## Numerical Measures of Central Tendency and Variability

Read these sections and complete the questions at the end of each section. First, we will define central tendency and introduce mean, median, and mode. We will then elaborate on median and mean and discusses their strengths and weaknesses in measuring central tendency. Finally, we'll address variability, range, interquartile range, variance, and the standard deviation.

### Measures of Central Tendency

#### Arithmetic Mean

The arithmetic mean is the most common measure of central tendency. It is simply the sum of the numbers divided by the number of numbers. The symbol "$\mu$" is used for the mean of a population. The symbol "$\mathrm{M}$" is used for the mean of a sample. The formula for $\mu$ is shown below:

$\mu=\Sigma \mathrm{X} / \mathrm{N}$

where $\Sigma \mathrm{X}$ is the sum of all the numbers in the population and $N$ is the number of numbers in the population.

The formula for $M$ is essentially identical:

$\mathrm{M}=\Sigma X / N$

where $\Sigma \mathrm{X}$ is the sum of all the numbers in the sample and $\mathrm{N}$ is the number of numbers in the sample.

As an example, the mean of the numbers $1,2,3,6,8$ is $20 / 5=4$ regardless of whether the numbers constitute the entire population or just a sample from the population.

Table 1 shows the number of touchdown (TD) passes thrown by each of the 31 teams in the National Football League in the 2000 season. The mean number of touchdown passes thrown is 20.4516 as shown below.

\begin{aligned} \mu &=\Sigma \mathrm{X} / \mathrm{N} \\ &=634 / 31 \\ &=20.4516 \end{aligned}

Table 1. Number of touchdown passes.

 37 33 33 32 29 28 28 23 22 22 22 21 21 21 20 20 19 19 18 18 18 18 16 15 14 14 14 12 12 9 6

Although the arithmetic mean is not the only "mean" (there is also a geometric mean), it is by far the most commonly used. Therefore, if the term "mean" is used without specifying whether it is the arithmetic mean, the geometric mean, or some other mean, it is assumed to refer to the arithmetic mean.