Question

Mean deviation of 6, 8, 12, 15, 10, 9 through mean is:
(a) 10
(b) 2.33
(c) 2
(d) None of these.

Answer 1

Hint: We solve this problem by finding the mean first and then we find mean deviation. The formula of mean and mean deviation through mean are given as
\[\bar{x}=\dfrac{\text{sum of observations}}{\text{number of observations}}\]
\[M.D=\dfrac{\sum{\left| {{x}_{i}}-\bar{x} \right|}}{\text{number of observations}}\]
Where,\[{{x}_{i}}\] is \[{{i}^{th}}\] observation. By using these formulas we find the mean deviation of given observations.

Complete step-by-step solution:
We are given with some observations as
6, 8, 12, 15, 10, 9
Here, we can see that there are 6 observations.
Let us assume that the mean of given observations is \[\bar{x}\].
We know that the mean of observations is given as
\[\bar{x}=\dfrac{\text{sum of observations}}{\text{number of observations}}\]
By using the above formula to given observations we get
\[\begin{align}
  & \Rightarrow \bar{x}=\dfrac{6+8+12+15+10+9}{6} \\
 & \Rightarrow \bar{x}=\dfrac{60}{6}=10 \\
\end{align}\]
Now let us find the mean deviation of given observations.
We know that the mean deviation formula is given as
\[M.D=\dfrac{\sum{\left| {{x}_{i}}-\bar{x} \right|}}{\text{number of observations}}\]
By using the above formula we get
\[\Rightarrow M.D=\dfrac{\left| 6-10 \right|+\left| 8-10 \right|+\left| 12-10 \right|+\left| 15-10 \right|+\left| 10-10 \right|+\left| 9-10 \right|}{6}\]
Now by applying the modulus to each term mentioned above equation we get
\[\begin{align}
  & \Rightarrow M.D=\dfrac{4+2+2+5+0+1}{6} \\
 & \Rightarrow M.D=\dfrac{14}{6} \\
 & \Rightarrow M.D=2.33 \\
\end{align}\]
Therefore, the mean deviation of given observations is 2.33.
So, option (b) is the correct answer.


Note: Students may make mistakes in taking the formula of mean deviation as wrong. The formula of mean deviation is given as
\[M.D=\dfrac{\sum{\left| {{x}_{i}}-\bar{x} \right|}}{\text{number of observations}}\]
But students may do mistake in taking the formula without modulus that is
\[M.D=\dfrac{\sum{\left( {{x}_{i}}-\bar{x} \right)}}{\text{number of observations}}\]
This gives the wrong answer. So, the application of the formula is important. If the question is asked to find the sum of deviations then there is no need to take the modulus, but if we are asked to find the mean deviation then we have to take the modulus for each deviation. So, this part has to be taken care of that we are finding either sum of deviations or mean deviation.