Question

Explain the difference between the quantities ${\left(\sum _{}{x}_i\right)}^2$and $\sum _{}{x^2}_i$Construct an example to show that in general those two quantities are unequal.

Short answer

${\left(\sum _{}{x}_i\right)}^2\ne \sum _{}{x_i}^2$

Step-by-step solution

 Step 1. Given information.  

${\left(\sum _{}{x}_i\right)}^2$and$\sum _{}{x^2}_i$

 Step 2. The difference between  ∑xi2 and ∑x2i also construct an example to show that those two quantities are unequal

${\left(\sum _{}{x}_i\right)}^2$is the square of sums, which means that all of the values in the data set are added first, and then the total sum is squared.

$\sum _{}{x^2}_i$is the sum of the squares, where each data set value is squared and then added to get the final sum.

Consider the set to be 1,2,3,4

$$\begin{gathered} {\left(\sum _{}{x}_i\right)}^2={\left(1+2+3+4\right)}^2 \\ {\left(\sum _{}{x}_i\right)}^2={\left(10\right)}^2 \\ {\left(\sum _{}{x}_i\right)}^2=100 \end{gathered}$$

and

$$\begin{gathered} \sum _{}{x^2}_i=1^2+2^2+3^3+4^2 \\ \sum _{}{x^2}_i=1+4+9+16 \\ \sum _{}{x^2}_i=30 \end{gathered}$$

As, resultrole="math" localid="1652990753066" ${\left(\sum _{}{x}_i\right)}^2\ne \sum _{}{x_i}^2$