## 1.1 Mean

The first measure we will discuss is very likely one you have heard of before: the mean, also known 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 (\$)
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?

8712.8