Question

The mean deviation from the median is _________ that is measured from any other value.
(a) equal to
(b) less than
(c) greater than
(d) none of these

Answer 1

Hint: The mean deviation or mean absolute deviation is defined as a statistical measure which is used to calculate the average deviation from the mean value of the given data set. We have to assume a dataset and find the mean deviation about mean and that about median using the formulas $M.A.D\left( \overline{x} \right)=\dfrac{\sum\limits_{i=1}^{n}{{{f}_{i}}\left| {{x}_{i}}-\overline{x} \right|}}{N}\text{ }$ and $M.A.D\left( M \right)=\dfrac{\sum\limits_{i=1}^{n}{{{f}_{i}}\left| {{x}_{i}}-M \right|}}{N}$ respectively. Then, we have to compare the values obtained.

Complete step-by-step solution:
Let us recollect what mean deviation is. The mean deviation or mean absolute deviation is defined as a statistical measure which is used to calculate the average deviation from the mean value of the given data set. Mean absolute deviation is given by the formulas
$\begin{align}
  & M.A.D\left( \overline{x} \right)=\dfrac{\sum\limits_{i=1}^{n}{{{f}_{i}}\left| {{x}_{i}}-\overline{x} \right|}}{N}...\left( i \right) \\
 & M.A.D\left( M \right)=\dfrac{\sum\limits_{i=1}^{n}{{{f}_{i}}\left| {{x}_{i}}-M \right|}}{N}...\left( ii \right) \\
\end{align}$
We will use formula (i) when the central tendency is mean and formula (ii) when the central tendency is median.
Let us consider a dataset 3, 4, 5, 10, 8. We have to find the mean deviation and mean deviation from the median.
We know that mean deviation is given by
$M.A.D\left( \overline{x} \right)=\dfrac{\sum\limits_{i=1}^{n}{{{f}_{i}}\left| {{x}_{i}}-\overline{x} \right|}}{N}$
Let us find the mean of the dataset.
$\begin{align}
  & \Rightarrow \overline{x}=\dfrac{3+4+5+10+8}{5} \\
 & \Rightarrow \overline{x}=\dfrac{30}{5} \\
 & \Rightarrow \overline{x}=6 \\
\end{align}$
Now, we have to find M.A.D.
\[\begin{align}
  & \Rightarrow M.A.D\left( \overline{x} \right)=\dfrac{\left| 3-6 \right|+\left| 4-6 \right|+\left| 5-6 \right|+\left| 10-6 \right|+\left| 8-6 \right|}{5} \\
 & \Rightarrow M.A.D\left( \overline{x} \right)=\dfrac{3+2+1+4+2}{5} \\
 & \Rightarrow M.A.D\left( \overline{x} \right)=\dfrac{12}{5} \\
 & \Rightarrow M.A.D\left( \overline{x} \right)=2.4 \\
\end{align}\]
Now, let us find mean deviation from the median. We have to arrange the dataset in the ascending order. The data can be written as 3, 4, 5, 8, 10.
Now, we have to find the median. We know that the number of data in the dataset is 5, which is an odd number. Therefore, the median will be the central value. Here, the median is $M=5$ .
Let us find the mean deviation from the median. We know that mean deviation from the median is given by
$M.A.D\left( M \right)=\dfrac{\sum\limits_{i=1}^{n}{{{f}_{i}}\left| {{x}_{i}}-M \right|}}{N}$
Let us substitute the values.
\[\begin{align}
  & \Rightarrow M.A.D\left( M \right)=\dfrac{\left| 3-5 \right|+\left| 4-5 \right|+\left| 5-5 \right|+\left| 8-5 \right|+\left| 10-5 \right|}{5} \\
 & \Rightarrow M.A.D\left( M \right)=\dfrac{2+1+0+3+5}{5} \\
 & \Rightarrow M.A.D\left( M \right)=\dfrac{11}{5} \\
 & \Rightarrow M.A.D\left( M \right)=2.2 \\
\end{align}\]
We can see that the value of mean deviation from the median is less than that from the mean. Therefore mean deviation from the median is less than that measured from any other value.
Hence, the correct option is b.

Note: Students must be thorough with the formulas of mean, median, mode and mean absolute deviation. We usually call mean deviation as mean absolute deviation because we are taking the absolute value of the difference of the value and mean(or median).