Question

Life of bulbs produced by two factories A and B are given below:

Length of life (in hours):	550–650	650–750	750–850	850–950	950–1050
Factory A: (Number of bulbs)	10	22	52	20	16
Factory B: (Number of bulbs)	8	60	24	16	12

The bulbs of which factory are more consistent from the point of view of length of life?  

 

Answer 1

For factory A
Let the assumed mean A = 800 and h = 100. 

Length of Life (in hours)	Mid-Values \[\left( x_i \right)\]	\[u_i = \frac{x_i - 800}{100}\]	\[u_i^2\]	Number of bulbs \[\left( f_i \right)\]	\[f_i u_i\]	\[f_i u_i^2\]
550–650	600	−2	4	10	−20	40
650–750	700	−1	1	22	−22	22
750–850	800	0	0	52	0	0
850–950	900	1	1	20	20	20
950–1050	1000	2	4	16	32	64
				\[\sum_{} f_i = 120\]	\[\sum_{} f_i u_i = 10\]	\[\sum_{} f_i u_i^2 = 146\]

Mean,

\[X_A = A + h\left( \frac{\sum_{} f_i u_i}{\sum_{} f_i} \right) = 800 + 100 \times \frac{10}{120} = 808 . 33\]

Standard deviation,

\[\sigma_A = h\sqrt{\left( \frac{\sum_{} f_i u_i^2}{\sum_{} f_i} \right) - \left( \frac{\sum_{} f_i u_i}{\sum_{} f_i} \right)^2} = 100\sqrt{\left( \frac{146}{120} \right) - \left( \frac{10}{120} \right)^2} = 100 \times 1 . 0998 = 109 . 98\]

∴ Coefficient of variation = \[\frac{\sigma_A}{X_A} \times 100 = \frac{109 . 98}{808 . 33} \times 100 = 13 . 61\]

For factory B
Let the assumed mean A = 800 and h = 100.

Length of Life (in hours)	Mid-Values \[\left( x_i \right)\]	\[u_i = \frac{x_i - 800}{100}\]	\[u_i^2\]	Number of bulbs \[\left( f_i \right)\]	\[f_i u_i\]	\[f_i u_i^2\]
550–650	600	−2	4	8	−16	32
650–750	700	−1	1	60	−60	60
750–850	800	0	0	24	0	0
850–950	900	1	1	16	16	16
950–1050	1000	2	4	12	24	48
				\[\sum_{}f_i = 120\]	\[\sum_{} f_i u_i = - 36\]	\[\sum_{} f_i u_i^2 = 156\]

Mean,

\[X_B = A + h\left( \frac{\sum_{} f_i u_i}{\sum_{} f_i} \right) = 800 + 100 \times \frac{\left( - 36 \right)}{120} = 800 - 30 = 770\]

Standard deviation,

\[\sigma_B = h\sqrt{\left( \frac{\sum_{} f_i u_i^2}{\sum_{} f_i} \right) - \left( \frac{\sum_{} f_i u_i}{\sum_{} f_i} \right)^2} = 100\sqrt{\left( \frac{156}{120} \right) - \left( \frac{- 36}{120} \right)^2} = 100 \times 1 . 1 = 110\]

∴ Coefficient of variation = \[\frac{\sigma_B}{X_B} \times 100 = \frac{110}{770} \times 100 = 14 . 29\]

Since the coefficient of variation of factory B is greater than the coefficient of variation of factory A, therefore, factory B has more variability than factory A. This means bulbs of factory A are more consistent from the point of view of length of life.