Question

The mode of the following frequency distribution of 165 observations is 34.5. Find the value of a and b.
Class
5-14
14-23
23-32
32-41
41-50
50-59
59-68

Frequency
5
11
a
53
b
16
10

Answer 1

Answer: frequency a= 43,b = 27

Solution:

Mode of grouped data is given by the formula

$Mode = l + ( \frac{f1 - fo}{2f1 - fo - f2} )h$
here l = lower limit of modal class

f1= frequency of modal class

fo= frequency of preceding class to modal class

f2= frequency of succeeding class to modal class

h= height/class interval

Given Mode= 34.5

hence modal class: 32-41

l= 32

h= 9

f1 = 53

fo = a

f2 = b

put all values in the formula

$34.5 = 32 + ( \frac{53 - a}{2 \times 53 - a - b}) \times 9 \\ \\ \frac{2.5}{9} = ( \frac{53 - a}{106 - a - b} ) \\ \\ = > 265 - 2.5a - 2.5b = 477 - 9a \\ \\ 7.5a - 2.5b = 212 \\ \\ 75a - 25b = 2120 \\ \\ \\ 13a - 5b = 424 \: \: \: eq1$
As total observation are 165,i.e. sum of all Frequencies are 165

so

$5 + 11 + a + 53 + b + 16 + 10 = 165 \\ \\ 95 + a + b = 165 \\ \\ a + b = 70 \: \: \: eq2$
as we can see that eq 1 and eq2 are pair of linear equations in two variables

substitute b = 70-a in eq1

$13a - 5(70 - a) = 424 \\ \\ 13a - 350 + 5a = 424 \\ \\ 18a = 424 + 350 \\ \\ 18a = 774 \\ \\ a = 43 \\ \\ b = 70 - 43 = 27$
So, frequency a= 43

b = 27