Question

The mean of 100 observations is 18.4 and the sum of squares of deviations from mean is 1444, find the coefficient of variation?

Answer 1

Hint: Write the sum of squares of deviations from mean in summation form as \[\sum\limits_{i=1}^{n}{{{\left( {{x}_{i}}-\bar{x} \right)}^{2}}}\] form n = 100 observations. Here \[\bar{x}\] is the notation of the mean of these observations. Now, calculate the standard deviation of the given observations by using the formula \[\sigma =\sqrt{\dfrac{1}{n}\times \sum\limits_{i=1}^{n}{{{\left( {{x}_{i}}-\bar{x} \right)}^{2}}}}\], where \[\sigma \] is the standard deviation. Finally, use the formula coefficient of variation = $\left( \dfrac{\sigma }{\overline{x}}\times 100 \right)$ to get the answer.

Complete step by step answer:
Here we have been provided with the mean of 100 observations and the sum of squares of deviations from the mean. We have been asked to calculate the coefficient of variation. But first we need to calculate the standard deviation.
Now, the provided mean is 18.4, so we have \[\bar{x}=18.4\] where \[\bar{x}\] denotes the mean. Also we have the sum of squares of the deviations from the mean equal to 1444, so mathematically we have,
\[\Rightarrow \sum\limits_{i=1}^{100}{{{\left( {{x}_{i}}-\bar{x} \right)}^{2}}}=1444\] …… (1)
Now, the standard deviation of n observations is given as \[\sigma =\sqrt{\dfrac{1}{n}\times \sum\limits_{i=1}^{n}{{{\left( {{x}_{i}}-\bar{x} \right)}^{2}}}}\], where \[\sigma \] denotes the standard deviation, so substituting the value obtained in equation (i) for n = 100 we get,
\[\begin{align}
  & \Rightarrow \sigma =\sqrt{\dfrac{1}{100}\times 1444} \\
 & \Rightarrow \sigma =\dfrac{38}{10} \\
 & \Rightarrow \sigma =3.8 \\
\end{align}\]
The coefficient of variation is given by the relation $\left( \dfrac{\sigma }{\overline{x}}\times 100 \right)$, so substituting the obtained value of the standard deviation and the given value of mean we get,
$\Rightarrow $ Coefficient of variation = $\left( \dfrac{3.8}{18.4}\times 100 \right)$
$\therefore $ Coefficient of variation = 20.65
Hence, the coefficient of variation of the provided 100 observations is 20.65.

Note: The coefficient of variation is a statistical measure of relative dispersion of data points in a data series around the mean. If you will square both the sides of the relation of the standard deviation then you will get the variance of the given observations. Variance is given as \[{{\sigma }^{2}}=\dfrac{1}{n}\times \sum\limits_{i=1}^{n}{{{\left( {{x}_{i}}-\bar{x} \right)}^{2}}}\]. Remember the formulas of mean, standard deviation, variance, etc. which are used frequently in statistics.