Skill: Construct and interpret box plots

To construct a box plot, first find the five-number summary:

1. Minimum
2. Lower quartile, $Q_1$
3. Median
4. Upper quartile, $Q_3$
5. Maximum

The box represents the middle $50\%$ of the data, from $Q_1$ to $Q_3$. The line inside the box is the median, and the whiskers extend to the minimum and maximum.

Example

Construct and interpret a box plot for this data set:

$$4,\ 5,\ 7,\ 8,\ 9,\ 10,\ 12,\ 13,\ 15,\ 16,\ 18,\ 22$$

The data are already in order.

Step 1: Find the median

There are $12$ values, so the median is the average of the 6th and 7th values:

$$\text{Median}=\frac{10+12}{2}=11$$

Step 2: Find $Q_1$

The lower half is:

$$4,\ 5,\ 7,\ 8,\ 9,\ 10$$

Its median is the average of the 3rd and 4th values:

$$Q_1=\frac{7+8}{2}=7.5$$

Step 3: Find $Q_3$

The upper half is:

$$12,\ 13,\ 15,\ 16,\ 18,\ 22$$

Its median is the average of the 3rd and 4th values:

$$Q_3=\frac{15+16}{2}=15.5$$

Step 4: Record the five-number summary

Step 5: Calculate the IQR

The interquartile range measures the spread of the middle half:

$$\text{IQR}=Q_3-Q_1=15.5-7.5=8$$

Step 6: Draw the box plot

• Draw a number line that includes $4$ through $22$.
• Place the left edge of the box at $7.5$.
• Place the right edge of the box at $15.5$.
• Draw a line inside the box at the median, $11$.
• Draw whiskers from the box to the minimum, $4$, and maximum, $22$.

A box plot for this data would have:

$$4 \longdash [\,7.5 \mid 11 \mid 15.5\,] \longdash 22$$

The actual graph should be drawn to scale.

Interpretation

• The typical value is represented by the median, which is $11$.
• The middle $50\%$ of the data lies between $7.5$ and $15.5$.
• The IQR is $8$, so the middle half of the data has a spread of $8$ units.
• The upper whisker is longer than the lower whisker, suggesting that the data may be slightly right-skewed.
• The box plot does not show every individual data value, and it cannot determine the mean.