Construct and interpret box plots

A box plot represents a numerical distribution with a scale and five-number summary: minimum, lower quartile, median, upper quartile, and maximum, with the interquartile range (IQR) showing the spread of the middle half of the data. Constructing one requires ordering data and locating quartiles consistently; interpreting one involves comparing center, variability, skewness, and possible outliers, while recognizing that a box plot does not display individual data values or determine the mean.

Detailed Explanation: Construct and interpret box plots

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

Minimum
Lower quartile, $Q_1$
Median
Upper quartile, $Q_3$
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.

Learn by doing: Construct and interpret box plots

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Statistics - Quartiles - Data Set (With Outliers) to Box Plot

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