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The sum and sum of the squares corresponding to length $x$ in(cm) and weight $y$ (in gm) of 50 plant products are given below;
$\begin{aligned}
&\sum_{i=1}^{50} x_i=212, \sum_{i=1}^{50} x_i^2=902.8, \sum_{i=1}^{50} y_i=261 \\
&\sum_{i=1}^{50} y_i^2=1457.6
\end{aligned}$
Which is more varying, the length or weight?
$\begin{aligned}
&\sum_{i=1}^{50} x_i=212, \sum_{i=1}^{50} x_i^2=902.8, \sum_{i=1}^{50} y_i=261 \\
&\sum_{i=1}^{50} y_i^2=1457.6
\end{aligned}$
Which is more varying, the length or weight?
Solution:
2551 Upvotes
Verified Answer
For length
$\mathrm{N}=50, \sum_{i=1}^{50} x_i=212, \sum_{i=1}^{50} x_i^2=902.8$
$\bar{x}=\frac{\sum x_i}{N}=\frac{212}{50}=4.24$
Standard deviation $=\sigma$
$=\frac{1}{N}\left[N \sum x_i^2-\left(\sum x_i\right)^2\right]$
$=\frac{1}{50} \sqrt{50 \times 902.8-(212)^2}$
$=\frac{\sqrt{196}}{50}=\frac{14}{50}=0.28$
Coefficients of variation $=\frac{\sigma}{\bar{x}} \times 100$
$=\frac{0.28}{4.24} \times 100=6.6$
For weight, $\mathrm{N}=50, \sum_{i=1}^{50} y_i=261, \sum_{i=1}^{50} y_i^2=1457.6$
Mean $\bar{x}=\frac{\sum y_i}{N}=\frac{261}{50}=5.22 \mathrm{gm}$.
Standard deviation, $\sigma$
$\begin{aligned}
&=\frac{1}{N} \sqrt{N \sum y_i^2-\left(\sum y_i\right)^2} \\
&=\frac{1}{50} \sqrt{50 \times 1457.6-(261)^2}=\frac{\sqrt{4759}}{50}=1.38 \\
&\therefore \text { Coefficients of variation }(\text { C.V })=\frac{\sigma}{\overline{\mathrm{x}}} \times 100 \\
&=\frac{1.38}{5.22} \times 100=26.4
\end{aligned}$
Coefficients of variation of weights is more than that of length
$\therefore$ Weight is more varying than length.
$\mathrm{N}=50, \sum_{i=1}^{50} x_i=212, \sum_{i=1}^{50} x_i^2=902.8$
$\bar{x}=\frac{\sum x_i}{N}=\frac{212}{50}=4.24$
Standard deviation $=\sigma$
$=\frac{1}{N}\left[N \sum x_i^2-\left(\sum x_i\right)^2\right]$
$=\frac{1}{50} \sqrt{50 \times 902.8-(212)^2}$
$=\frac{\sqrt{196}}{50}=\frac{14}{50}=0.28$
Coefficients of variation $=\frac{\sigma}{\bar{x}} \times 100$
$=\frac{0.28}{4.24} \times 100=6.6$
For weight, $\mathrm{N}=50, \sum_{i=1}^{50} y_i=261, \sum_{i=1}^{50} y_i^2=1457.6$
Mean $\bar{x}=\frac{\sum y_i}{N}=\frac{261}{50}=5.22 \mathrm{gm}$.
Standard deviation, $\sigma$
$\begin{aligned}
&=\frac{1}{N} \sqrt{N \sum y_i^2-\left(\sum y_i\right)^2} \\
&=\frac{1}{50} \sqrt{50 \times 1457.6-(261)^2}=\frac{\sqrt{4759}}{50}=1.38 \\
&\therefore \text { Coefficients of variation }(\text { C.V })=\frac{\sigma}{\overline{\mathrm{x}}} \times 100 \\
&=\frac{1.38}{5.22} \times 100=26.4
\end{aligned}$
Coefficients of variation of weights is more than that of length
$\therefore$ Weight is more varying than length.
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