Measures of Central Location

The first measure of central location is the usual "average" that is familiar to everyone. In the formula in the following definition we introduce the standard summation notation $\Sigma$, where $\Sigma$ is the capital Greek letter sigma. In general, the notation $\Sigma$ followed by a second mathematical symbol means to add up all the values that the second symbol can take in the context of the problem. Here is an example to illustrate this.

EXAMPLE 1

Find $\Sigma x, \Sigma x^{2}$, and $\Sigma(x-1)^{2}$ for the data set

$1 \quad 3 \quad 4$

Solution:

$\begin{aligned} \Sigma x &=1+3+4=8 \\ \Sigma x^{2} &=1^{2}+3^{2}+4^{2}=1+9+16=26 \\ \Sigma(x-1)^{2} &=(1-1)^{2}+(3-1)^{2}+(4-1)^{2}=0^{2}+2^{2}+3^{2}=13 \end{aligned}$

In the definition we follow the convention of using lowercase $\mathrm{n}$ to denote the number of measurements in a sample, which is called the sample size.

Definition

The sample mean of a set of $n$ sample data is the number $\bar{x}$ defined by the formula

$\bar{x}=\frac{\Sigma x}{n}$

EXAMPLE 2

$2 \quad −1 \quad 0 \quad 2$

Solution:

$\bar{x}=\frac{\Sigma x}{n}=\frac{2+(-1)+0+2}{4}=\frac{3}{4}=0.75$

EXAMPLE 3

Solution:

$\begin{aligned} \bar{x} &=\frac{\Sigma x}{n}=\frac{1.90+3.00+2.53+3.71+2.12+1.76+2.71+1.39+4.00+3.33}{10} \\ &=\frac{26.45}{10}=2.645 \end{aligned}$

EXAMPLE 4

$\begin{array}{l|lllllll} x & 0 & 1 & 2 & 3 & 4 \\ \hline f & 3 & 6 & 6 & 3 & 1 \end{array}$

Find the sample mean.

Solution:

$\begin{array}{lllllllllllllllllll}0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 2 & 2 & 2 & 2 & 2 & 2 & 3 & 3 & 3 & 4\end{array}$

The sample size can be read directly from the table, without first listing the entire data set, as the sum of the frequencies: $n=3+6+6+3+1=19$. The sample mean can be computed directly from the table as well:

$\bar{x}=\frac{\Sigma x}{n}=\frac{0 \times 3+1 \times 6+2 \times 6+3 \times 3+4 \times 1}{19}=\frac{31}{19}=1.6316$

In the examples above the data sets were described as samples. Therefore the means were sample means, denoted by $\bar{x}$. If the data come from a census, so that there is a measurement for every element of the population, then the mean is calculated by exactly the same process of summing all the measurements and dividing by how many of them there are, but it is now the population mean and is denoted by $\mu$, the lower case Greek letter mu.

Definition