Construct box-and-whisker plots

Constructing a box-and-whisker plot represents an ordered data set with its five-number summary: minimum, lower quartile, median, upper quartile, and maximum, using the median of the lower and upper halves to determine quartiles. The plot shows how data are distributed, including the middle 50% through the interquartile range, and supports comparisons of center and spread; the box edges are quartiles, not averages, and advanced variants, formal inference, and alternative quartile conventions are not included.

Detailed Explanation: Construct box-and-whisker plots

A box-and-whisker plot shows the five-number summary of a data set:

minimum
lower quartile, $Q_1$
median
upper quartile, $Q_3$
maximum

The box represents the middle $50\%$ of the data. The whiskers extend to the minimum and maximum.

Example

Construct a box-and-whisker plot for this data set:

3,\ 5,\ 6,\ 7,\ 8,\ 10,\ 12,\ 13,\ 15,\ 16,\ 20

Step 1: Make sure the data are in order

The data are already arranged from least to greatest.

Step 2: Find the median

There are $11$ values, so the median is the middle value:

3,\ 5,\ 6,\ 7,\ 8,\ \boxed{10},\ 12,\ 13,\ 15,\ 16,\ 20

Thus, the median is $10$ .

Step 3: Find the lower quartile

Use the values below the median:

3,\ 5,\ 6,\ 7,\ 8

The middle value is $6$ , so:

Q_1=6

Step 4: Find the upper quartile

Use the values above the median:

12,\ 13,\ 15,\ 16,\ 20

The middle value is $15$ , so:

Q_3=15

Do not include the median in either half.

Step 5: Identify the five-number summary

\begin{aligned} \text{Minimum} &=3\\ Q_1&=6\\ \text{Median}&=10\\ Q_3&=15\\ \text{Maximum}&=20 \end{aligned}

Step 6: Draw the plot

Mark the five values on a number line. Draw:

a box from $Q_1=6$ to $Q_3=15$ ,
a line inside the box at the median, $10$ ,
whiskers from the box to the minimum, $3$ , and maximum, $20$ .

3          6          10          15          20
|----------[-----------|-----------]-----------|
minimum   Q₁        median        Q₃        maximum

The box, from $6$ to $15$ , shows the middle half of the data. The box edges are quartiles, not averages.

Learn by doing: Construct box-and-whisker plots

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Statistics - Quartiles - Data Set (No Outliers) to Box Plot

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