Explains about mean, median and mode

1. Central Tendency
Prof Vivek Katare
Shantiniketan Business School
Nagpur

2. Center: a representative or average
value that indicates where the
middle of the data set is located
Center of all colours
BLACK

3.  The most common characteristic to measure
is the center the dataset. Often people talk
about the AVERAGE.
◦ The average American man is six feet, one inches
tall

4.  “Average” Is ambiguous, since several
different methods can be used to obtain an
average
 Loosely stated, the average means the center
of the distribution or the most typical case
 Measures of Average are also called the
Measures of Central Tendency
 Mean
 Median
 Mode

5.  Measures of Center is the data value(s) at
the center or middle of a data set
 Mean
 Median
 Mode
◦ We will consider the definition, calculation
(formula), advantages, disadvantages,
properties, and uses for each measure of central
tendency

6.  Notation  Mean of a set of
◦ ∑ (sigma) denotes the sample values (read
sum of a set of values as x-bar)
◦ x is the variable usually
used to represent the x1 x2 x3 ......... xn
individual data values
◦ n represents the
x
number of values in a n
sample
◦ N represents the x
number of values in a x
population n

7.  Ex: The number of highway miles per gallon
of the 10 worst vehicles is given:
12 15 13 14
15 16 17 16
17 18
12 15 13 14 15 16 17 16 17 18
x
10
153
x 15.3
10

8. Example (2):
Given the following frequency
distribution of first year students
of a particular School.
Age (Years) 13 14 15 16 17
Number of
Students 2 5 13 7 3

9. Solution:
The given distribution belongs to a grouped data and the variable involved is
ages of first year students. While the number of students represent frequencies.
Ages (Years) Number of f .x
X Students (f)
13 2 26
14 5 70
15 13 195
16 7 112
17 3 51
Total f 30 fx 454
fx 454
x 15.13
f 30

10. Example (3):
The following data shows distance
covered by 100 persons to perform their
routine job
Distance (Km) : 0-10 10-20 20-30 30-40
No. of Persons : 10 20 40 30

11. Solution………..
Distance(Km) No. of Mid f.x
Persons Points
(f) x
0-10 10 5 50
10-20 20 15 300
20-30 40 25 1000
30-40 30 35 1050
f 100 fx 2400
fx 2400
x 24
f 100

12.  Is the middle value when the raw data values are
arranged in order from smallest to largest or vice
versa
 Is used when one must find the center or midpoint of
a data set
 Is used when one must determine whether the data
values fall into the upper half or lower half of the
distribution
 Does not have to be an original data value
 Various notations: MD, Med

13.  Arrange data in order  Arrange data in order
from smallest to from smallest to
largest largest
 Find the data value in  Find the mean of the
the “exact” middle TWO middle numbers
(there is no “exact”
middle)
Odd Number of Data Even Number of Data
Values (n is odd) Values (n is even)

14.  The number of highway miles per gallon of the
10 worst vehicles is given:
12 15 13 14
15 16 17 16
17 18
 Find the median.

15.  Solution: Arrange data in the ascending
order…
We get
12 13 14 15
15 16 16 17
17 18
15 16 31
Med 15.5
2 2

16. Example 2:
Find the Median of the following
distribution
X : 1 2 3 4 5 6
f : 7 12 17 19 21 24
Solution………..

17. X f Cumulative
Frequency
1 7 7
2 12 19
3 17 36
4 19 55
5 21 76
6 24 100
N=100
N 1
Med = Value of 2
th item
100 1 i.e 50.5 th item , which nearly belongs to c.f. of 55
Med
2 Hence Median = 4

18. Example 3:
Find the Median and Median Class of the
following distribution
X : 15-25 25-35 35-45 45-55 55-65 65-75
f : 4 11 19 14 0 2
Solution………..

19. Class f Cumulative Freq.
interval
15-25 4 4
25-35 11 15
35-45 19 34
45-55 14 48
55-65 0 48
65-75 2 50
N=50
N 50
Median Number = Value of 2 th element i.e 25th element,
2
and this value lies in cumulative freq. (34) for the class interval (35-45).
So the median class is (35-45)

20. Hence N
c
Median Med Lm ( 2 ).i
f
Where
Lm = 35
c = 15
f =19
i =10 put these values in above formula

21. 25 15
Med 35 [ ].10
19
100
35 35 5.26 40.26
19

22. Example 4:
Find the Median for following data
Value : 0-4 5-9 10-19 20-29 30-39 40-49 50-59 60-69
Freq. : 328 350 720 664 598 524 378 244
Solution………..

23. Class Class f Cumulative
interval Distribution Freq.
0-9 -0.5-9.5 678 678
10-19 9.5-19.5 720 1398
20-29 19.5-29.5 664 2062
30-39 29.5-39.5 598 2660
40-49 39.5-49.5 524 3184
50-59 49.5-59.5 378 3562
60-69 59.5-69.5 244 3806
N
Median Number = Value of 2 th element i.e 3806 1903rd element,
2
and this value lies in cumulative freq. (2062) for the
class distribution (19.5-29.5).
So the median class is(19.5-29.5)

24. Hence N
c
Median Med Lm ( 2 ).i
f
Where
Lm = 19.5
C = 1398
f =664
I =10 put these values in above formula

25. 1903 1398
Med 19.5 [ ].10
664
505 5050
19.5 [ ].10 19.5
664 664
19.5 7.60 27.1

26. Example 5:
Find missing frequency from the following
data, given that the median marks is 23
Marks : 0-10 10-20 20-30 30-40 40-50
No. of
Students : 5 8 ? 6 3

27. Solution………..
Let, the missing frequency is β.
Marks (x) No. of students (f) Cumulative
Frequency
0-10 5 5
10-20 8 13
20-30 β 13+β
30-40 6 19+β
40-50 3 22+β
Given median is 23. hence the median class is (20-30).
and N= (22+β)

28. Hence N
c
Median Med Lm ( 2 ).i
f
Where
Lm = 20
c = 13
f =β
i =10 put these values in above formula

29. 22
13
Med 20 [ 2 ].10
22 26
23 20 [ ].5
110 5 130
23 20
110 25 130
23

30. 23 110 25 130
2 20
10
So, the missing frequency is 10

31.  Is the data value(s) that occurs most often in
a data set
 Is not always unique. A data set can have
more than one mode, or the mode may not
exist for a data set
 Has no “special” symbol
 Look for the number(s) that occur the most
often in the data set

32. Example 1)
Find mode of following data
2, 7, 10, 15, 10, 17, 8, 10, 2
Solution:
Size of element : 2 7 8 10 15 17
No. of times it occur : 2 1 1 3 1 1
Number 10 is observed three times in above
data set
Hence 10 is the mode of above data set.

33. To find the Mode of grouped data we have following
formula
fm f1
Mode Lm [ ]i
2 fm f1 f2
Where
• Lm is the lower class limit of the modal class
• fm is the frequency of the modal class
• f1 is the frequency of the class before the modal class in
the frequency table
• f2 is the frequency of the class after the modal class in
the frequency table
• i is the class interval of the modal class

34. Example 2:
Find the Mode of following data
Weight (in Kg.) : 30-35 35-40 40-45 45-50 50-55 55-60
No. of Students : 5 9 14 22 16 4
Solution………..
Prof Vivek Katare
Shantiniketan Business School
Nagpur

35. Here the class interval (45-50) has the maximum frequency,
i.e. 22, therefore, modal class is (45-50).
Hence
fm f 1
Mode Lm [ ]i
2 fm f 1 f 2
We have,
• Lm =45
• fm =22
• f1 =14
• f2 =16
• i =5 put these values in above formula
22 14
Mode 45 [ ]5
2(22) 14 16

36. 8
Mode 45 [ ]5
44 14 16
8
45 [ ]5
14
20
45
7
45 2.85
47.85

37. Example 3:
The following are the marks obtained by students in a
class test. Find the modal marks.
Marks Students
32-35 10
36-39 37
40-43 65
44-47 80
48-51 51
52-55 35
56-59 18
60-63 4

38. Solution:
Class(Marks) Class frequency
Distribution
32-35 31.5-35.5 10
36-39 35.5-39.5 37
40-43 39.5-43.5 65
44-47 43.5-47.5 80 Modal class
48-51 47.5-51.5 51
52-55 51.5-55.5 35
56-59 55.5-59.5 18
60-63 59.5-63.5 4
It is clear from the data that the mode lies in the class 43.5-47.5

39. Hence
fm f 1
Mode Lm [ ]i
2 fm f 1 f 2
We have
• Lm =43.5
• fm =80
• f1 =65
• f2 =51
• i =4 put these values in above formula
80 65
Mode 43.5 [ ]4
2(80) 65 51

40. 15
Mode 43.5 [ ]4
160 65 51
15
43.5 [ ]4
44
15
43.5
11
43.5 1.36
44.86

41. Assignment
1) Calculate Mean, Median and Mode of the data given below:
X 0.5 1.2 0.9 1 0.6 0.8 1.8 0.9 2.2 1.1
2) Calculate Mean, Median and Mode of the information given below:
Class 0-4 5-9 10- 15- 20- 25- 30- 35- 40-
Interval 14 19 24 29 34 39 44
Freq. 8 10 11 13 25 13 10 5 5
3) Find Mean, Median of the following data:
Age Belo 20 30 40 50 60 70 80
w 10
No. of 5 25 60 105 180 250 275 320
Persons

42. 4) The mean marks of 100 students were found to be 40. Later on, it
was discovered that a score of 53 was misread as 83. Find the
correct mean corresponding to the correct score.
5) Mean of 200 observations is found to be 50. If at the time of
computation, two items are wrongly taken as 40 and 28 instead of 4
and 82. Find the correct mean.
6) Calculate Median and Mode for the distribution of the weights of 150
students from the data given below:
Class 30-40 40-50 50-60 60-70 70-80 80-90
Interval
Freq. 18 37 45 27 15 8

43. 7) Construct the Frequency table for the following data regarding
annual profits, in thousands of rupees in 50 firms by taking class
intervals as per your convenience.
Find Mean, Median and Mode.
28 35 61 29 36 48 57 67 69
50 48 40 47 42 41 37 51 62
63 33 31 31 34 40 38 37 60
51 54 56 37 46 42 38 61 59
58 44 39 57 38 44 45 45 47
38 44 47 47 64
Prof Vivek Katare
Shantiniketan Business School
Nagpur