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

Table 1.1: Income per person ($) in different countries.
Country Income per person ($)
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 population mean is usually denoted \(\mu\). Usually, we do not know what the true value of \(\mu\) is, but we can use the sample mean, \(\overline{x}\), to try and estimate it.

Your turn: Consider the following five values:

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

What is the mean of these values?