Question

If the sum of the deviations of 50 observations from 30 is 50, then the mean of these observations is
(a) 31
(b) 50
(c) 51
(d) 30

Answer 1

Hint: To solve this question, we will assume the values of the elements in the data set and write the relation for the sum of the deviation and equate it to the given sum. Then, we will try to find the sum of the values of the elements and divide it to 50 to get the mean of the data set.

Complete step-by-step answer:
Now, let ${{x}_{1}},{{x}_{2}},{{x}_{3}},....{{x}_{50}}$ be the observations of a data set with 50 observations.
Thus, ${{x}_{1}}-30,{{x}_{2}}-30,{{x}_{3}}-30,....{{x}_{50}}-30$ will be the deviations of the 50 observations from 30.
Therefore, sum of the deviations will be given as follows:
$\Rightarrow {{x}_{1}}-30+{{x}_{2}}-30+{{x}_{3}}-30+....{{x}_{50}}-30$
It is given that this sum of deviations is equal to 50.
$\Rightarrow {{x}_{1}}-30+{{x}_{2}}-30+{{x}_{3}}-30+....{{x}_{50}}-30=50$
Each of the 50 terms in this summation has a ─30.
$\Rightarrow {{x}_{1}}+{{x}_{2}}+{{x}_{3}}+....{{x}_{50}}-50\times 30=50$
We will take this 50(30) to the other side of equal to sign.
$\begin{align}
  & \Rightarrow {{x}_{1}}+{{x}_{2}}+{{x}_{3}}+....{{x}_{50}}=50+50\left( 30 \right) \\
 & \Rightarrow {{x}_{1}}+{{x}_{2}}+{{x}_{3}}+....{{x}_{50}}=50\left( 31 \right) \\
\end{align}$
Therefore, the sum of the values of all the observations of the data set is equal to 50(31). We will divide both sides by 50.
$\Rightarrow \dfrac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}+....{{x}_{50}}}{50}=31$
But we know that the mean of the data set is $\dfrac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}+....{{x}_{50}}}{50}$.
 $\Rightarrow \text{Mean}=31$

So, the correct answer is “Option (a)”.

Note: The deviation of an element or observation of a data set from a given number is the difference in the value of the observation and the number. So, the sum of the deviations will be equal to the sum of deviation of each individual observation in the data set. The mean of a data set is defined as the quotient of the sum of value of all the elements or observations in the data set and the number of elements or observations in a data set.