Calculate arithmetic mean using Assumed Mean Method.

Here are the calculations for the arithmetic mean using the specified methods.

1. Calculate arithmetic mean using Assumed Mean Method.
Given:
Size ($x$): 20, 19, 18, 17, 16, 15, 14, 13, 12, 11
Frequency ($f$): 1, 2, 4, 8, 11, 10, 7, 4, 2, 1
Assumed Mean ($A$) = 15

Step 1: Create a table to calculate $d = x - A$ and $fd$.

$$\begin{array}{|c|c|c|c|} \hline x & f & d = x - 15 & fd \\ \hline 20 & 1 & 5 & 5 \\ 19 & 2 & 4 & 8 \\ 18 & 4 & 3 & 12 \\ 17 & 8 & 2 & 16 \\ 16 & 11 & 1 & 11 \\ 15 & 10 & 0 & 0 \\ 14 & 7 & -1 & -7 \\ 13 & 4 & -2 & -8 \\ 12 & 2 & -3 & -6 \\ 11 & 1 & -4 & -4 \\ \hline \sum & \sum f = 50 & & \sum fd = 27 \\ \hline \end{array}$$

Step 2: Apply the formula for the Assumed Mean Method.
The formula for the arithmetic mean using the Assumed Mean Method is $\bar{x} = A + \frac{\sum fd}{\sum f}$.

$$\bar{x} = 15 + \frac{27}{50}$$

Step 3: Calculate the arithmetic mean.

$$\bar{x} = 15 + 0.54 \\ \bar{x} = 15.54$$

The arithmetic mean is $\boxed{\textbf{15.54}}$.

2. Calculate arithmetic mean using Direct and Assumed Mean Methods.
Given:
Marks ($x$): 4, 5, 6, 7, 8
No. of Students ($f$): 12, 10, 18, 20, 24