Chapter 12 The Summation Operator
The summation operator, \(\sigma\), is an indispensable element of statistics because it is used in almost all formulas. Most students will encounter it first when they see the formula for the arithmetic mean. In this chapter I outline the basic rules we use to work with the summation operator. Make sure you work through them to understand when and how a rule might ease computation.
12.1 The Single Summation Operator
\[\sum^{n}_{i=1}x_i\] is how most single summation operators will be shown in a textbook. This is basically telling you to each value \(i\) of the variable \(x\). For example, if \(x = 1, 2, 3, 5, 6\), then \[\sum^{n}_{i=1}x_i = 1 + 2 + 3 + 5 + 6 = 17\]
For the same values of \(x\), if the expression is \[\sum^{n}_{i=1}x_i^{2}\]
we calculate \[\sum^{n}_{i=1}x_i^{2} = (1)^2 + (2)^2 + (3)^2 + (5)^2 + (6)^2 = 1 + 4 + 9 + 25 + 36 = 75\]
On the other hand, if the expression were \[\left(\sum^{n}_{i=1}x_i\right)^2\] then the answer would be \((75)^2 = 289\). Obviously then, \[\sum^{n}_{i=1}x_i^{2} \neq \left(\sum^{n}_{i=1} x_i\right)^2\]
12.2 The Double Subscript
In some of the chapters you will encounter the double subscript \(x_{ij}\). For example, when we build contingency tables with one categorical variable as the rows and the other categorical variable as the columns, the table might look as follows:
This might be represented as follows:
The important thing to note is that one subscript \((i)\) indexes the row being referenced in the calculation while the other \((j)\) indexes the column being referenced in the calculation.
12.3 The Constant Rule
This rule says that \[\sum^{n}_{i=1}ax_i = a\sum^{n}_{i=1}x_i\]
For example, with \(x = 1, 2, 3, 5, 6\), \[\sum^{n}_{i=1}3x_i\] would be \[\sum^{n}_{i=1}3x_i = 3 \sum^{n}_{i=1}x_i = 3 (1 + 2 + 3 + 5 + 6) = 3(17) = 51\]
Note that this is the same thing as doing \[3(1) + 3(2) + 3(3) + 3(5) + 3(6) = 3 + 6 + 9 + 15 + 18 = 51\] but this would be the more laborious way of calculating the answer.
Similarly, \[\sum^{n}_{i=1}\dfrac{x_i}{2} = \dfrac{1}{2}\sum^{n}_{i=1}x_i\]
Say for the same values of \(x\), \[\sum^{n}_{i=1}\dfrac{x_i}{2}\] would be \[\dfrac{1}{2}\sum^{n}_{i=1}x_i = \dfrac{1}{2}17 = 8.5\]
The same answer would result if you calculated \[\sum^{n}_{i=1}\dfrac{x_i}{2} = \dfrac{1}{2} + \dfrac{2}{2} + \dfrac{3}{2} + \dfrac{5}{2} + \dfrac{6}{2} = 0.5 + 1 + 1.5 + 2.5 + 3 = 8.5\]
12.4 The Distributive Rule
\[\sum^{n}_{i=1}(x_i + y_i) = \sum^{n}_{i=1}x_i + \sum^{n}_{i=1}y_i\]. Say \(x = 1, 2, 3, 5, 6\) and \(y = 4, 8, 7, 9, 11\), \[\sum^{n}_{i=1}(x_i + y_i) = \sum^{n}_{i=1}x_i + \sum^{n}_{i=1} y_i = (1 + 2 + 3 + 5 + 6) + (4 + 8 + 7 + 9 + 11) = (17) + (39) = 56\]
This is the same as \[\sum^{n}_{i=1}x_i + \sum^{n}_{i=1}y_i = 17 + 39 = 56\]
Of course, \[\sum^{n}_{i=1}(x_i - y_i) = \sum^{n}_{i=1}x_i - \sum^{n}_{i=1}y_i\]
Try this for yourself with the given values of \(x\) and \(y\).
12.5 The Dot Notation
Given the \(x_{ij}\) double subscripts, note that some times you may see the dot notation in action. For example, \[x_{\bullet j} = \sum^{R}_{i=1}x_{ij}\] and \[x_{i \bullet} = \sum^{C}_{j=1}x_{ij}\]
What does this mean? Assume the data are as follows:
What if I want the total of Column 1? I could represent this as \[x_{\bullet i} = \sum^{n_j}_{i=1}x_{ij}\] where \(n_j\) represents the total number of values we see in row \(i\). So for Column 1, this would be \[x_{\bullet 1} = \sum^{n_1}_{i=1}x_{ij} = 1 + 4 + 6 = 11\]
What about Column 2? Here we have one blank cell but still, \[x_{\bullet 2} = \sum^{n_2}_{i=1}x_{ij} = 2 + 7 = 9\]
Similarly \[x_{\bullet 3} = \sum^{n_3}_{i=1}x_{ij} = 3 + 5 + 8 = 16\]
What about the row sums? \[x_{i \bullet} = \sum^{C}_{j=1}x_{ij}\]
So if I want the sum of the third row I would have \[x_{i \bullet} = \sum^{n_3}_{j=1}x_{ij} = 6 + 7 + 8 = 21\]
And so on for the sum of the second row and the sum of the first row, respectively.
12.6 The Basic Rules of Summation
- \(\sum x \cdots\) sum all the values of \(x\)
- \(\sum x^2 \cdots\) square each \(x\) and then sum the values of \(x^2\)
- \(\sum xy \cdots\) multiple each pair of \(x\) and \(y\) to obtain \(xy\) and then sum the values of \(xy\)
- \(\left(\sum x\right)^2 \cdots\) sum all the values of \(x\) and then square \(\sum x\)
- \(\sum x \sum y \cdots\) sum all the values of \(x\), sum all the values of \(y\), and then multiple the two sums \(\sum x\) and \(\sum y\). Note that \(\sum xy \neq \sum x \sum y\)
- \(\sum(x + 2) \cdots\) add 2 to each \(x\) and then sum
- \(\sum(x - y) \cdots\) subtract \(y\) from each \(x\) and then sum. This is the same as doing \(\sum x - \sum y\)
- \(\sum(x + y) \cdots\) add \(y\) to each \(x\) and then sum. This is the same as doing \(\sum x + \sum y\)
- \(\sum(2x) \cdots\) add each \(x\) and then multiply the sum by 2. Note that \(\sum(2x) = 2\sum x\)