Question

Find the Mean from the Following Table

Given data:

\[ \begin{array}{|c|c|} \hline \text{Class-interval} & \text{Frequency (f)} \\ \hline 0-10 & 3 \\ 10-20 & 10 \\ 20-30 & 11 \\ 30-40 & 9 \\ 40-50 & 7 \\ \hline \end{array} \]

Hint:

When calculating the mean for a frequency distribution, always calculate the class marks first, then multiply each class mark by the corresponding frequency, and finally divide the sum of these products by the total frequency.

Answer 1

Step 1: Formula for the Mean

To find the mean from the frequency distribution, we use the formula:

\[ \text{Mean} = \frac{\sum f_i x_i}{\sum f_i} \]

Where \( f_i \) is the frequency and \( x_i \) is the class mark (midpoint) of each class interval.

Step 2: Find the Class Marks

First, find the class marks \( x_i \) for each class interval. The class mark is calculated as the average of the lower and upper limits of each interval:

\[ x_1 = \frac{0 + 10}{2} = 5, \quad x_2 = \frac{10 + 20}{2} = 15, \quad x_3 = \frac{20 + 30}{2} = 25, \quad x_4 = \frac{30 + 40}{2} = 35, \quad x_5 = \frac{40 + 50}{2} = 45. \]

Step 3: Create the Table

Now, create a table with \( f_i \), \( x_i \), and \( f_i x_i \):

\[ \begin{array}{|c|c|c|c|} \hline \text{Class-interval} & \text{Frequency} (f_i) & \text{Class mark} (x_i) & f_i x_i \\ \hline 0-10 & 3 & 5 & 15 \\ 10-20 & 10 & 15 & 150 \\ 20-30 & 11 & 25 & 275 \\ 30-40 & 9 & 35 & 315 \\ 40-50 & 7 & 45 & 315 \\ \hline \end{array} \]

Step 4: Calculate the Sums

Now, calculate the sum of \( f_i x_i \) and \( f_i \):

\[ \sum f_i x_i = 15 + 150 + 275 + 315 + 315 = 1070, \quad \sum f_i = 3 + 10 + 11 + 9 + 7 = 40. \]

Step 5: Calculate the Mean

The mean is:

\[ \text{Mean} = \frac{1070}{40} = 26.75 \]

Step 6: Conclusion

Conclusion: The mean of the given frequency distribution is \( \mathbf{26.75} \).