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,

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

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.