Question

From the following data of marks, calculate standard deviation. What will be the value of standard deviation, if marks obtained by each student is increased by one?

Marks Obtained	1	2	3	4	5	6	7	8	9
No. of students	32	41	57	98	123	83	46	17	3

Answer 1

Marks (X)	Frequency (f)	fX	$\mathit{x}\mathbf{=}\left(\mathit{X}\mathbf{-}\overline{\mathit{X}}\right)$	x2	fx2
1 2 3 4 5 6 7 8 9	32 41 57 98 123 83 46 17 3	32 82 171 392 615 498 322 136 27	−3.55 −2.55 −1.55 −0.55 0.45 1.45 2.45 3.45 4.45	12.60 6.50 2.40 0.30 0.20 2.10 6.00 11.90 19.80	403.2 266.5 136.8 29.4 24.6 174.3 276 202.3 59.4
	Σf = 500	Σfx = 2275			Σfx2 = 1572.5

$$\begin{gathered} \overline{X}=\frac{\Sigma fX}{\Sigma f}=\frac{2275}{500}=4.55 \\ \mathrm{S}.\mathrm{D}.\left(\mathrm{\sigma }\right)=\sqrt{\frac{\Sigma f{x}^2}{N}}=\sqrt{\frac{1572.5}{500}}=\sqrt{3.145}=1.77marks \end{gathered}$$

As, standard deviation is independent of the change in origin, i.e. it will not get affected if the value of the series is increased or decreased by a constant quantity. Therefore it will remain the same.