Question

The following figures are the heights in cms of 7 children chose at random:
64, 59, 67, 69, 65, 70, 68
Calculate the simple arithmetic mean of the heights by (i) Direct method, (ii) Short-c ut Method, and (iii) Step Deviation Method.

Answer 1

S.N.	Height (X)	Deviation from assumed mean d=(X − A)	$\mathit{d}\mathbf{\text{'}}\mathbf{=}\frac{{\mathit{d}}_{}}{\mathit{i}}\mathbf{=}\frac{{\mathit{d}}_{}}{\mathbf{2}}$
1 2 3 4 5 6 7	64 59 67 $\boxed{69=A}$ 65 70 68	−5 −10 −2 0 −4 1 −1	−2.5 −5 −1 0 −2 0.5 −0.5
N=7	$\sum _{}\mathit{X}$=462	Σd=−21	Σd'= −10.5

(i) Calculating mean height using direct method:

$$\begin{gathered} \overline{X}=\frac{{\displaystyle \sum _{}}X}{N} \\ \mathrm{or},\overline{X}=\frac{64+59+67+69+65+70+68}{7} \\ \mathrm{or},\overline{X}=\frac{462}{7} \\ \Rightarrow \overline{X}=66cms \end{gathered}$$

(ii) Calculating mean height using short cut method:

$\overline{X}=A+\frac{{\displaystyle \sum _{}^{}}d{\displaystyle }}{N}$

Here,

A represents assumed mean.
d represents the deviation of the values from the assumed mean.

Here, we take 69 as the assumed mean. So, we take deviations of each item in the series from 69.

$$\begin{gathered} \overline{X}=69-\frac{21}{7} \\ \mathrm{or},\overline{X}\mathit{}=69-3 \\ \Rightarrow \overline{X}=66cms \end{gathered}$$

(iii) Calculating mean height using step-deviation method:

$\overline{X}=A+\frac{\mathrm{\Sigma }d\text{'}}{N}\times i$

$$\begin{gathered} \overline{X}=69+\frac{\left(-10.5\right)}{7}\times 2 \\ \mathrm{or},\overline{X}=69-\frac{21}{7} \\ \Rightarrow \overline{X}\mathit{}=66\mathrm{cms} \end{gathered}$$

Hence, the arithmetic mean of the heights is 66 cms.