Measure of Central Tendency
Mode Median Mean 3
众数 中位数 平均数
The The middle The
number value of a arithmetic
that set of average
appears ordered of a set of
most often numbers
data
极值
a value that is too small or
too large in a set of data
Data set:
2,5,7,4,5,8,9,1,7,4,3,4,350
4
TP 2 Identify the extreme value in
the following set of data and
explain your answer.
The extreme value is −58 .
The value of −58 is much smaller
than other data. 5
TP 2 Identify the extreme value in 6
the following set of data and
explain your answer.
5 , 8 , 27 , 4 , 3 , 2
The extreme value is 27 .
The value of 27 is much larger
than other data.
The impact of extreme value :
Data set : 4, 5, 5, 6, 6, 6, 7, 7
Mean
4+5+5+6+6+6+7+7
=
46 8
=8
= 5.75
7
The impact of extreme value :
Data set : 4, 5, 5, 6, 6, 6, 7, 7
Median = 6
Mode = 6
Mean = 5.75
8
The impact of extreme value :
Data set : 4, 5, 5, 6, 6, 6, 7, 7,80
Mean
4 + 5 + 5 + 6 + 6 + 6 + 7 + 7 + 80
= 126
=9 9
= 14 9
The impact of extreme value :
Data set : 4, 5, 5, 6, 6, 6, 7, 7,80
Median= 6
Mode= 6
Mean= 14
10
The impact of extreme value :
Data set : Data set :
4, 5, 5, 6, 6, 6, 7, 7 4, 5, 5, 6, 6, 6, 7, 7,80
Median= 6 Mode= 6 Median= 6 Mode= 6
Mean= . Mean=
When an extreme value exists in a set of data,
it will affect the value of mean.
While the value of median and mode
do not change with extreme values. 11
The effect of changing a set of data
to the mode, mean and median
Data is changed
uniformly
12
The effect of changing a set of data to the mode, mean and median
Data set : 3 , 4 , 9 , 3 , 8 , 2 , 4 , 3
Data set : 2 , 3 , 3 , 3 , 4, 4 , 8 , 9
2+3+3+3+4+4+8+9
Mean =
8
36
=8
= 4.5
13
The effect of changing a set of data to the mode, mean and median
Data set : 3 , 4 , 9 , 3 , 8 , 2 , 4 , 3
Data set : 2 , 3 , 3 , 3 , 4, 4 , 8 , 9
3+4
Median= 2 Mode = 3
= 3.5
14
The effect of changing a set of data to the mode, mean and median
Data set : 2 , 3 , 3 , 3 , 4, 4 , 8 , 9
Each number in the set of data is added by 2.
Data is changed uniformly
Data set : 4 , 5 , 5 , 5 , 6, 6 , 10 , 11
Mean 4 + 5 + 5 + 5 + 6 + 6 + 10 + 11
= 52
= 8 = 6.5 8
15
The effect of changing a set of data to the mode, mean and median
Data set : 2 , 3 , 3 , 3 , 4, 4 , 8 , 9
Each number in the set of data is added by 2.
Data set : 4 , 5 , 5 , 5 , 6, 6 , 10 , 11
5+6 16
Median= 2 Mode = 5
= 5.5
Data set : 2 , 3 , 3 , 3 , 4 , 4 , 8 , 9
Mean=4.5 Median=3.5 Mode=3
+2 : 4 , 5 , 5 , 5 , 6 , 6 ,10,11
Mean=6.5 Median=5.5 Mode=5
A uniform change in data will result in a
uniform change in values
for mean, median and mode.
17
The effect of changing a set of data to the mode, mean and median
Data set : 2 , 3 , 3 , 3 , 4, 4 , 8 , 9
Each number in the set of data is multiplied by 4.
Data is changed uniformly
Data set : 8 , 12 , 12 , 12 , 16, 16 , 32 , 36
Mean8 + 12 + 12 + 12 + 16 + 16 + 32 + 36
=
144 8
= 8 = 18
18
The effect of changing a set of data to the mode, mean and median
Data set : 2 , 3 , 3 , 3 , 4, 4 , 8 , 9
Each number in the set of data is multiplied by 4.
Data set : 8 , 12 , 12 , 12 , 16, 16 , 32 , 36
Median = 12 + 16 Mode = 12
2
= 14
19
Data set : 2 , 3 , 3 , 3 , 4 , 4 , 8 , 9
Mean=4.5 Median=3.5 Mode=3
×4 : 8 ,12,12,12,16,16,32,36
Mean=18 Median=14 Mode=12
A uniform change in data will result in a
uniform change in values
for mean, median and mode.
20
The effect of changing a set of data
to the mode, mean and median
Data is changed in a
non-uniform manner
21
The masses of Ali, Ben and Kent are 43 kg,
49 kg and 49 kg respectively.
Determine the mode, mean and median of their
masses. 43 + 49 + 49
Mode Mean= 3 Median
=49 kg
=49 kg 141
=3 22
= 47
The masses of Ali, Ben and Kent are 43
kg, 49 kg and 49 kg respectively.
After one year, Ali’s mass increased 2 kg,
Ben’s mass decreased 1 kg and Kent’s
mass increased 3 kg.
Data is changed in a non-uniform manner
Determine the new mode, mean and
median of their masses. 23
The masses of Ali, Ben and Kent are 43 kg, 49 kg and 49 kg
respectively. After one year, Ali’s mass increased 2 kg, Ben’s
mass decreased 1 kg and Kent’s mass increased 3 kg.
Determine the new mode, mean and median of their masses.
43+2 49−1 49+3 Mean
=45 kg =48 kg =52 kg
45 + 48 + 52
No Median =
Mode =48 kg 145 3
=3
= 48.3 24
Initial masses of Ali, Ben and Kent are 43 kg, 49 kg and 49 kg
respectively. Mode=49 kg Mean=47 kg Median=49 kg
After one year, Ali, Ben and Kent are 45 kg, 48 kg and 52 kg
respectively. No Mode Mean=48.3 kg Median=48 kg
When every value of the data
changes non-uniformly,
the new mode, mean and median will
also change non-uniformly. 25
When we have to deal with
a large amount of data, we may
condense the data into several groups,
by the concept of grouping of data.
Grouped Data
26
Organising data in
frequency tables for grouped data
Frequency table for
grouped data
27
Example 1: TP 3
The marks obtained by forty students
in an examination are listed below:
16, 17, 18, 3, 7, 23, 18, 13, 10, 21,
7, 1, 13, 21, 13, 15, 19, 24, 16, 2,
23, 5, 12, 18, 8, 12, 6, 8, 16, 5,
3, 5, 0, 7, 9, 12, 20, 10, 2, 23 28
The marks obtained by forty students in an examination are listed below:
16, 17, 18, 3, 7, 23, 18, 13, 10, 21, 7, 1, 13,
21, 13, 15, 19, 24, 16, 2, 23, 5, 12, 18, 8, 12,
6, 8, 16, 5, 3, 5, 0, 7, 9, 12, 20, 10, 2, 23
Divide the data into five groups,
0−5, 6−10, 11−15, 16−20 and 21−25
Uniform class interval
~to prevent the data from overlapping
29
16, 17, 18, 3, 7, 23, 18, 13, 10, 21, 7, 1, 13,
21, 13, 15, 19, 24, 16, 2, 23, 5, 12, 18, 8, 12,
6, 8, 16, 5, 3, 5, 0, 7, 9, 12, 20, 10, 2, 23
Frequency Distribution Table
Marks Tally Frequency
0−5 9
6 − 10 9
11 − 15 7
16 − 20 9
21 − 25 6 30
TP 2
31
32
TP 2
33
34
Interpreting the frequency table
for grouped data
Modal Class of a set
of grouped data
35
Determining the modal class of a set of grouped data.
Modal Class Class Frequency
10−15 4
the class interval 16−20 7
with the 21−25 20
highest 26−30 8
31−35 1
frequency Highest frequency= 20
Modal class=21−25 36
Determining the modal class of a set of grouped data.
TP 2 The table shows the number of members in 20 minutes.
Number of members Frequency
1−5 5
6−10 8
11−15 5
16 −20 2
Highest frequency= 8 37
Modal class= 6−10
Determining the modal class of a set of grouped data.
TP 2 The table shows the test scores of 25 students.
Number of members Frequency
40 −49 5
50 −59 4
60 −69 7
70 −79 5
80−89 4
Highest frequency=7 Modal class= 60−69
38
Determining the modal class of a set of grouped data.
TP 2 The table shows the body weight of 20 students.
Number of members Frequency
35 −39 3
40 −44 5
45 −49 6
50 −54 4
Highest frequency= 6
Modal class=45 −49 39
Revision:
Interpreting the frequency table
Mean
for a set
of ungrouped data
40
The mean of a set of Data TP 3
Find the mean of the following data.
Score Frequency
54 #1 :
10 5
15 9 Find the
20 4 sum of data
25 2
30 6 41
The mean of a set of Data
Find the mean of the following data.
Score Frequency Sum of data
5× 4 20
50
10 × 5 135
80
15 × 9 50
180
20 × 4
25 × 2
30 × 6 42
Find the mean of the following data.
Number of frequency
4+5+9+4+2+6
= 30
Mean
20 + 50 + 135 + 80 + 50 + 180 515
30 = 30
= 17.17 43
Interpreting the frequency table
for grouped data
Mean for a set of
grouped data
44
The mean of a set of Grouped Data
Find the mean of the following grouped data.
#1 : Score Frequency
Determine the 0−4 4
5−9 5
midpoint of 10−14 9
the class 15−19 4
intervals. 20−24 2
25−29
6 4455
The mean of a set of Grouped Data
Find the mean of the following grouped data.
Score Midpoint Frequency
0−4 4
5−9 5
10−14 9
15−19 4
20−24 2
25−29 6 46
The mean of a set of Grouped Data
Find the mean of the following grouped data.
Midpoint=
+
2
47
The mean of a set of Grouped Data
Find the mean of the following grouped data.
Score Midpoint Frequency Midpoint
0−4 4
5−9 2 0+4
10−14
15−19 5 =2
20−24 9 4
25−29 4
2 =2
6 =2 48
The mean of a set of Grouped Data
Find the mean of the following grouped data.
Score Midpoint Frequency Midpoint
0−4 2 4 5+9
5−9 7 5 =2
10−14 9
15−19 14
20−24 4 =2
25−29 2
6 =7 49
The mean of a set of Grouped Data
Find the mean of the following grouped data.
Score Midpoint Frequency
0−4 2 4 50
5−9 7 5
10−14 9
15−19 12 4
20−24 2
25−29 17 6
22
27