Question

Find the coefficient of mean deviation from the median of the observation $40,62,54,90,68$ and $76$.

Answer 1

Hint: irst arrange the given term in ascending order and then find the median of the observations. Use it to find the mean deviation of the data and then use the formula given below to find the coefficient of mean deviation:
Coefficient of mean deviation$ = \dfrac{{{\text{Mean deviation}}}}{{{\text{Median}}}}$

Complete step-by-step answer:
The given observations are $40,62,54,90,68$ and $76$. We have to find the coefficient of mean deviation from the median of the given observations.
First we arrange the given observations in ascending orders,
$40,54,62,68,76,98$
We can observe that there are 6 observations, so the number of terms is taken as 6, which is an even number.
Number of terms$\left( n \right) = 6\left( {{\text{Even number}}} \right)$
We know that the median of the even number is given using the formula:
Median$\left( M \right) = \dfrac{{{{\left( {\dfrac{n}{2}} \right)}^{th}}{\text{term}} + {{\left( {\dfrac{n}{2} + 1} \right)}^{th}}{\text{term}}}}{2}$
Substitute $n = 6$into the equation:
Median$\left( M \right) = \dfrac{{{{\left( {\dfrac{6}{2}} \right)}^{th}}{\text{term}} + {{\left( {\dfrac{6}{2} + 1} \right)}^{th}}{\text{term}}}}{2}$
Median$\left( M \right) = \dfrac{{{3^{rd}}{\text{term}} + {4^{th}}{\text{term}}}}{2}$
We can observe in the observation that the ${3^{rd}}$ term of the sequence is $62$ and the ${4^{th}}$ term of the sequence is $68$. After the substitution of the values of the terms, then median is given as:
Median$\left( M \right) = \dfrac{{62 + 68}}{2}$
Median$\left( M \right) = \dfrac{{130}}{2} = 65$
So, the median of the given data is $65$.
Now, we create a table to find the mean deviation.

${x_i}$	$\left| {{x_1} - M} \right|$
40	25
54	11
62	3
68	3
76	11
90	25
	$\sum {\left| {{x_i} - M} \right|} = 78$

We know that the mean deviation is given using the formula:
Mean deviation from median$ = \dfrac{{\sum {\left| {{x_i} - M} \right|} }}{n}$
Substitute$\sum {\left| {{x_i} - M} \right|} = 78$ and $n = 6$ into the formula:
Mean deviation from median$ = \dfrac{{78}}{6} = 13$
If the deviation are taken from the median, then the coefficient of mean deviation is given as:
Coefficient of mean deviation$ = \dfrac{{{\text{Mean deviation}}}}{{{\text{Median}}}}$
Now, we substitute mean deviation as $13$ and median as $65$ into the above formula:
Coefficient of mean deviation$ = \dfrac{{13}}{{65}}$
Coefficient of mean deviation$ = 0.2$
So, the coefficient of mean deviation is $0.2$
Therefore, the option (D) is correct.

Note: As we can see in the given data that there are 6 observations, which is even in number but in case if the number of observations are odd, then we use another formula to find the median of the data.
Median$\left( M \right) = {\left( {\dfrac{n}{2}} \right)^{th}}{\text{term}}$