Question

Consider the following statements:
1. The sum of the deviations from mean is always zero.
2. The sum of absolute deviations is minimum when taken around the median.
Which of the above statements is/are correct?
A. 1 only
B. 2 only
C. Both 1 and 2
D.Neither 1 nor 2

Answer 1

Hint: To find out which of the statements is correct or incorrect, we have to know what mean and median are. Here we also have to know about deviations and absolute deviations. Mean and median are statistical values, mean is just the average of the given set of observations, which is the ratio of sum of all the observations to total no. of observations, whereas the median is the middle value of the observations when all the observations are arranged in the increasing order.

Complete step by step solution:
Let there be a set of observations: ${x_1},{x_2},{x_3},......{x_n}$, and let their mean be $\overline x $.
Now deviation from mean is the difference of the mean of the data from an observation is called the deviation from mean. As given below:
\[ \Rightarrow ({x_i} - \overline x )\]
The absolute deviation is the modulus of the difference of the mean of the data from an observation, as the difference of mean from an observation can be positive or negative, hence applying the modulus makes this deviation always positive which is called the absolute deviation. As given below:
\[ \Rightarrow |{x_i} - \overline x |\]
Now the sum of the deviations from mean is given below:
$ \Rightarrow \sum {({x_i} - \overline x )} $
Let us take an example of observations which are $2,3,6,9,10,11,15$
The total no. of observations is 7 and the mean is given by:
$ \Rightarrow \overline x = \dfrac{{2 + 3 + 6 + 9 + 10 + 11 + 15}}{7}$
$\therefore \overline x = 8$
Now finding the deviations from mean is as given below:
$ \Rightarrow \sum\limits_{i = 1}^7 {({x_i} - \overline x )} = (2 - 8) + (3 - 8) + (6 - 8) + (9 - 8) + (10 - 8) + (11 - 8) + (15 - 8)$
$ \Rightarrow \sum\limits_{i = 1}^7 {({x_i} - \overline x )} = - 6 - 5 - 2 + 1 + 2 + 3 + 7$
$ \Rightarrow \sum\limits_{i = 1}^7 {({x_i} - \overline x )} = - 13 + 13$
$\therefore \sum\limits_{i = 1}^7 {({x_i} - \overline x )} = 0$
Hence the sum of the deviations from mean is always zero.
$\therefore $The first statement is true.
Here the median of the observations is 9.
Now the sum of absolute deviations from median is given by:
$ \Rightarrow \sum\limits_{i = 1}^7 {|{x_i} - median|} = \sum\limits_{i = 1}^7 {|{x_i} - 9|} $
Here the median is greater than the mean of the observations.
Median = 9, Mean =8
$\because $Median > mean
\[ \Rightarrow \sum\limits_i {({x_i} - 8)} > \sum\limits_i {({x_i} - 9)} \]
\[\therefore \sum\limits_i {({x_i} - 9)} \] is the least or minimum.
Hence the sum of absolute deviations is minimum when taken around the median.
$\therefore $ The second statement also holds true.

So, the correct answer is “Option C”.

Note: Note that the definition of the mean is the value with minimum deviation about a set of observations whereas the median is the central value in the data set, and it is true that the absolute deviation is minimum.