Question

From the data given below state which group is more variable, G₁ or G₂?

Marks	10-20	20-30	30-40	40-50	50-60	60-70	70-80
Group G1	9	17	32	33	40	10	9
Group G2	10	20	30	25	43	15	7

Answer 1

Marks	\[f_i\]	Midpoint    \[\left( x_i \right)\]	\[u_i = \frac{x_i - 45}{10}\]	\[f_i u_i\]	\[f_i {u_i}^2\]
10−20	9	15	−3	−27	81
20−30	17	25	−2	−34	68
30−40	32	35	−1	−32	32
40−50	33	45	0	0	0
50−60	40	55	1	40	40
60−70	10	65	2	20	40
70−80	9	75	3	27	81
	N=150			\[\sum f_i u_i = - 6\]	\[\sum f_i u_i = - 342\]

\[h = 5, a = 45\]

For group 1: \[\bar{X} = a + h\left( \frac{\sum f_i u_i}{N} \right) = 45 + 10\left( \frac{- 6}{150} \right) = 44 . 6\]

 

\[\sigma^2 = h^2 \left[ \frac{\sum f_i {u_i}^2}{N} - \left( \frac{\sum f_i u_i}{N} \right)^2 \right] = 100\left[ \frac{342}{150} - \frac{36}{22500} \right] = 227 . 84\]
\[\sigma = \sqrt{227 . 84} = 15 . 09\]

Marks	\[f_i\]	Midpoint \[\left( x_i \right)\]	\[u_i = \frac{x_i - 45}{10}\]	\[f_i u_i\]	\[f_i {u_i}^2\]
10−20	10	15	−3	−30	90
20−30	20	25	−2	−40	80
30−40	30	35	−1	−30	30
40−50	25	45	0	0	0
50−60	43	55	1	43	43
60−70	15	65	2	30	60
70−80	7	75	3	21	63
	\[\sum f_i = 150\]			\[\sum f_iu_i = 6\]	\[\sum f_i {u_i}^2 = 366\]

For group 2:

\[\bar{X} = a + h\left( \frac{\sum f_i u_i}{N} \right) = 45 + 10\left( \frac{- 6}{150} \right) = 44 . 6\]

\[\sigma^2 = h^2 \left[ \frac{\sum f_i {u_i}^2}{N} - \left( \frac{\sum f_i u_i}{N} \right)^2 \right] = 100\left[ \frac{366}{150} - \frac{36}{22500} \right] = 243 . 84\]
\[\sigma = \sqrt{243 . 84} = 15 . 62\]

Mean of both the groups are same and SD of group 2 is greater than that of group 1.
So, group 2 will be more variable.