Question

From the data given below state which series is more consistent:

Variable	:	10-20	20-30	30-40	40-50	50-60	60-70
Series A	:	10	18	32	40	22	18
Series B	:	18	22	40	32	18	10

Answer 1

For Series A

Variable	Mid value (m)	f	fm	Deviation from mean $x=m-{\overline{\mathit{X}}}_{\mathrm{A}}$ ${\overline{\mathit{X}}}_{\mathrm{A}}=42.14$	${\mathit{x}}_{\mathbf{A}}^{\mathbf{2}}$	$\mathit{f}{\mathit{x}}_{\mathbf{A}}^{\mathbf{2}}$
10−20	15	10	150	−27.14	736.58	7365.8
20−30	25	18	450	−17.14	293.78	5288.04
30−40	35	32	1120	−7.14	50.98	1631.36
40−50	45	40	1800	2.86	8.18	327.2
50−60	55	22	1210	12.86	165.38	3638.36
60−70	65	18	1170	22.86	522.58	9406.44
		Σf=140	Σfm=5900			$\sum f{x}_A^2=27657.2$

$$\begin{gathered} {\overline{\mathit{X}}}_{\mathrm{A}}=\frac{\mathrm{\Sigma }fm}{\mathrm{\Sigma }f}=\frac{5900}{140}=42.14 \\ S{D}_{\mathrm{A}}=\sqrt{\frac{\mathrm{\Sigma }f{x}^2}{\mathrm{\Sigma }f}}=\sqrt{\frac{27657.2}{140}}=14.06 \\ \mathrm{Coefficient}\mathrm{of}\mathrm{variation}=\frac{S{D}_{\mathrm{A}}}{{\overline{\mathit{X}}}_{\mathrm{A}}}\times 100 \\ =\frac{14.06}{42.14}\times 100=33.36\% \end{gathered}$$

For Series B

Variable	Mid value (m)	f	fm	Deviation from mean $x=m-{\overline{\mathrm{X}}}_{\mathrm{B}}$ ${\overline{\mathrm{X}}}_{\mathrm{B}}=37.86$	${\mathit{x}}_{\mathbf{B}}^{\mathbf{2}}$	$\mathit{f}{\mathit{x}}_{\mathbf{B}}^{\mathbf{2}}$
10−20	15	18	270	−22.86	522.58	9406.44
20−30	25	22	550	−12.86	165.38	3638.36
30−40	35	40	1400	−2.86	8.18	327.2
40−50	45	32	1440	7.14	50.98	1631.36
50−60	55	18	990	17.14	293.78	5288.04
60−70	65	10	650	27.14	736.58	7365.8
		Σf=140	Σfm=5300			$\mathrm{\Sigma }f{x}_{\mathrm{B}}^2=27657.2$

$$\begin{gathered} {\overline{\mathit{X}}}_{\mathrm{B}}=\frac{\mathrm{\Sigma }fm}{\mathrm{\Sigma }f}=\frac{5300}{140}=37.86 \\ S{D}_{\mathrm{B}}=\sqrt{\frac{\mathrm{\Sigma }f{x}_{\mathrm{B}}^2}{\mathrm{\Sigma }f}}=\sqrt{\frac{27657.2}{140}}=\sqrt{197.55}=14.06 \\ \mathrm{Coefficient}\mathrm{of}\mathrm{variation}=\frac{S{D}_{\mathrm{B}}}{{\overline{X}}_{\mathrm{B}}}\times 100 \\ =\frac{14.06}{37.86}\times 100=37.14\% \end{gathered}$$

As coefficient of variation for series 'A' is less than series 'B', therefore, series A is more consistent.