### Theory:

The mode of the grouped frequency distribution can be determined using the formula:

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

The class interval with maximum frequency is called the modal class.

Where $$l$$ is the lower limit of the modal class,

$$f_1$$ is the frequency of the modal class,

$$f_0$$ is the frequency of the class preceding the modal class,

$$f_2$$ is the frequency of the class succeeding the modal class, and

$$h$$ is the width of the class interval.
Example:
Find the mode of the following data:

 Class interval $$130 - 140$$ $$140 - 150$$ $$150 - 160$$ $$160 - 170$$ $$170 - 180$$ Frequency $$5$$ $$36$$ $$14$$ $$28$$ $$1$$

Solution:

The maximum frequency is $$36$$, and the modal class is $$140 - 150$$.
The mode of the grouped frequency distribution can be determined using the formula:

Mode $$= l + \left(\frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h$$
Here, $$l = 140$$, $$f_1 = 36$$, $$f_0 = 5$$, $$f_2 = 14$$, $$h = 10$$

Substituting the known values in the above formula, we have;

Mode $$= 140 + \left(\frac{36 - 5}{2(36) - 5 - 14} \right) \times 10$$

$$= 140 + \left(\frac{36 - 5}{72 - 5 - 14} \right) \times 10$$

$$= 140 + (\frac{31}{53}) \times 10$$

$$= 140 + 0.585 \times 10$$

$$= 140 + 5.85$$

$$= 145.85$$

Therefore, the mode of the given data is $$145.85$$.