Question

Find the mean deviation about the median for the following data.
\[18\], \[23\],\[9\], \[11\],\[26\], \[4\], \[14\], \[21\].

Answer 1

Hint: We will be using the formula for calculating the median and mean deviation about median formulas. The Median of any series is the middle term of the ordered data. The formula of the median is as given below:
\[{\text{Median}}\left( {\text{M}} \right) = {\left( {\dfrac{{n + 1}}{2}} \right)^{th}}{\text{term}}\], where \[n\] is the number of terms in the series.
The formula for calculating the mean deviation about the median is as below:
\[{\text{Mean deviation about median}}\left( {\text{M}} \right) = \dfrac{{\sum\nolimits_{i = 1}^n {\left| {{x_i} - {\text{M}}} \right|} }}{{{\text{number of terms}}}}\] , where
\[{x_i}\] are the terms of the series and \[{\text{M}}\] is the median.

Complete step-by-step solution:
For calculating the median of the given series, first of all, we will be arranging the series into order as shown below:
\[4\], \[9\], \[11\], \[14\], \[18\], \[21\], \[23\], \[26\].
Now by using the formula of median we get:
\[ \Rightarrow {\text{Median}}\left( {\text{M}} \right) = {\left( {\dfrac{{8 + 1}}{2}} \right)^{th}}{\text{term}}\]
By solving the above expression, we get:
\[ \Rightarrow {\text{Median}}\left( {\text{M}} \right) = {4.5^{th}}{\text{term}}\]
Step 2: Now as we can see that the series is having an even number of terms so we will be using below the average formula for calculating the exact middle value of the series:
\[ \Rightarrow {\text{Average}} = \dfrac{{\left( {{\text{value below median}} + {\text{value above median}}} \right)}}{2}\]
Now by substituting the values of the term below-median which is
\[14\] and the term which is above the median is \[18\] in the above expression:
\[ \Rightarrow {\text{Average}} = \dfrac{{\left( {14 + 18} \right)}}{2}\]
By simplifying the above expression, we get:
\[ \Rightarrow {\text{Average}} = \dfrac{{32}}{2}\]
By doing the final division in the above expression we get:
\[ \Rightarrow {\text{Average}} = 16\]
So, the median will be equal \[16\].
Step 3: Now, by using the formula of mean deviation we get:
\[ \Rightarrow {\text{Mean deviation }}\left( {\text{M}} \right) = \dfrac{{\left| {4 - 16} \right| + \left| {9 - 16} \right| + \left| {11 - 16} \right| + \left| {14 - 16} \right| + \left| {18 - 16} \right| + \left| {21 - 16} \right| + \left| {23 - 16} \right| + \left| {26 - 16} \right|}}{8}\]
By doing the simple subtraction in the above expression we get:
\[ \Rightarrow {\text{Mean deviation }}\left( {\text{M}} \right) = \dfrac{{\left| { - 12} \right| + \left| { - 7} \right| + \left| { - 5} \right| + \left| { - 2} \right| + \left| 2 \right| + \left| 5 \right| + \left| 7 \right| + \left| {10} \right|}}{8}\]
Now, by opening the mode of the terms we get:
\[ \Rightarrow {\text{Mean deviation }}\left( {\text{M}} \right) = \dfrac{{12 + 7 + 5 + 2 + 2 + 5 + 7 + 10}}{8}\] , because
\[\left| { - 1} \right| = 1\].
Now by doing the addition of the terms in the above expression we get:
\[ \Rightarrow {\text{Mean deviation }}\left( {\text{M}} \right) = \dfrac{{50}}{8}\]
By solving the above expression, we get:
\[ \Rightarrow {\text{Mean deviation }}\left( {\text{M}} \right) = 6.25\]

Mean deviation about the median is \[6.25\].

Note: Students need to remember the formulas for calculating the median. If the series is odd then the formula will be as below:
\[{\text{Median}}\left( {\text{M}} \right) = {\left( {\dfrac{{n + 1}}{2}} \right)^{th}}{\text{term}}\], but if the series is even then we need to calculate the final median by using the average formula as below:
\[ \Rightarrow {\text{Average}} = \dfrac{{\left( {{\text{value below median}} + {\text{value above median}}} \right)}}{2}\]