Understanding the Problem

The question provides a grouped frequency distribution table showing the number of students (frequency) within certain marks ranges (class intervals). We are also given that the mean marks of the students from this data is 27.5. One of the frequencies in the table is represented by the variable x, which we need to find.

To solve this, we will use the formula for calculating the mean of a grouped frequency distribution. The mean is calculated as the sum of the products of the class mark and frequency for each class, divided by the total number of frequencies.

Calculating Mean of Grouped Data

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

\(\bar{x} = \frac{\sum f_i x_i}{\sum f_i}\)

Where:

• \(\sum f_i x_i\) is the sum of the product of frequency ($\(f_i\)$) and class mark ($\(x_i\)$) for each class interval.
• \(\sum f_i\) is the sum of the frequencies, which represents the total number of students.

Steps to Find the Value of x

1. Calculate the class mark (\(x_i\)) for each class interval. The class mark is the midpoint of the class interval.

2. Calculate the product of the frequency (\(f_i\)) and the class mark (\(x_i\)) for each class interval.

3. Find the sum of frequencies (\(\sum f_i\)).

4. Find the sum of the products (\(\sum f_i x_i\)).

5. Substitute the known values (the given mean, \(\sum f_i\), and \(\sum f_i x_i\)) into the mean formula and solve for x.

Detailed Calculation Table

Let's create a table to organize the data and calculations:

Summation of Frequencies and Products

Sum of frequencies: \(\sum f_i = 10 + 15 + x + 35 + 15 = 75 + x\)

Sum of products: \(\sum f_i x_i = 50 + 225 + 25x + 1225 + 675 = 2175 + 25x\)

Solving for x

We are given that the mean (\(\bar{x}\)) is 27.5.

Using the mean formula:

\(27.5 = \frac{2175 + 25x}{75 + x}\)

Now, we solve this equation for x:

\(27.5 \times (75 + x) = 2175 + 25x\)

\(27.5 \times 75 + 27.5x = 2175 + 25x\)

\(2062.5 + 27.5x = 2175 + 25x\)

Subtract 25x from both sides:

\(2062.5 + 27.5x - 25x = 2175\)

\(2062.5 + 2.5x = 2175\)

Subtract 2062.5 from both sides:

\(2.5x = 2175 - 2062.5\)

\(2.5x = 112.5\)

Divide by 2.5:

\(x = \frac{112.5}{2.5}\)

\(x = \frac{1125}{25}\)

\(x = 45\)

The value of x is 45.

Conclusion

Based on the given mean marks of 27.5 and the grouped frequency distribution, the calculated value for the missing frequency x is 45.