If the mode of the following data is 165, then what is the value of x?

​Class	150-155	155-160	160-165	165-170	170-175
Frequency	23	25	37	x	28

Correct option is A

Given:

Mode = 165

Formula Used:

In a grouped frequency distribution, mode lies in the modal class — the class with the highest frequency.

The mode formula is:

Mode = $$l + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h$$​​

Where:

l = lower boundary of modal class

$$f_1$$​= frequency of modal class

$$f_0$$​ = frequency of class before modal class

$$f_2$$​ = frequency of class after modal class

h = class width

Solution:

Given mode = 165, so modal class = 160 – 165

​$$l = 160, \quad f_1 = 37, \quad f_0 = 25, \quad f_2 = x, \quad h = 5$$​​

Using Mode formula:

​$$165 = 160 + \left( \frac{37 - 25}{2 \times 37 - 25 - x} \right) \times 5 \\ \ \\165 - 160 = \left( \frac{12}{74 - 25 - x} \right) \times 5 \\ \ \\5 = \left( \frac{12}{74 - 25 - x} \right) \times 5 \\ \ \\1 = \frac{12}{49 - x}\\ \ \\49 - x = 12\\ \ \\x = 37$$​​