# Summation notation

Summation notation is used to denote the sum of values. Suppose you have a sample consisting of the ages of 5 students in a middle school. Suppose the ages of these five students are 12, 14, 15, 10, and 9

The variable *age of a student* can be denoted by x

You could write the ages of the five students as follows.

Age of first student = x_{1} = 12

Age of second student = x_{2} = 14

Age of third student = x_{3} = 15

Age of fourth student = x_{4} = 10

Age of fifth student = x_{5} = 9

In the notation above, x represents the price and the subscript denotes a particular student.

Suppose you want to add the ages of all five students.

x_{1} + x_{2} + x_{3} + x_{4} + x_{5} = 12 + 14 + 15 + 10 + 9 = 60

The upper case Greek letter Σ is used to denote the sum of all values (2 values, 5, values, 200 values, or more). Σ is pronounced sigma.

Using Σ notation, we can instead write Σx = x_{1} + x_{2} + x_{3} + x_{4} + x_{5 } = 60

In the example above, you only had 5 terms to add. Imagine you had 100. You could just use Σx to represent the addition of 100 terms. Therefore, Σx is a handy shortcut because it makes sense to write Σx as opposed to writing the following 100 terms down.

x_{1} + x_{2} + x_{3} + x_{4} + … + x_{50} + … + x_{100}

**Meaning of Σx ^{2}**

Square each value of x and then take the sum.

Σx^{2} = x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2} + x_{5}^{2} + … + x_{n}^{2}

**Meaning of (Σx) ^{2}**

Add the values of x and then square the result.

(Σx)^{2} = (x_{1} + x_{2} + x_{3} + x_{4} + x_{5} + … + x_{n} ) ^{2}

**Meaning of Σxy**

Multiply each value of x by each value of y and then take the sum.

Σxy = x_{1} × y_{1} + x_{2} × y_{2} + x_{3} × y_{3} + … + x_{n} × y_{n}

**Meaning of Σxy ^{2}**

Multiply each value of x by the square of each value of y and then take the sum.

Σxy^{2} = Σxy = x_{1} × y_{1}^{2} + x_{2} × y_{2}^{2} + x_{3} × y_{3}^{2} + … + x_{n} × y_{n}^{2}

At this point, most likely you understood how to work with summation notation.