Here are the steps to solve this problem: 1) Form a frequency distribution table with class intervals of width 1: Marks Frequency 0-1 8 1-2 9 2-3 8 3-4 6 4-5 5 5-6 3 2) To find the mean: Add all values and divide by total number of values = ∑fx/N = 0*8 + 1*9 + 2*8 + 3*6 + 4*5 + 5*3 = 84/35 = 2.4 3) To find the median: The median is the middle value when the data is arranged in ascending order. Since there are 35

1. Mode (Z)
› Dr. Mamatha S Upadhya

2. In everyday life we often apply the concept of majority. For example, the
one who got majority of votes in your class become the class leader.
Sometimes majority represents the data. Mode is the measure which
expressing the concept of majority.

3. Mode (Z)
› The Mode (Z) or M0
› The Mode of a set of numbers is the value that occurs with the
greatest frequency.
› It is possible that there is no Mode in a series of numbers or to
have more than one value.
› The Mode is used when a quick and approximate measure of the
central tendency is desired.

4. › Unimodal Data
› A series of numbers is known to be unimodal if it has only
one Mode.
› Example: 3,3,4,5,5,5,7 has a mode of 5

5. › Bimodal Data
› A series of numbers is known to be bimodal if it has
two Modes.
› Example: 105,105, 105, 107, 108, 109, 109, 109,110,
112 has two Modes 105 and 109

6. › Multimodal Data
› A series of numbers is known to be multimodal if it has more
than two Modes.
› Example: 105, 105, 105, 107, 108, 109, 109, 109, 110, 112, 115,
115, 115 has three Modes 105, 109, 115.

7. › Define Mode.
› ‘The value of the variable which occurs most frequently in the distribution is called the
mode.’
›Definition: The mode is that value (or size) of the variate for which the frequency is
maximum or the point of maximum frequency or the point of maximum density.
Calculation of mode in the case of Individual Observations.
For Individual Observations , the value of mode is the item with the highest frequency.
Example Find the mode from the following data.
› 10,20,10,30,20,10,40,40,20,20,30,10,40,10,50.
Value Frequency
10 5
20 4
30 2
40 3
50 1
Since the value 10 has the highest frequency the modal value is 10.

8. Example: The ages of six persons who participated in an interview were
20,21,21,24,25,24,21 and 27 years. Find the mode of the data.
Here the observation 21 appears three times, 24 appears two and all others are
appear in a single time. So the value which appears a maximum number of
times is 21.
∴ Mode = 21 years.
Example: Mr. Vijayakumar, the physical education teacher of a school is
trying to determine the average height of students in the cricket team of the
school. The height of the players in inches are 70,72,72,74,74,74,75,76,76,76
and 77.Calculate the mode of the heights.
Here the data has two values, 74 and 76, which appears 3 times.
All the other values appear less than 3 times. So the data set has two distinct
modes 74 and 76.

9. Mode from a discrete frequency distribution
› The mode of a discrete frequency is the observation which appears a
maximum number of times. ie, the observation having the highest frequency.
Example: The following distribution shows the sizes of shirts sold on a textile
shop in Bangalore on a month. Calculate the mode.
Size (in inches) : 36 38 40 42 44
No of shirts sold :15 22 31 30 20
Solution. In the given frequency distribution, the observation having the
maximum frequency is 40. Mode is 40.

10. Mode from a continuous frequency distribution.
› As in the case of median, here also we have to locate a class called modal
class to find mode in continuous frequency series. Modal class is the class
having highest frequency. Mode can be determined by the formula:
𝑀0= L+
𝑓1−𝑓0
2𝑓1−𝑓0−𝑓2
x i
› L= lower limit of the modal class
› f1=frequency of the modal class
› f0= frequency of the class preceding the modal class.
› f2= frequency of the class Succeeding the modal class

11. › Example 2.Compute the mode of the following distribution:
Class : 0-7 7-14 14-21 21-28 28-35 35-42 42-49
Frequency: 19 25 36 72 51 43 28
Solution: Here maximum frequency 72 lies in the class-interval 21-28. Therefore 21-28 is the modal
class.
l= 21, 1
f = 72, 0
f = 36, 2
f = 51, i = 7
i
f
f
f
f
f
l
Mode 





2
0
1
0
1
2
103
.
25
7
51
36
)
72
(
2
36
72
21 






Mode
i
f
f
f
f
f
l
Mode 





2
0
1
0
1
2

12. Example: The production per day of a company (in Tons) on 60 days are given
below. Calculate the mode.
Production per day : 21-22 23-24 25-26 27-28 29-30
Number of days : 7 13 22 10 8
Solution. Here the classes are of inclusive type. We have to convert into
exclusive class before determining the mode.
X f
20.5-22.5 7
22.5-24.5 13
24.5-26.5 22
26.5-28.5 10
28.5-30.5 8
The maximum frequency is 22 so the modal
class is 24.5-26.5
Mode= L+
𝑓1−𝑓0
2𝑓1−𝑓0 − 𝑓2
x i
Mode= 24.5+
22 −13
2𝑥22 −13−10
x 2
Mode= 25.36 tons.

13. Example: Calculate the missing frequency from the following data, if mode is given as 26.
Marks 0-10 10-20 20-30 30-40 40-50
No.of Students 3 5 8 ? 3
Let the missing frequency be A
It is given that mode=26
So the modal Class is 20-30.
Mo=L+
𝑓1−𝑓0
2𝑓1−𝑓0−𝑓2
x i
26=20+
8−5
2(8)−5−𝐴
x 10
A=3.5

14. Merits and Demerits of Mode:
› Merits:
(i) Mode is readily comprehensible and easy to calculate.
(ii)Mode is not at all affected by extreme values.
› Demerits:
(i) Mode is ill-defined. It is not always possible to find a clearly defined mode.
(ii)It is not based upon all the observations.

15. Empirical relationship among Mean, Median and Mode
› Mode= 3median-2mean
The importance of this relation is that we can estimate the value of any one of them by
knowing the values of the other two. However, this relationship is true if the distribution is
moderately asymmetric.
In the asymmetrical curve the area on the left of mode is greater than area on the right then
› Mean < median < mode
Example If Z=400, and 𝑥=300 find median
400= 3M-2x300
M.=333.3
Example: For a given data the mean=39.23 and median =36.67 . Hence find mode.
Mode= 3 (36.67)-2(39.23)= 31.55

16. Graphical location of Mode
› Mode from Histogram
The steps involved in obtaining mode from histogram are:
Step 1: Draw a histogram to the given data.
Step 2: Locate modal class (highest bar of histogram).
Step 3: Join diagonally the upper end points of the highest bar to the end points of the adjacent
bars.
Step 4: Mark the point of intersection of the diagonals.
Step 5: Draw perpendicular from this point of intersection to the x-axis.
Step 6: The point where the perpendicular meets the x-axis gives the modal value.

17. Example: Locate mode graphically:
Age : 0-10 10-20 20-30 30-40 40-50 50-60
Number of People : 12 18 27 20 17 6
Mode = 25.6

18. Example 1: Locate Mode graphically for the following distribution.
Class
Interval
10-19 20-29 30-39 40-49 50-59 60-69 70-79
Frequency 10 12 18 30 16 6 8
Inclusive
Series
Exclusive
Series
Frequency
10-19 9.5-19.5 10
20-29 19.5-29.5 12
30-39 29.5-39.5 18
40-49 39.5-49.5 30
50-59 49.5-59.5 16
60-69 59.5-69.5 6
70-79 69.5-79.5 8

19. Example 2: The weights in grams of 50 apples picked at random from a market are as follows:
131,113,82,75,204,81,84,118,104,110,80,107,111,141,136,123,90,78,90,115, 110,98, 106, 99,
107,84,76,186,82,100,109,128,115,107,115,119,93,187,139,129,130,68,195,123,125,111,92,8
6,70,126.
Form the grouped frequency table by dividing the variate range into intervals of equal width,
each corresponding to 20 gms in such a way that the mid-value of the first class corresponding
to 70 gms. Hence locate mode graphically.
Solution: Mid-value of first class =70, Width of each class =20 (given)
Weight in
grams
Frequency
60-80 5
80-100 13
100-120 17
120-140 10
140-160 1
160-180 0
180-200 3
200-220 1
Total 50
Mode is 107.27 gms

20. A psychologist estimates the I.Q. of 60 children. The values are as follows:
103, 98, 87, 85, 67, 96, 115, 109, 127, 103, 95, 123,94, 88,102, 76, 73, 80,
84, 102, 115, 93, 76, 81, 132, 90, 119, 84, 97, 120, 114, 101, 153, 98, 99,
105, 110, 107, 110, 128, 89, 112, 118, 101, 122, 146, 96, 109, 72, 97, 94,
94, 79, 79, 100, 54, 102, 89, 43, 111.
Form a frequency distribution having class intervals with width 15. Hence
locate mode graphically.

21. From the following data prepare frequency distribution table and
hence determine the mean , median and mode.
0, 1,2,3,4,5,6,6,5,4,4,5,5,4,4,3,3,3,3,2,2,2,3,2,3,2,2,2,1,1,1,0,0,1,0.