2.5: Sigma Notation and Calculating the Arithmetic Mean
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This unit is here to remind you of material that you once studied and said at the time “I am sure that I will never need this!”
Here are the formulas for a population mean and the sample mean.
Formula for Population Mean
\[\boldsymbol{\mu}=\frac{1}{N} \sum_{i=1}^{N} x_{i}\nonumber\]
Formula for Sample Mean
\[\overline{x}=\frac{1}{n} \sum_{i=1}^{n} x_{i}\nonumber\]
The Greek letter \(\mu\) is the symbol for the population mean and \(\overline{x}\) is the symbol for the sample mean. Both formulas have a mathematical symbol that tells us how to make the calculations. It is called Sigma notation because the symbol is the Greek capital letter sigma: \(\Sigma\). Like all mathematical symbols it tells us what to do: just as the plus sign tells us to add and the \(\times\) sign tells us to multiply. These are called mathematical operators. The \(\Sigma\) symbol tells us to add a specific list of numbers.
Let’s say we have a sample of animals from the local animal shelter and we are interested in their average age. If we list each value, or observation, in a column, you can give each one an index number. The first number will be number 1 and the second number 2 and so on.
| Animal | Age |
| 1 | 9 |
| 2 | 1 |
| 3 | 8.5 |
| 4 | 10.5 |
| 5 | 10 |
| 6 | 8.5 |
| 7 | 12 |
| 8 | 8 |
| 9 | 1 |
| 10 | 9.5 |
Each observation represents a particular animal in the sample. Purr is animal number one and is a 9 year old cat, Toto is animal number 2 and is a 1 year old puppy and so on.
To calculate the mean we are told by the formula to add up all these numbers, ages in this case, and then divide the sum by 10, the total number of animals in the sample.
Animal number one, the cat Purr, is designated as \(x_1\), animal number 2, Toto, is designated as \(x_2\) and so on through Dundee who is animal number 10 and is designated as \(x_{10}\).
The \(i\) in the formula tells us which of the observations to add together. In this case it is \(x_1\) through \(x_{10}\) which is all of them. We know which ones to add by the indexing notation, the \(i = 1\) and the \(n\) or capital \(N\) for the population. For this example the indexing notation would be \(i = 1\) and because it is a sample we use a small \(n\) on the top of the \(\Sigma\) which would be 10.
\[\overline{x} = \frac{1}{10}\sum^{10}_{i=1}x_i = \frac{1}{10}(9 + 1 + 8.5 + 10.5 + 10 + 8.5 + 12 + 8 + 1 + 9.5)\nonumber\]
The sum of the ages is found to be 78 and dividing by 10 gives us the sample mean age as 7.8 years.