Mean, Median, Mode, and Range

Idea · Beginner

Problem Solving

The mean is the average of the values.

\text{Mean}=\frac{\text{Sum of Values}}{\text{Number of Values}}

In many contest problems, it is more useful to rewrite the mean formula as

\boxed{\text{Sum}=\text{Mean}\times\text{Number of Values}}

Example. The mean of 8 numbers is 15. Find their sum.

Solution.

\text{Sum}=15\cdot8=120

The median is the middle value after the values are arranged in increasing or decreasing order.

If there are two middle values, the median is their mean.

Example.

2,\ 5,\ 7,\ 9,\ 12

The median is

\boxed{7}

For

2,\ 5,\ 7,\ 9

the median is

\frac{5+7}{2}=\boxed{6}

The mode is the value that occurs most frequently.

Example 1: One Mode

2,\ 3,\ 3,\ 5,\ 7

The value 3 occurs more often than any other value, so

\text{Mode}=\boxed{3}

Example 2: More Than One Mode

2,\ 2,\ 4,\ 5,\ 5,\ 7

Both 2 and 5 occur more often than any other value, so

\text{Modes}=\boxed{2\text{ and }5}

Example 3: No Mode

2,\ 2,\ 4,\ 4,\ 6,\ 6

Every value occurs the same number of times, so there is no mode.

The range measures the difference between the largest and smallest values.

\boxed{\text{Range}=\text{Maximum}-\text{Minimum}}

Example.

4,\ 7,\ 9,\ 13

Then

\text{Range}=13-4=\boxed{9}

Operation	Mean	Median	Mode	Range
$+k$	$+k$	$+k$	$+k$	Same
$-k$	$-k$	$-k$	$-k$	Same
$\times k$	$\times k$	$\times k$	$\times k$	$\times \|k\|$
$\div k,\ k\ne0$	$\div k$	$\div k$	$\div k$	$\div \|k\|$