Mean, median, mode, range, and factorials

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:

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

Mean

Definitions

Mean

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

To find the mean:

• 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 $7$.

$$\begin{array}{rl}\frac{(-2+3+3+4+13+21+26)}{7}& =\frac{68}{7}\\ & =9\frac{5}{7}\end{array}$$

The mean of this data set is $\frac{68}{7}$, or $9\frac{5}{7}$. Since no calculator is allowed on the exam, it’s often easiest to leave the mean as a fraction or mixed number rather than converting it to a decimal, or to estimate it — here, $68÷7$ is a little under $70÷7=10$ — to match the closest answer choice.

Example: Finding a missing value from the mean

The mean of five numbers is $8$. Four of the numbers are $5,7,9,$ and $10$. What is the fifth number?

Since mean $\times$ count = total, the five numbers add up to $8\times 5=40$. Subtract the four known numbers: $40-5-7-9-10=9$.

Answer: $9$

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.

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.

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 a positive integer and all positive integers less than it, down to $1$. By definition, $0!=1$.

• 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:

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

Example 2:

Notice that both the numerator and denominator of $\frac{8!}{5!}$ contain $5\times 4\times 3\times 2\times 1$, so those factors cancel (since $\frac{5}{5}=1$, $\frac{4}{4}=1$, and so on).

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

Steps taken:

• Expand factorials
• Cancel common factors
• Multiply remaining values

Shortcut: For $\frac{n!}{k!}$ where $n>k$, you don’t need to expand both factorials fully — just write the factors from $n$ down to $k+1$. For example, $\frac{8!}{5!}=8\times 7\times 6$. This keeps the arithmetic small, which matters since no calculator is allowed.

Knowledge check:

Simplify:

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

(spoiler)

$$210$$