Mean, median, mode, and range

Sometimes you’re given a set of numbers and asked to describe it using mean, median, mode, and range. Each one tells you something different about the data:

• Mean — The average
• Median — The middle value
• Mode —The most common value
• Range — How spread out the values are

For the practice problems, we’ll use this data set:

$$-2,3,3,4,13,21,26$$

We will use this set to find the mean, then the median, then the mode, and finally the range.

Each step shows a different way to describe the same group of numbers. :::

Mean

Definitions

Mean

The average: add all values and divide by the number of values

Data set:

$$-2,3,3,4,13,21,26$$

The mean is the same as the average. To find it:

• Add all the values in the data set
• Divide the total by the number of values
  This data set has $7$ numbers, so divide the total by $7$.

$$\begin{array}{rl}\frac{(-2+3+3+4+13+21+26)}{7}& =\frac{63}{7}\\ & =9\end{array}$$

The mean, or average, of this data set is $9$.

Median

Definitions

Median

The middle value after the numbers are in order.

• If there’s an odd number of values, there’s one middle number.
• If there’s an even number of values, there are two middle numbers, so you average them.

The median is the middle value in a set of numbers once they are arranged in order. It helps show the central position of the data, especially when values are spread out.

Let’s find the median of our data set. First, list the numbers from least to greatest:

$$-2,3,3,4,13,21,26$$

Next, cross off values from the left and right until only the middle number remains:

$$\cancel{-2},3,3,4,13,21,\cancel{26}$$

$$\cancel{-2},\cancel{3},3,4,13,\cancel{21},\cancel{26}$$

$$\cancel{-2},\cancel{3},\cancel{3},4,\cancel{13},\cancel{21},\cancel{26}$$

The value left in the middle is $4$, so the median is $4$!

Sidenote

Median for an even data set

Let’s say our data set was $(-2,3,4,13,21,26)$ instead.

We would start the same way by listing the numbers in order:

$$-2,3,4,13,21,26$$

Now cross off from both ends:

$$\cancel{-2},3,4,13,21,\cancel{26}$$

$$\cancel{-2},\cancel{3},4,13,\cancel{21},\cancel{26}$$

Now $4$ and $13$ are in the middle, so we average them:

$$\frac{4+13}{2}=\frac{17}{2}=8.5$$

So the median is $8.5$.

Mode

Definitions

Mode

The value (or values) that occur most often.

The mode is the value that appears most frequently in a data set.

In the data set:

$$-2,3,3,4,13,21,26$$

The number $3$ appears twice, while all other numbers appear only once.

So the mode is $3$.

Sidenote

Multiple modes

If multiple numbers repeat the same number of times, there can be more than one mode.

For example:

$$4,4,4,5,5,8,8,8$$

Both $4$ and $8$ appear three times, so the modes are $4$ and $8$.

Range

Definitions

Range

The largest value minus the smallest value.

The range shows how spread out the data is by finding the difference between the largest and smallest values.

In our data set:

$$-2,3,3,4,13,21,26$$

The largest value is $26$ and the smallest value is $-2$.

$$26--2$$

$$26+2$$

$$28$$

So the range is $28$.

Knowledge check: Given the list $3,3,4,6,8,12,$ what is the mean, the median, the mode, and the range?

(spoiler)

Mean: $6$
Median: $5$
Mode: $3$
Range: $9$

Factorials

Definitions

Factorial

A compact way to write the product of an integer and all positive integers less than it, down to $1$.

Factorials grow very quickly, so it’s important to simplify them efficiently. A factorial is a way to write repeated multiplication of a number and all positive integers below it down to $1$.

• Start with the given number and multiply by each whole number below it
• Continue multiplying until you reach $1$

Let’s walk through these examples step-by-step.

Example 1:

$$5!=5\ast 4\ast 3\ast 2\ast 1=120$$

$$8!=8\ast 7\ast 6\ast 5\ast 4\ast 3\ast 2\ast 1=40320$$

You can use this idea to simplify factorial fractions by canceling common factors. Notice that both the numerator and denominator contain $5\ast 4\ast 3\ast 2\ast 1$, so those factors cancel (since $\frac{5}{5}=1$, $\frac{4}{4}=1$, etc.).

Example 2:

$$\begin{array}{rl}\frac{8!}{5!}& =\frac{8\ast 7\ast 6\ast 5\ast 4\ast 3\ast 2\ast 1}{5\ast 4\ast 3\ast 2\ast 1}\\ & =\frac{8\ast 7\ast 6\ast \cancel{5\ast 4\ast 3\ast 2\ast 1}}{\cancel{5\ast 4\ast 3\ast 2\ast 1}}\\ & =336\end{array}$$

Steps taken:

• Expand factorials
• Cancel common factors
• Multiply remaining values

Knowledge check:

Simplify:

$$\frac{7!}{4!}$$

(spoiler)

$$210$$