Q. 64.6( 9 Votes )

The mode of the f

Answer :

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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