Skill: Create box plots informally

A box plot shows an ordered data set using five numbers:

1. Minimum: the smallest value
2. Lower quartile ($Q_1$): the middle of the lower half
3. Median: the middle value of the whole data set
4. Upper quartile ($Q_3$): the middle of the upper half
5. Maximum: the largest value

Use this consistent method:

• If there is an odd number of data values, do not include the median in either half when finding the quartiles.
• If there is an even number of data values, split the data into two equal halves.

Example

Create a box plot for this data set:

$$4,\ 6,\ 7,\ 8,\ 9,\ 10,\ 12,\ 13,\ 15,\ 18$$

The data is already in order.

1. Find the minimum and maximum

• Minimum: $4$
• Maximum: $18$

2. Find the median

There are $10$ values, so the median is the average of the two middle values, $9$ and $10$:

$$\text{Median}=\frac{9+10}{2}=9.5$$

3. Find the lower quartile

The lower half is:

$$4,\ 6,\ 7,\ 8,\ 9$$

The middle value is $7$, so:

$$Q_1=7$$

4. Find the upper quartile

The upper half is:

$$10,\ 12,\ 13,\ 15,\ 18$$

The middle value is $13$, so:

$$Q_3=13$$

The five-number summary is:

$$\boxed{4,\ 7,\ 9.5,\ 13,\ 18}$$

These are:

• Minimum: $4$
• Lower quartile: $7$
• Median: $9.5$
• Upper quartile: $13$
• Maximum: $18$

5. Draw the box plot

Place all five values on one number line. Draw:

• A box from $7$ to $13$
• A line inside the box at the median, $9.5$
• A whisker from $4$ to $7$
• A whisker from $13$ to $18$

A simple sketch is:

4 -------- [ 7 -------- | 9.5 -------- 13 ] -------- 18
            whisker          box          whisker

The box represents the middle half of the data, from $Q_1$ to $Q_3$. The whiskers show the lower and upper portions of the data.