Find the mean and mode of the following data Class: 15-20 | 20-25 | 25-30 | 30-35 | 35-40 | 40 45 Frequency: 12 | 10 | 15 | 11 | 7 | 5

Class: 15-20 | 20-25 | 25-30 | 30-35 | 35-40 | 40 45

Frequency:   12 | 10 | 15 | 11 | 7 | 5

15-20, 20-25, 25-30, 30-35, 35-40, 40-45

$ \text{Frequencies: } 12, 10, 15, 11, 7, 5 $

$ \text{Mean Calculation:} $

$ x_i = \frac{\text{Lower Bound} + \text{Upper Bound}}{2} $

Class marks: 

$17.5, 22.5, 27.5, 32.5, 37.5, 42.5 $

$ \text{Now, calculate } f_i x_i: $

$ 12 \times 17.5 = 210, $
$ 10 \times 22.5 = 225, $
$ 15 \times 27.5 = 412.5, $
$ 11 \times 32.5 = 357.5, $
$ 7 \times 37.5 = 262.5, $
$ 5 \times 42.5 = 212.5 $

$ \sum f_i x_i = 1680, \quad \sum f_i = 60 $

$ \text{Mean: } \bar{x} = \frac{\sum f_i x_i}{\sum f_i} = \frac{1680}{60} = 28 $

$ \text{Mode Calculation:} $

The modal class is the class with the highest frequency, which is 25-30.

$ \text{Formula for mode: }$

$\text{Mode} = L + \frac{(f_1 - f_0)}{(2f_1 - f_0 - f_2)} \times h $

$ L = 25, \quad f_1 = 15, \quad $

$f_0 = 10, \quad f_2 = 11, \quad h = 5 $

$ \text{Substituting values:} $
$ \text{Mode} = 25 + \frac{(15 - 10)}{(2 \times 15 - 10 - 11)} \times 5 $
$ = 25 + \frac{5}{9} \times 5 $
$ = 25 + 2.78 $
$ = 27.78 $

$ \text{Final Answer:} $

$ \text{Mean} = 28, \quad \text{Mode} = 27.78 $