Learning Objectives
- Use summation notation to express the sum of all numbers
- Use summation notation to express the sum of a subset of numbers
- Use summation notation to express the sum of squares
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 Table \(\PageIndex{1}\).
Table \(\PageIndex{1}\): Weights of \(4\) grapes.
| Grape |
X |
| 1 |
4.6 |
| 2 |
5.1 |
| 3 |
4.9 |
| 4 |
4.4 |
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 capital sigma (\(\sum\)) 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
\[ \sum X^2 = 4.62 + 5.12 + 4.92 + 4.42 = 21.16 + 26.01 + 24.01 + 19.36 = 90.54 \]
Notice that:
\[ \left(\sum X \right)^2 \neq \sum X^2 \]
because the expression on the left means to sum up all the values of \(X\) and then square the sum (\(19^2 = 361\)), 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. Table \(\PageIndex{2}\) shows the data for variables \(X\) and \(Y\). The cross products (\(XY\)) are shown in the third column. The sum of the cross products is \(3+4+21 = 28\).
Table \(\PageIndex{2}\): Cross Products.
| X |
Y |
XY |
| 1 |
3 |
3 |
| 2 |
2 |
4 |
| 3 |
7 |
21 |
In summation notation, this is written as:
\[\sum XY = 28.\]
