Question

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

Short answer

$\sum _{}{x}_i{y}_i\ne \left(\sum _{}{x}_i\right)\left(\sum _{y_i}\right)$

Step-by-step solution

 Step 1. Given information.  

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

 Step 2. The difference between ∑xiyi  and ∑xi∑yi also provide an example to show that those two quantities are unequal

$\sum _{}{x}_i{y}_i$is the sum of products, where the individual corresponding values of two data sets are multiplied first, followed by the total sum.

$\left(\sum _{}{x}_i\right)\left(\sum _{}{y}_i\right)$is a sum product, which means that the individual data sets are added separately and then the final sum of the two data sets is multiplied.

Consider the following set:

$\left(x_i\right)=1,2,3,4$and $\left(y_i\right)=4,3,2,1$

role="math" localid="1652992532305" $$\begin{gathered} \sum _{}{x}_i{y}_i=\left(1\times 4\right)+\left(2\times 3\right)+\left(3\times 2\right)+\left(1\times 4\right) \\ \sum _{}{x}_i{y}_i=4+6+6+4 \\ \sum _{}{x}_i{y}_i=20 \end{gathered}$$

And

role="math" localid="1652992556394" $$\begin{gathered} \left(\sum _{}{x}_i\right)\left(\sum _{}{y}_i\right)=\left(1+2+3+4+\right)\left(4+3+2+1\right) \\ \left(\sum _{}{x}_i\right)\left(\sum _{}{y}_i\right)=\left(10\right)\left(10\right) \\ \left(\sum _{}{x}_i\right)\left(\sum _{}{y}_i\right)=100 \end{gathered}$$

role="math" localid="1652992583289" $\sum _{}{x}_i{y}_i\ne \left(\sum _{}{x}_i\right)\left(\sum _{y_i}\right)$

As a result, $\sum _{}{x}_i{y}_i$ is the product of sums and $\left(\sum _{}{x}_i\right)\left(\sum _{}{y}_i\right)$ is the sum of sums.