# The mode of the f

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.

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