Introduction to Statistics (Online Edition)/Introduction/Summation Notation

by David M. Lane, David Scott, Mikki Hebl, Rudy Guerra, Dan Osherson, Heidi Zimmer

Summation Notation
by 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.

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 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,

$\sum _{i=1}^{4}X_{i}=X_{1}+X_{2}+X_{3}+X_{4}=4.6+5.1+4.9+4.4=19$

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, 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:

$\left(\sum {X}\right)^{2}\neq \sum {}X^{2}$

because the expression on the left means to sum up all the values of X and then square the sum (19² = 361), whereas the expression on the right means to square 28 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.

In summation notation, this is written as:

$\sum {}XY=28$