Solution :- 

Calculating Mean from Marks and Frequency Data

To find the mean of a frequency distribution, we need to sum the product of each value (Marks) and its corresponding frequency (No. of Students) and then divide by the total number of observations (total frequency).

The formula for the mean ($\bar{x}$) of a frequency distribution is:

$$ \bar{x} = \frac{\sum (f \cdot x)}{\sum f} $$

Where:

• $x$ represents the values (Marks).
• $f$ represents the frequencies (No. of Students).
• $\sum (f \cdot x)$ is the sum of the products of each value and its frequency.
• $\sum f$ is the sum of all frequencies (total number of students).

Data Organization and Calculation

First, let's organize the given data into a table and calculate the product ($f \cdot x$) for each row:

Summing the Values

Next, we calculate the sum of the 'No. of Students' column ($\sum f$) and the sum of the 'Product ($f \cdot x$)' column ($\sum (f \cdot x)$).

• Total No. of Students ($\sum f$): $$ \sum f = 6 + 3 + 6 + 2 + 5 = 22 $$
• Sum of Products ($\sum (f \cdot x)$): $$ \sum (f \cdot x) = 114 + 108 + 360 + 138 + 425 = 1145 $$

Mean Calculation

Finally, we apply the formula using the sums calculated above:

$$ \bar{x} = \frac{\sum (f \cdot x)}{\sum f} = \frac{1145}{22} $$

Performing the division:

$$ \bar{x} \approx 52.045 $$

Rounding the result to the nearest whole number, the mean marks are approximately 52.