Question

Lives of two models of refrigerators in a recent survey are as under:

Life No fo Years	:	0-2	2-4	4-6	6-8	8-10	10-12
No of Refrigerators	:
Model A	:	5	16	13	7	5	4
Model B	:	2	7	12	19	9	1

What is the average life of each model of refrigerators? Which model has more uniformity?

Answer 1

Model A:

Years	Mid Value (m)	f	fm	Deviation from mean $$\begin{gathered} x_{\mathrm{A}}=\left(m-\overline{\mathit{X}}\right) \\ \left(\overline{\mathit{X}}=5.12\right) \end{gathered}$$	$x_{\mathrm{A}}^2$	$f{x}_{\mathrm{A}}^2$
0 − 2 2 − 4 4 − 6 6 − 8 8 − 10 10 − 12	1 3 5 7 9 11	5 16 13 7 5 4	5 48 65 49 45 44	−4.12 −2.12 −.12 1.88 3.88 5.88	16.97 4.49 0.01 3.53 15.05 34.57	84.85 71.84 0.13 24.71 75.25 138.28
		Σf = 50	Σfm = 256			$\sum f{x}_{\mathrm{A}}^2=395.06$

$$\begin{gathered} {\overline{\mathit{X}}}_{\mathrm{A}}=\frac{\mathrm{\Sigma }fm}{\mathrm{\Sigma }f}=\frac{256}{50}=5.12 \\ SD=\sqrt{\frac{\mathrm{\Sigma }f{x}_A^2}{\mathrm{\Sigma }f}}=\sqrt{\frac{395.06}{50}}=\sqrt{7.90}=2.81 \\ \mathrm{Coefficient}\mathrm{of}\mathrm{variation}=\frac{SD}{{\overline{\mathit{X}}}_{\mathit{A}}}=\frac{2.81}{5.12}\times 100=54.9\% \end{gathered}$$
Average life of model A is 5.12 year.

Model B:

Years	Mid Value (m)	f	fm	Deviation from mean $$\begin{gathered} x_B=\left(m-\overline{\mathit{X}}\right) \\ \left(\overline{\mathit{X}}=6.16\right) \end{gathered}$$	$x_B^2$	$f{x}_B^2$
0 − 2 2 − 4 4 − 6 6 − 8 8 − 10 10 − 12	1 3 5 7 9 11	2 7 12 19 9 1	2 21 60 133 81 11	−5.16 −3.16 −1.16 0.84 2.84 4.84	26.62 9.98 1.34 0.70 8.06 23,42	53.24 69.86 16.08 13.3 72.54 23.42
		Σf = 50	Σfm = 308			$\sum f{x}_B^2=248.44$

$$\begin{gathered} {\overline{\mathit{X}}}_{\mathrm{B}}=\frac{\mathrm{\Sigma }fm}{\mathrm{\Sigma }f}=\frac{308}{50}=6.16 \\ SD=\sqrt{\frac{\mathrm{\Sigma }f{x}_B^2}{\mathrm{\Sigma }f}}=\sqrt{\frac{248.44}{50}}=\sqrt{4.96}=2.23 \\ \mathrm{Coefficient}\mathrm{of}\mathrm{variation}=\frac{\mathit{S}\mathit{D}}{{\overline{\mathit{X}}}_{\mathit{B}}}=\frac{2.23}{6.16}\times 100=36.2\% \end{gathered}$$
Average life of model B is 6.16 year.

Since the coefficient of variation of model B is less than that of A, thus model B has more uniformity.