Question

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

Answer 1

Hint: First we will use formula to calculate mean by adding up all the numbers and then divide by how many numbers there are. Then we will compute the algebraic sum of deviations from the above mean and simplify the equation to find the required value.

Complete step-by-step answer:
Let us assume that the values are \[{x_1}\], \[{x_2}\], \[{x_3}\],…, \[{x_{50}}\].
We are given that the sum of deviations of 50 observations from 30 is 50.
We know that the formula to calculate the mean is by adding up all the numbers and then dividing them by total numbers.
First, we will add the marks obtained by students, we get
\[
   \Rightarrow {x_1} + {x_2} + {x_3} + ... + {x_{50}} \\
   \Rightarrow \sum\limits_{i = 0}^{50} {{x_i}} \\
 \]
Dividing the above value by 50 to find the mean, we get
\[ \Rightarrow {\text{Mean}} = \dfrac{{\sum\limits_{i = 0}^{50} {{x_i}} }}{{50}}{\text{ ......eq(1)}}\]
Using the given conditions, we will be computing the value of algebraic sum of deviations from the above mean, we get
\[ \Rightarrow \sum\limits_{i = 1}^{50} {\left( {{x_i} - 30} \right) = 50} \]
Taking 30 common from the sum of left hand side of the above equation, we get
\[
   \Rightarrow \sum\limits_{i = 1}^{50} {{x_i} - 50\left( {30} \right) = 50} \\
   \Rightarrow \sum\limits_{i = 1}^{50} {{x_i} - 1500 = 50} \\
 \]
Rearranging the terms of the above equation, we get
\[
   \Rightarrow \sum\limits_{i = 1}^{50} {{x_i}} - 1500 + 1500 = 50 + 1500 \\
   \Rightarrow \sum\limits_{i = 1}^{50} {{x_i}} = 1550 \\
 \]
Dividing the above equation by 50 on both sides, we get
\[
   \Rightarrow \dfrac{{\sum\limits_{i = 1}^{50} {{x_i}} }}{{50}} = \dfrac{{1550}}{{50}} \\
   \Rightarrow \dfrac{{\sum\limits_{i = 1}^{50} {{x_i}} }}{{50}} = 31 \\
 \]
Using equation (1), we have that the value of mean is 31.
Therefore, the required value is 31.
Hence, option C is correct.

Note: We need to know that the arithmetic mean is the average of the numbers, which is a calculated central value of a set of numbers. There are other types of mean such as geometric and harmonic mean. We will compare the values of our mean to find the required value. We know that a summation is just a representation of the sum of a number of terms, like how we wrote in equation (1).