Summation Notation

David M. Lane

Prerequisites

None

Learning Objectives
1. Use summation notation to express the sum of all numbers
2. Use summation notation to express the sum of a subset of numbers
3. 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 1.

Table 1. Weights of 4 grapes.
Grape X
1
2
3
4
4.6
5.1
4.9
4.4

We label Grape 1's weight X1, Grape 2's weight X2, etc. The following formula means to sum up the weights of the four grapes: The Greek letter capital sigma (Σ) indicates summation. The "i = 1" at the bottom indicates that the summation is to start with X1 and the 4 at the top indicates that the summation will end with X4. The "Xi" indicates that X is the variable to be summed as i goes from 1 to 4. Therefore, = X1 + X2 + X3 + X4 = 4.6 + 5.1 + 4.9 + 4.4 = 19.0.

The symbol 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: 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

ΣX² = 4.62 + 5.12 + 4.92 + 4.42 = 21.16 + 26.01 + 24.01 + 19.36 = 90.54.

Notice that: because the expression on the left means to sum up all the values of X and then square the sum (19² = 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 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 2. Cross Products.
X Y XY
1
2
3
3
2
7
3
4
21

In summation notation, this is written as: ΣXY = 28.

Please answer the questions: