Question

If different values of variables $x$ are $9.8, 5.4, 3.7, 1.7, 1.8, 2.6, 2.8, 8.6, 10.5$ and $11.1$. Find the value of $\sum {\left( {x - \bar x} \right)}$.
A) 0
B) 1
C) 2
D) 3

Answer 1

Here we will firstly find the mean of the given numbers by using the formula of the mean. Then we will put the value of the mean in the function $\sum {\left( {x - \bar x} \right)}$. Then by putting the value of mean in the equation and expanding the equation we will get the required value. Mean is equal to the ratio of sum of the total numbers and total count of the numbers.

Formula used:
We will use the formula Mean $=$ Sum of all numbers $\div$ Total numbers

Complete Step by Step Solution:
The given values of $x$ are $9.8,5.4,3.7,1.7,1.8,2.6,2.8,8.6,10.5$ and $11.1$
Firstly we will find the value of the mean of all the value by using the formula Mean $=$ Sum of all numbers $\div$ Total numbers. Therefore, we get
$\bar x = \dfrac{{9.8 + 5.4 + 3.7 + 1.7 + 1.8 + 2.6 + 2.8 + 8.6 + 10.5 + 11.1}}{{10}}$
Now by solving this we will get the value of mean of the given numbers.
Adding the terms in the numerator, we get
$\Rightarrow \bar x = \dfrac{{58}}{{10}} = 5.8$
Now we will simply put the value of the mean in the formula which we have to find and then expand the equation. Therefore, we get
$\begin{array}{l}\sum {\left( {x - \bar x} \right)} = \left( {9.8 - 5.8} \right) + \left( {5.4 - 5.8} \right) + \left( {3.7 - 5.8} \right) + \left( {1.7 - 5.8} \right) + \left( {1.8 - 5.8} \right) + \\\\\left( {2.6 - 5.8} \right) + \left( {2.8 - 5.8} \right) + \left( {8.6 - 5.8} \right) + \left( {10.5 - 5.8} \right) + \left( {11.1 - 5.8} \right)\end{array}$
By solving this equation we will get the value of the $\sum {\left( {x - \bar x} \right)}$. Therefore, we get
$\Rightarrow \sum {\left( {x - \bar x} \right)} = 4 - 0.4 - 2.1 - 4.1 - 4.0 - 3.2 - 3.0 + 2.8 + 4.7 + 5.3$
$\Rightarrow \sum {\left( {x - \bar x} \right)} = 0$
Hence the value of the function $\sum {\left( {x - \bar x} \right)}$ is equal to 0.

So, option A is the correct option.

Note:
Mean is also known as the average of the numbers. We should remember while expanding the equation put the sum sign in between the terms because this $\sum {}$ sign is the representation of the summation of the terms of that equation while expanding. While solving the equation firstly solve the equation inside the brackets and then perform the addition operation on the terms. We should not confuse the mean with the median. Median is the middle value of the given list of numbers or it is the value which is separating the data into two halves i.e. upper half and lower half.