# 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 = x1 = 12

Age of second student = x2 = 14

Age of third student = x3 = 15

Age of fourth student = x4 = 10

Age of fifth student = x5 = 9

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

Meaning of Σx

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

x1 + x2 + x3 + x4 + x5 = 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 = x1 + x2 + x3 + x4 + x5  = 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.

x1 + x2 + x3 + x4 + ... + x50 + ... + x100

Meaning of Σx2

Square each value of x and then take the sum.

Σx2 = x12 + x22 + x32 + x42 + x52 + ... + xn2

Meaning of (Σx)2

Add the values of x and then square the result.

(Σx)2 = (x1 + x2 + x3 + x4 + x5 + ... + xn )2

Meaning of Σxy

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

Σxy = x1 × y1 + x2 × y2 + x3 × y3 + ... + xn × yn

Meaning of Σxy2

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

Σxy2 = x1 × y12 + x2 × y22 + x3 × y32 + ... + xn × yn2

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

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