Summation Notation

Many statistical formulas involve summing numbers. Fortunately there is a convenient notation for expressing summation. This section covers the basics of this summation notation.

Let's say we have a variable $X$ that represents the weights (in grams) of $4$ grapes. The data are shown in [link].

We label Grape $1$'s weight $X_1$, Grape $2$'s weight $X_2$, etc. The following formula means to sum up the weights of the four grapes: $$\sum_{i=1}^4 X_i$$ The Greek letter $\Sigma$ indicates summation. The "$i=1$" at the bottom indicates that the summation is to start with $X_1$ and the $4$ at the top indicates that the summation will end with $X_4$. The $X_i$ indicates that $X$ is the variable to be summed as $i$ goes from $1$ to $4$. Therefore, $$\sum_{i=1}^4 X_i=X_1+X_2+X_3+X_4=4.6+5.1+4.9+4.4=19.0$$ The symbol $$\sum_{i=1}^3 X_i$$ indicates that only the first $3$ scores are to be summed. The index variable $i$ goes from $1$ to $3$. When all the scores of a variable (such as $X$) are to be summed, it is often convenient to use the following abbreviated notation: $$\sum X$$ Thus when no values of $i$ are shown, it means to sum all the values of $X$.

Many formulas involve squaring numbers before they are summed. This is indicated as $$\sum X^2=4.6^2+5.1^2+4.9^2+4.4^2=21.16+26.01+24.01+19.36=90.54$$ Notice that: $$\sum X^2\neq \sum X^2$$ because the expression on the left means to sum up all the values of $X$ and then square the sum ($19^2=381$) whereas the expression on the right means to square the numbers and then sum the squares ( $90.54$, as shown).

Some formulas involve the sum of cross products. [link] shows the data for variables $X$ and $Y$. The totals are shown in the bottom row and the cross products are shown in the third column. The sum of the cross products is $28$.

In summation notation, this is written as: $$\sum XY=28$$

Self Test Questions

For the following data