The key idea is to calculate the Mean and Standard Deviation for a frequency distribution.

1. First, calculate the total frequency ($\sum f_i$) and the sum of products of marks and frequencies ($\sum f_i x_i$).

   $\sum f_i = 1+6+6+8+8+2+2+3+0+2+1+0+0+0+1 = 40$

   $\sum f_i x_i = 2(1)+3(6)+4(6)+5(8)+6(8)+7(2)+8(2)+9(3)+11(2)+12(1)+16(1) = 239$

2. Determine the mean ($\bar{x}$) using the formula:

3. Next, calculate the sum of squared products ($\sum f_i x_i^2$). $\sum f_i x_i^2 = 2^2(1)+3^2(6)+4^2(6)+5^2(8)+6^2(8)+7^2(2)+8^2(2)+9^2(3)+11^2(2)+12^2(1)+16^2(1) = 1753$ …

To determine the mean and standard deviation for a frequency distribution, we need to understand what these measures represent. The mean ($\bar{x}$) is the average value of the data, taking into account how often each value appears. It's the central point around which the data balances. The standard deviation ($\sigma$) measures the typical spread or dispersion of the data points around this mean. A small standard deviation indicates data points are clustered closely around the mean, while a large standard deviation means they are more spread out.

For a frequency distribution where $x_i$ represents the individual data points (marks) and $f_i$ represents their corresponding frequencies, the formulas are:

Let's calculate these values step-by-step.

1. Organize the data and calculate necessary sums.

   To efficiently calculate the mean and standard deviation, we create an extended table. This table includes columns for the marks ($x_i$), frequencies ($f_i$), the product $f_i x_i$, the square of the marks $x_i^2$, and the product $f_i x_i^2$. The sums of the $f_i$, $f_i x_i$, and $f_i x_i^2$ columns are essential for our formulas.

   Marks ($x_i$)	Frequency ($f_i$)	$f_i x_i$	$x_i^2$	$f_i x_i^2$
   2	1	2	4	4
   3	6	18	9	54
   4	6	24	16	96
   5	8	40	25	200
   6	8	48	36	288
   7	2	14	49	98
   8	2	16	64	128
   9	3	27	81	243
   10	0	0	100	0
   11	2	22	121	242
   12	1	12	144	144
   13	0	0	169	0
   14	0	0	196	0
   15	0	0	225	0
   16	1	16	256	256
   Total	$\sum f_i = 40$	$\sum f_i x_i = 239$		$\sum f_i x_i^2 = 1753$