## 1.1 Mean

The first measure we will discuss is very likely one you have heard of before: the * mean*, often referred to as the

*. Consider the below sample of \(n=5\) countries and their associated income per person (GDP per capita, PPP$ inflation-adjusted).*

**average**Country | Income per person ($) |
---|---|

Chad | 1740 |

Lao | 6940 |

Turkey | 25000 |

Mozambique | 1170 |

Norway | 66300 |

We can calculate the mean of these values simply by adding them up, and then dividing by the number of values, \(n = 5\): \[\begin{align} \text{mean income per person} &= (1740 + 6940 + 25000 + 1170 + 66300) \div 5 \\ &= 101150 \div 5 \\ &= 20230 \end{align}\]

Let's now introduce a little bit more notation. If we denote any one of these income values to be \(x_i\), where \(i\) can take any value from \(1\) to \(n = 5\), that means we can denote each value as follows:

- \(x_1 = 1740\)
- \(x_2 = 6940\)
- \(x_3 = 25000\)
- \(x_4 = 1170\)
- \(x_5 = 66300\)

The mean of a sample such as this, the * sample mean*, is usually denoted \(\overline{x}\), pronounced "x bar".
Using this notation, we can succinctly define the sample mean to be:

\[\begin{align} \overline{x} = \dfrac{1}{n} \sum_{i=1}^{n} x_i. \end{align}\]

Note that \(\sum\) is a summation sign, so that if we read out \(\displaystyle \sum_{i=1}^{n}\) in words, we would say, "the sum from \(i=1\) to \(n\)". In other words, this formula is telling us to add up the values \(x_1\) up to \(x_n\), and then divide that sum by \(n\). Remembering that we had \(n=5\) in our example, hopefully you can see that is exactly what we have done when calculating the sample mean of \(20230\) above.

As mentioned, the * sample mean* is usually denoted \(\overline{x}\). The

*is usually denoted \(\mu\). Usually, we do not know what the true value of \(\mu\) is, but we can use the*

**population mean***, \(\overline{x}\), to try and estimate it.*

**sample mean****Your turn:** Consider the following five values:

\[7770, 10200, 954, 1640, 23000. \]

What is the mean of these values?

8712.8