Calculating Mean with Assumed Mean Method
The question in the image asks to calculate the mean of the given numbers using an assumed mean of 75. The data is presented in tabulated form, presumably representing frequencies corresponding to certain data points (or "marks").
Let's solve it step by step:
1. Use the Assumed Mean formula which is:
\[
\text{Mean} = A + \left( \frac{\sum f_i d_i}{\sum f_i} \right)
\]
where \( A \) is the assumed mean, \( f_i \) are the frequencies, and \( d_i \) are the deviations from the assumed mean for each class.
2. Calculate \( d_i \) which is the difference between data points and the assumed mean (75 in this case). Then multiply each \( d_i \) by the corresponding frequency \( f_i \) to find \( f_i d_i \).
The table with frequencies and deviations will look like this:
\[
\begin{align*}
\text{Marks (X)} &\text{ } \text{Freq. (f)} &\text{ } \text{Assumed mean (A = 75)} &\text{ } \text{Deviation (d = X - A)} &\text{ } f \times d \\
\hline
5 &\text{ } 6 &\text{ } 75 &\text{ } 5 - 75 = -70 &\text{ } 6 \times (-70) = -420 \\
7 &\text{ } 9 &\text{ } 75 &\text{ } 7 - 75 = -68 &\text{ } 9 \times (-68) = -612 \\
12 &\text{ } 12 &\text{ } 75 &\text{ } 12 - 75 = -63 &\text{ } 12 \times (-63) = -756 \\
14 &\text{ } 14 &\text{ } 75 &\text{ } 14 - 75 = -61 &\text{ } 14 \times (-61) = -854 \\
24 &\text{ } 7 &\text{ } 75 &\text{ } 24 - 75 = -51 &\text{ } 7 \times (-51) = -357 \\
\end{align*}
\]
3. Calculate the sum of the column \( f_i d_i \):
\[
\sum (f_i d_i) = -420 - 612 - 756 - 854 - 357 = -2999
\]
4. Calculate the sum of the frequencies \( \sum f_i \):
\[
\sum f_i = 6 + 9 + 12 + 14 + 7 = 48
\]
5. Now, plug the values into the formula to calculate the mean:
\[
\text{Mean} = 75 + \frac{-2999}{48} = 75 + (-62.48) \approx 75 - 62.5 = 12.5
\]
Therefore, the mean of the given numbers using an assumed mean of 75 is approximately 12.5.
