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? [NCERT EXEMPLAR]

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, ${\overline{X}}_A=A+h\left(\frac{\sum _{}{f}_i{u}_i}{\sum _{}{f}_i}\right)=800+100\times \frac{10}{120}=808.33$

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 = $\frac{{\sigma }_A}{{\overline{X}}_A}\times 100=\frac{109.98}{808.33}\times 100=13.61$

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, ${\overline{X}}_B=A+h\left(\frac{\sum _{}{f}_i{u}_i}{\sum _{}{f}_i}\right)=800+100\times \frac{\left(-36\right)}{120}=800-30=770$

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 = $\frac{{\sigma }_B}{{\overline{X}}_B}\times 100=\frac{110}{770}\times 100=14.29$

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.