## S1 Summation notation

Summation notation or sigma notation is a shorthand method of writing the sum or addition of a string of similar terms. This module explains the use of this notation.

### The Basic Idea

We use the Greek symbol sigma $$\Sigma$$ to denote summation. $$\Sigma$$ is called the summation sign.

A typical element of the sequence which is being summed appears to the right of the summation sign as shown in the figure below:

This is written as $${\textstyle \mathop{{\textstyle \sum_{i=1}^{5}}}}2i$$.

To expand and work out it’s value, we replace $$i$$ by its starting value (below the sigma symbol) and obtain each successive term by adding $$1$$ to the previous value until we reach the final value of $$i$$ (above the sigma symbol) and then we evaluate.

For the above sequence: \begin{align*} {\textstyle \mathop{{\textstyle \sum_{i=1}^{5}}}}2i & =2\times1+2\times2+2\times3+2\times4+2\times5=30\\ & =2+4+6+8+10\\ & =30. \end{align*}

#### Example

Expand and evaluate $${\textstyle \mathop{{\textstyle \sum_{i=0}^{3}}}}(i^{2}-3)$$

Note that only the $$i$$ value changes in each term.

\begin{align*} {\textstyle \mathop{{\textstyle \sum_{i=0}^{3}}}}(i^{2}-3) & =\underbrace{(0^{2}-3)}_{i=0}+\underbrace{(1^{2}-3)}_{i=1}+\underbrace{(2^{2}-3)}_{i=2}+\underbrace{(3^{2}-3)}_{i=3}\\ & =(-3)+(-2)+1+6\\ & =2. \end{align*}

### Data Sets

Subscripted variables are used to indicate values in a data set. $$x_{1},x_{2},x_{3}...$$ refers to first value, second value, third value and so on. Formulae used to calculate summary measures of a data set make use of summation notation.

#### Example

Given the set of data $$x_{1}=1,x_{2}=2,x_{3}=4,x_{4}=5$$ evaluate

1. $$\overline{x}=\frac{{\textstyle \mathop{{\textstyle \sum_{i=1}^{n}x_{i}}}}}{n}$$ where $$n$$ is the number of items in the data set

This formula calculates the mean or average of a set of data.

\begin{align*} \overline{x} & =\frac{{\textstyle \mathop{{\textstyle \sum_{i=1}^{4}x_{i}}}}}{4}\\ & =\frac{x_{1}+x_{2}+x_{3}+x_{4}}{4}\\ & =\frac{1+2+4+5}{4}\\ & =3 \end{align*}

1. $$s^{2}=$$ $$\frac{{\textstyle \mathop{{\textstyle \sum_{i=1}^{4}(x_{i}-}\overline{x})^{2}}}}{n-1}$$

This formula calculates the variance (a measure of spread around the mean) of a sample of data.

\begin{align*} s^{2} & =\frac{{\textstyle \mathop{{\textstyle \sum_{i=1}^{4}(x_{i}-}\overline{x})^{2}}}}{n-1}\\ & =\frac{{\textstyle \mathop{{\textstyle (x_{1}-}\overline{x})^{2}}}+{\textstyle \mathop{{\textstyle (x_{2}-}\overline{x})^{2}}}+{\textstyle \mathop{{\textstyle (x_{3}-}\overline{x})^{2}}}+{\textstyle \mathop{{\textstyle (x_{4}-}\overline{x})^{2}}}}{3}\\ & =\frac{{\textstyle \mathop{{\textstyle (1-}3)^{2}}}+{\textstyle \mathop{{\textstyle (2-}3)^{2}}}+{\textstyle \mathop{{\textstyle (4-}3)^{2}}}+{\textstyle \mathop{{\textstyle (5-}3)^{2}}}}{3}\\ & =\frac{4+1+1+4}{3}\\ & =\frac{10}{3}. \end{align*}

### Exercise 1

Find

1. $${\textstyle \mathop{{\textstyle \sum_{i=1}^{5}}}}3i$$

2. $${\textstyle \mathop{{\textstyle \sum_{i=1}^{3}}}}(5i-2)$$

3. $${\textstyle \mathop{{\textstyle \sum_{i=1}^{3}}}(}5i)-2$$

1. $$45\;$$ b. $$24\;$$ c. 28

### Exercise 2

Given $$x_{1}=-2,\,x_{2}=0,\,x_{3}=1,\,x_{4}=3,\,x_{5}=3$$ evaluate

1. $${\textstyle \mathop{{\textstyle \sum_{i=1}^{5}}}}10x_{i}$$

2. $${\textstyle \mathop{{\textstyle 10\sum_{i=1}^{5}}}}x_{i}$$

3. $${\textstyle \mathop{{\textstyle \sum_{i=1}^{5}}}}(x_{i})^{2}$$

4. $$\left({\textstyle \mathop{{\textstyle \sum_{i=1}^{5}}}}x_{i}\right)$$ $$^{2}$$

5. $${\textstyle \mathop{{\textstyle \sum_{i=1}^{5}}}}i\times(x_{i})$$

6. $$\overline{x}=\frac{{\textstyle \mathop{{\textstyle \sum_{i=1}^{n}x_{i}}}}}{n}$$

1. $$50\;$$ b. $$50\;$$ c. $$23$$
$$\qquad\quad$$d. $$25\;$$ e. $$28\;$$ f. $$1$$