Question

Calculate the mean, variance and standard deviation of the following frequency distribution. 

Class:	1–10	10–20	20–30	30–40	40–50	50–60
Frequency:	11	29	18	4	5	3

Answer 1

Let the assumed mean A = 25. 

Class	Mid-Values \[\left( x_i \right)\]	\[d_i = x_i - A\] \[ = x_i - 25\]	\[d_i^2\]	Frequency \[\left( f_i \right)\]	\[f_i d_i\]	\[f_i d_i^2\]
1–10	5.5	−19.5	380.25	11	−214.5	4182.75
10–20	15	−10	100	29	−290	2900
20–30	25	0	0	18	0	0
30–40	35	10	100	4	40	400
40–50	45	20	400	5	100	2000
50–60	55	30	900	3	90	2700
				\[\sum_{}f_i =70\]	\[\sum_{} f_i d_i= −274.5\]	\[\sum_{} f_i d_i^2=12182.75\]

\[A + \frac{\sum_{} f_i d_i}{\sum_{} f_i} = 25 + \left( \frac{- 274 . 5}{70} \right) = 25 - 3 . 92 = 21 . 08\]

Variance =\[\sigma^2 = \left( \frac{1}{N} \sum_{} f_i d_i^2 \right) - \left( \frac{1}{N} \sum_{} f_i d_i \right)^2 = \frac{12182 . 75}{70} - \left( \frac{- 274 . 5}{70} \right)^2 = 174 . 02 - 15 . 37 = 158 . 65\] 

Standard deviation = \[\sigma = \sqrt{158 . 65} = 12 . 6\]