A.2 Summations

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This is the upper-case Greek letter sigma. A sigma tells us that we need to sum (i.e., add) a series of numbers.

\[\sum\]

For example, four children are comparing how many pieces of candy they have:

ID Child  Pieces of Candy
1 Marty 9
2 Harold 8
3 Eugenia 10
4 Kevi 8

We could say that: \(x_{1}=9\), \(x_{2}=8\), \(x_{3}=10\), and \(x_{4}=8\).

If we wanted to know how many total pieces of candy the group of children had, we could add the four numbers. The notation for this is:

\[\sum x_{i}\]

So, for this example, \(\sum x_{i}=9+8+10+8=35\)

To conclude, combined, the four children have 35 pieces of candy.

In statistics, some equations include the sum of all of the squared values (i.e., square each item, then add). The notation is:

\[\sum x_{i}^{2}\]

or

\[\sum (x_{i}^{2})\]

Here, \(\sum x_{i}^{2}=9^{2}+8^{2}+10^{2}+8^{2}=81+64+100+64=309\).

Sometimes we want to square a series of numbers that have already been added. The notation for this is:

\[(\sum x_{i})^{2}\]

Here,\( (\sum x_{i})^{2}=(9+8+10+8)^{2}=35^{2}=1225\)

Note that \(\sum x_{i}^{2}\) and \((\sum x_{i})^{2}\) are different.

Summations 

Here is a brief review of summations as they will be applied in STAT 200: