1.11: Summation Notation

Skills to Develop

• 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:

$\sum XY = 28.$

Contributors

• Online Statistics Education: A Multimedia Course of Study (http://onlinestatbook.com/). Project Leader: David M. Lane, Rice University.

• David M. Lane