Question

If $n=10$, $\overline{x}=12$ and $\sum{{{x}^{2}}}=1530$, then calculate the coefficient of variation.
A. 20
B. 25
C. 30
D. 35

Answer 1

Hint: We first define the notion of coefficient of variation both as explanation and mathematical notion. We use the formula to find the value of standard deviation. We use all the given parameters to find the solution of the problem using the formula.

Complete step by step answer:
The coefficient of variation is the standard deviation expressed as a percentage of the mean.
Let us consider for given data ${{x}_{i}}$, variance = v, standard deviation = $\sqrt{v}=s$, mean = $\overline{x}=12$.
It’s also given ${{x}_{i}}$, $i=1(1)10$.
The mean can be explained as $\overline{x}=\dfrac{1}{n}\sum{{{x}_{i}}}$.
The variance is defined as \[v=\dfrac{1}{n}\sum{{{x}_{i}}^{2}}-{{\left( \overline{x} \right)}^{2}}\].
We also know standard deviation = \[s=\sqrt{\dfrac{1}{n}\sum{{{x}_{i}}^{2}}-{{\left( \overline{x} \right)}^{2}}}\] and $\sum{{{x}_{i}}^{2}}=1530$
So, we express coefficient of variation (c.v) in mathematical notation as
$c.v=\left( \dfrac{s}{\overline{x}}\times 100 \right)\%$.
We first try to find the value of standard deviation.
This gives by putting values
\[\begin{align}
  & s=\sqrt{\dfrac{1}{n}\sum{{{x}_{i}}^{2}}-{{\left( \overline{x} \right)}^{2}}} \\
 & \Rightarrow s=\sqrt{\dfrac{1}{10}\left( 1530 \right)-{{12}^{2}}} \\
\end{align}\]
We solve to get value of s as
\[\begin{align}
  & s=\sqrt{\dfrac{1530}{10}-{{12}^{2}}} \\
 & \Rightarrow s=\sqrt{153-144} \\
 & \Rightarrow s=\sqrt{9}=3 \\
\end{align}\].
Now we put the values of the given values to get coefficient of variation (c.v)
\[\begin{align}
  & c.v=\left( \dfrac{s}{\overline{x}}\times 100 \right)\% \\
 & \Rightarrow c.v=\left( \dfrac{3}{12}\times 100 \right)\% \\
 & \Rightarrow c.v=\left( \dfrac{100}{4} \right)\% \\
 & \Rightarrow c.v=25\% \\
\end{align}\]
So, the coefficient of variation is $25\%$.

So, the correct answer is “Option B”.

Note: We have to remember that from coefficient of variation we are finding the ratios of standard deviation and mean and its percentage value. It is a dependent parameter.