40 leaves taken from plants of a certain species. This is the table of frequency
distribution. Calculate the mean.
Lengths (mm)
Frequency ( f i )
25 – 29
30 – 34
35 – 39
40 – 44
45 – 49
50 – 54
55 – 59
2
4
7
10
8
6
3
Solution
2
Lengths (mm)
Frequency ( f i )
25 – 29
30 – 34
35 – 39
40 – 44
45 – 49
50 – 54
55 – 59
2
4
7
10
8
6
3
x
 f .x
f
i
i

i
f
i
Mid-point ( xi )
f i .xi
 f .x

i
i


 

2. By Assumed Mean
In order to calculate the mean of grouped data by deviation, you need to:
 Find the mid-point of each interval ( xi )






Find the assumed mean = A
Find the difference between A with xi , we call the deviation (= d i )
Multiply the frequency of each interval by its deviation ( f i .d i )
Find the sum of all the products f i .d i
Find the sum of all the frequencies
Divide the sum of the products f i .d i by the sum of the frequencies, then add it to
A.
Mean = x  A 
 f .d
f
i
i
i
Example 30
The following set of raw data shows the lengths, in millimeters, measured to the nearest
mm, of 40 leaves taken from plants of a certain species. This is the table of frequency
distribution.
Lengths (mm)
Frequency ( f i )
25 – 29
30 – 34
35 – 39
40 – 44
45 – 49
50 – 54
55 – 59
2
4
7
10
8
6
3
Solution
3
Lengths (mm)
Frequency ( f i )
25 – 29
30 – 34
35 – 39
40 – 44
45 – 49
50 – 54
55 – 59
2
4
7
10
8
6
3
x  A
f
 f .d
f
i
i
i
  
i
Mid-point ( xi )
Deviation ( d i )
f i .d i
 f .d

i
i


     

3. By Coding Method
In order to calculate the mean of grouped data by Coding Method, you need to:
 Find the mid-point of each interval ( xi )






Find the assumed mean = A
Fill the u i with zero (=0) in the class of A , then fill the u i with -1, -2, -3, …to the
upper, 1, 2, 3, … to the below of the class of A .
Multiply he frequency of each interval by its deviation ( f i .ui )
Find the sum of all the products f i .ui
Find the sum of all the frequencies
Divide the sum of the products f i .ui by the sum of the frequencies, multiply it with
C , then add it to A .
Mean = x  A 
 f .u
f
i
i
.C
i
Example 31
The following set of raw data shows the lengths, in millimeters, measured to the nearest
mm, of 40 leaves taken from plants of a certain species. This is the table of frequency
distribution.
Lengths (mm)
Frequency ( f i )
25 – 29
30 – 34
35 – 39
40 – 44
45 – 49
50 – 54
55 – 59
2
4
7
10
8
6
3
Solution
4
Lengths (mm)
Frequency ( f i )
25 – 29
30 – 34
35 – 39
40 – 44
45 – 49
50 – 54
55 – 59
2
4
7
10
8
6
3
x  A
f
 f .u
f
i
i
i
Deviation ( u i )
f i .ui
 f .u

.C   
i
Mid-point ( xi )
i
i


.      

Example 32
The table below shows the length of 50 pieces of wire used in a physics laboratory. Lengths
have been measured to the nearest centimetre. Find the mean by usual method and
Coding Method.
Lengths (mm)
Frequency ( f i )
26 – 30
31 – 35
36 – 40
41 – 45
46 – 50
4
10
12
18
6
Solution
5
The Mode of grouped frequency distribution

In order to calculate the mode of grouped data, you need to:
 Find the modal class. The modal class is the class interval that has the largest
frequency.
 Find the lower class boundary of the modal class (  Lb )



Find the difference of frequency between the modal class to its upper class (  a ).
Find the difference of frequency between the modal class to its lower class (  b ).
Add the Lb to products
Mode = Mo  LbMo 
a
by C , then add it to A .
ab
a
.C
ab
Example 33
The following set of raw data shows the lengths, in millimeters, measured to the nearest
mm, of 40 leaves taken from plants of a certain species. This is the table of frequency
distribution.
Lengths (mm)
Frequency ( f i )
25 – 29
30 – 34
35 – 39
40 – 44
45 – 49
50 – 54
55 – 59
2
4
7
10
8
6
3
Solution
The modal class is 40 – 44, so LbMo  
a      and b      
a
.C
ab

Mo   
.
  
Thus Mo  LbMo 
Mo    
Mo  
6
CE = …… - …… = ……
By histogram
DF = …… - …… = ……
AP : PB = CE : DF
C
10-
P
A
8Frequency
6-
AP : PB = …… : …….
D
B
AP : AB = …… : (…. + .…)
F
AP : …… = …… : …….
E
AP =
4-
  

AP = …….
2-
 Mo = 39.5 + AP
24.5 29.5 34.5 39.5 44.5 49.5 54.5 59.5
Mo = 39.5 + …… = ……
Lengths (mm)
Example 34
The weight, in kg, of 50 boys were recorded as shown in the table below:
Weight ( x kg)
40  x  45
45  x  50
50  x  55
55  x  60
60  x  65
65  x  70
70  x  75
Number of boys
4
5
10
14
8
6
3
Find the Mode.
Solution
7
Exercise 5
1. The following table shows the distribution of marks of some students who took part in
science quiz.
Marks
Tally
56 – 60
61 – 65
66 – 70
71 – 75
76 – 80
81 – 85
86 – 90
91 – 95
96 – 100
//// //
//// //
////
//// ////
////
////
//
///
///
Lower class
boundary
Upper class
boundary
Frequency
a. Copy and complete the table
b. Calculate the mean and the mode.
2. The length, in mm, of 48 rubber tree leaves are given below.
137
146
163
145
152
142
133
154
127
162
148
144
147
169
150
126
141
149
136
139
157
135
127
126
132
166
162
158
153
157
152
147
166
141
143
136
147
146
138
144
136
147
142
159
134
148
153
161
Copy and complete the following table:
Lengths ( x mm)
Tally
Frequency
125  x  130
130  x  135
135  x  140
140  x  145
145  x  150
150  x  155
155  x  160
160  x  165
165  x  170
a. Calculate the mean and the mode.
b. Use the histogram in exercise 4) to calculate the mode.
3. The waiting times, x minutes, for 60 patients at a certain clinic are as follows:
25
6
13
98
29
12
21
37
23
20
53
14
11
45
32
8
19
51
22
62
26
12
39
7
80
5
15
32
9
41
19
13
30
26
58
73
36
47
35
17
67
36
6
27
54
a. Using the frequency table in exercise 4), calculate the mean.
8
18
16
22
48
15
87
72
68
58
14
42
36
25
56
74
b. Using the histogram in exercise 4), calculate the mode.
4.
The weights, in kg, of 80 members of a
sports club were measured and recorded
as shown in the table.
a. Calculate the mean.
b. Calculate the mode.
Weight ( x kg)
Number of members
7
10
14
27
12
6
4
40  x  50
50  x  60
60  x  70
70  x  80
80  x  90
90  x  100
100  x  110
5. The marks scored in a test by 500 children are given in the following table:
Marks ( x )
Number of children
81
60  x  80
103
80  x  100
127
100  x  120
99
120  x  140
90
140  x  160
a. Using an assumed mean of 110, calculate the mean mark.
b. Calculate the mode.
6. Thirty bulbs were life-tested and their lifespan to the nearest hour are as follows:
167
177
172
171
169
164
179
171
175
167
177
179
171
173
179
165
165
174
175
175
174
179
167
168
169
174
171
171
177
168
a. Find the mean of lifespan by dividing their sum by 30.
b. Find the mean of lifespan by grouping the lifespan using class intervals 164 – 166,
167 – 169, and so on.
c. Find the mode of lifespan by looking the data.
d. Find the mode of lifespan by grouping data at b).
7. In an examination taken by 400 students, the scores were as shown in the following
distribution table:
Marks
1 – 10
11 – 20
21 – 30
31 – 40
41 – 50
51 – 60
61 – 70
71 – 80
81 – 90
91 – 100
Frequency
8
14
32
56
102
80
54
30
16
8
Find :
a. The mode
b. The mean
9