# Mean, Median, and Mode

## The Mean of a Data Set

The mean of a set of numbers, sometimes simply called
**
the average
**
, is the sum of the data divided by the total number of data.

**
Example 1
**
:

Find the mean of the set $\left\{2,5,5,6,8,8,9,11\right\}$ .

There are $8$ numbers in the set. Add them all, and then divide by $8$ .

$\begin{array}{l}\frac{2\text{\hspace{0.17em}}+\text{\hspace{0.17em}}5\text{\hspace{0.17em}}+\text{\hspace{0.17em}}5\text{\hspace{0.17em}}+\text{\hspace{0.17em}}6\text{\hspace{0.17em}}+\text{\hspace{0.17em}}8\text{\hspace{0.17em}}+\text{\hspace{0.17em}}8\text{\hspace{0.17em}}+\text{\hspace{0.17em}}9\text{\hspace{0.17em}}+\text{\hspace{0.17em}}11}{8}=\frac{54}{8}\\ =6.75\end{array}$

So, the mean is $6.75$ .

## The Median of a Data Set

The median of a set of numbers is the middle number in the set (after the numbers have been arranged from least to greatest) -- or, if there are an even number of data, the median is the average of the middle two numbers.

**
Example 1
**
:

Find the median of the set $\left\{2,5,8,11,16,21,30\right\}$ .

There are $7$ numbers in the set, and they are arranged in ascending order. The middle number (the $4$ th one in the list) is $11$ . So, the median is $11$ .

**
Example 2
**
:

Find the median of the set $\left\{3,10,36,255,79,24,5,8\right\}$ .

First, arrange the numbers in ascending order.

$\left\{3,5,8,10,24,36,79,255\right\}$

There are $8$ numbers in the set -- an even number. So, find the average of the middle two numbers, $10$ and $24$ .

$\frac{10\text{\hspace{0.17em}}+\text{\hspace{0.17em}}24}{2}=\frac{34}{2}=17$

So, the median is $17$ .

## The Mode of a Data Set

The mode of a set of numbers is the number which occurs most often.

**
Example 1
**
:

Find the mode of the set $\left\{2,3,5,5,7,9,9,9,10,12\right\}$ .

$2$ , $3$ , $7$ , $10$ and $12$ each occur once.

$5$ occurs twice and $9$ occurs three times.

So, $9$ is the mode.

**
Example 2
**
:

Find the mode of the set $\left\{2,5,5,6,8,8,9,11\right\}$ .

In this case, there are two modes -- $5$ and $8$ both occur twice, whereas the other numbers only occur once.