Question

The sum of squares of deviations for $10$ observations taken from mean $50$ is $250$. Then coefficient of variation is
A) $10\% $
B) $40\% $
C) $50\% $
D) None

Answer 1

Hint: For solving this problem students need to find the standard deviation and coefficient of variation. Here it is given that the sum of squares of deviations for some observation taken from the mean value is given.

Formula used: Standard deviation $\left( \sigma \right)$ =$\sqrt {\dfrac{1}{n}\sum {{{\left| {{X_i} - \overline X } \right|}^2}} } $
Co- efficient of variation = $\dfrac{{{\text{standard deviation}}}}{{{\text{mean}}}}$×$100$
That is, Coefficient of variation = $\dfrac{\sigma }{{\overline X }} \times 100$

Complete step-by-step solution:
Given that the sum of deviation for $10$ observation taken from mean $50$ is $250$.
Here n=$10$, Mean $\left( {\overline X } \right)$= $50$
And ${\left| {{X_1} - \overline X } \right|^2}$+${\left| {{X_2} - \overline X } \right|^2}$+$...$+${\left| {{X_9} - \overline X } \right|^2}$+${\left| {{X_{10}} - \overline X } \right|^2}$=$250$, where${X_1}$$,$${X_2}$…${X_{10}}$be the $10$observations.
That is, $\sum\limits_{i = 1}^{10} {{{\left| {{X_i} - \overline X } \right|}^2}} $=$250$
We know that, σ=$\sqrt {\dfrac{1}{n}\sum {{{\left| {{X_i} - \overline X } \right|}^2}} } $
Substitute the given values, we get
$ \Rightarrow $σ=$\sqrt {\dfrac{1}{{10}} \times 250} $
$ \Rightarrow $σ=$\sqrt {25} $
Taking square root, we get
$ \Rightarrow $σ=$5$
Therefore standard deviation $\left( \sigma \right)$= $5$
Now students, our next aim is to find coefficient of variation
We know that, coefficient of variation = $\dfrac{\sigma }{{\overline X }} \times 100$
Now substitute σ = $5$ and mean $\left( {\overline X } \right)$=$50$ in the formula of coefficient of variation.
$\therefore $ Coefficient of variation = $\dfrac{5}{{50}} \times 100$
$ \Rightarrow \dfrac{1}{{10}} \times 100$
$ \Rightarrow 0.1 \times 100$
$ \Rightarrow 10\% $

Therefore, Coefficient of variation = $10\% $

Note: Standard deviation is the measure of dispersion of a set of data from its mean Standard Deviation is also known as root mean square deviation as it is the square root of means of the squared deviation from the arithmetic mean.
Students may get confused while working on Standard Deviation. So following steps will help students to find $\sigma $
Step $1$: Find the mean.
Step $2$: For each data point, find the square of its distance to the mean.
Step $3$: Sum the values from step $2$.
Step $4$: Divide by the number of observations.
Step $5$: Finally take square root.