Question

The sum of the deviations of 50 observations from 30 is 50, then the mean of these observations is:
A. 50
B. 30
C. 31
D. 51

Answer 1

Given that the sum of the deviations of 50 observations form 30 is equal to 50, which means that each value of the observation is subtracted from 30 and summated which gives the value of 50. Now we have to find the mean of the 50 observations which is given by the ratio of sum of the 50 observations to the total no. of observations.

Complete step-by-step solution:
Here the deviation is nothing but finding the difference between each value from 30 as given.
So summating all these deviations of 50 observations gives the value 50.
The mean of 50 observations is given by:
$\Rightarrow \dfrac{{\sum\limits_{i = 1}^{50} {{x_i}} }}{{50}}$
Now expressing it mathematically:
Deviation of an observation from 30 is given by:
$\Rightarrow ({x_i} - 30)$, where ${x_i}$ is one of the observations from 50 observations.
Sum of the deviations of 50 observations from 30 is given by:
$\Rightarrow \sum\limits_{i = 1}^{50} {({x_i} - 30)}$
Now given that the sum of the deviations of 50 observations from 30 is equal to 50:
$\Rightarrow \sum\limits_{i = 1}^{50} {({x_i} - 30)} = 50$
Now splitting the summation (L.H.S) of the above expression is given by:
$\Rightarrow \sum\limits_{i = 1}^{50} {({x_i} - 30)} = ({x_1} - 30) + ({x_2} - 30) + ({x_3} - 30) + \cdot \cdot \cdot \cdot + ({x_{50}} - 30)$
$\Rightarrow \sum\limits_{i = 1}^{50} {({x_i} - 30)} = ({x_1} + {x_2} + {x_3} + \cdot \cdot \cdot \cdot + {x_{50}}) - 50(30)$
The above expression can be re-written as :
$\Rightarrow \sum\limits_{i = 1}^{50} {({x_i} - 30)} = \sum\limits_{i = 1}^{50} {{x_i}} - 50(30)$
Substituting the above expression in the $\sum\limits_{i = 1}^{50} {({x_i} - 30)} = 50$, as given below:
$\Rightarrow \sum\limits_{i = 1}^{50} {{x_i}} - 50(30) = 50$
$\Rightarrow \sum\limits_{i = 1}^{50} {{x_i}} = 50 + 50(30)$
$\Rightarrow \sum\limits_{i = 1}^{50} {{x_i}} = 50(1 + 30)$
$\Rightarrow \sum\limits_{i = 1}^{50} {{x_i}} = 50(31)$
$\Rightarrow \dfrac{{\sum\limits_{i = 1}^{50} {{x_i}} }}{{50}} = 31$
Now the mean of the 50 observations is given by:
$\therefore \dfrac{{\sum\limits_{i = 1}^{50} {{x_i}} }}{{50}} = 31$
The mean of 50 observations is 31.

Option C is the correct answer.

Note: Please note that here we are asked to find the sum of deviations from a given number but not from the sum of deviations from the mean. We have to be careful while solving such kinds of problems.