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

Given that,

Total number of observations = 165

⇒ Sum of frequencies = 165

⇒ 5 + 11 + a + 53 + b + 16 + 10 = 165

⇒ 95 + a + b = 165

⇒ a + b = 165 – 95

⇒ a + b = 70 …(i)

Also, given that

Mode of the data = 34.5

Corresponding to the modal value, the class it lies between is 32 – 41. [∵ 34.5 lies between 32 – 41]

⇒ Modal class = 32 – 41

Now,

Mode of such grouped frequency distribution is given by,

Where,

l = lower limit of the modal class = 32

f₀ = frequency of the class preceding the modal class = a

f₁ = frequency of the modal class = 53

f₂ = frequency of the class succeeding the modal class = b

c = size of class interval (the class intervals are same) = 9

∴ Substituting the values l = 32, f₀ = a, f₁ = 53, f₂ = b and c = 9 in the formula of mode. We get

⇒

⇒

⇒

⇒ 2.5 × (106 – a – b) = 477 – 9a

⇒ 265 – 2.5a – 2.5b = 477 – 9a

⇒ 9a – 2.5a – 2.5b = 477 – 265

⇒ 6.5a – 2.5b = 212

⇒

⇒

⇒ 65a – 25b = 2120

⇒ 13a – 5b = 424 …(ii)

Multiply 5 by equation (i),

a + b = 70 [× 5

⇒ 5a + 5b = 350 …(iii)

Solving equation (i) and (iii), we get

13a – 5b = 424

5a + 5b = 350

18a + 0 = 774

⇒ 18a = 774

⇒

⇒ a = 43

Putting value a = 43 in equation (i),

a + b = 70

⇒ 43 + b = 70

⇒ b = 70 – 43

⇒ b = 27

Thus, a = 43 and b = 27.