1.11 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 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:

\sum_{i = 1}^{4} X_{i}

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 "X_i" indicates that X is the variable to be summed as i goes from 1 to 4. Therefore,

$\sum_{i = 1}^{4} X_{i}$ = X₁ + X₂ + X₃ + X₄ = 4.6 + 5.1 + 4.9 + 4.4 = 19.0.

The symbol

∑ sub i equals 1 cubed X sub 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:

$∑ X$

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

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

Notice that:

$open parenthesis ∑ X close parenthesis squared not equal to ∑ X squared$

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.

Adapted from Online Statistics Education: A Multimedia Course of Study (onlinestatbook.com), Project Leader: David M. Lane, Rice University. Developed with NSF support. The original work is in the public domain; it is cited here at the authors' request. Changes were made: reformatted as an accessible XYZ web edition with native MathML. License: Public-Domain.