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Five-Number Summary and Boxplot Construction

Practice finding the five-number summary for 15 values and using it to draw a clear, correctly scaled boxplot by hand.

Beginner 9 min read

What You'll Learn

  • Sort 15 observations and retain repeated values as separate observations.
  • Find the median from the middle position in an ordered list of 15 values.
  • Find the first and third quartiles by taking medians of the two halves.
  • Record the minimum, quartiles, median, and maximum as the five-number summary.
  • Draw and label a boxplot using the five-number summary and a consistent numerical scale.
  • Check that the box, median line, and whiskers match the five summary values.

From Ordered Data to a Boxplot

In Using Technology to Draw a Histogram, you used bins to display a quantitative distribution. A boxplot gives a different, compact view: it marks five important values along a number line. In this tutorial, you will calculate those values for a data set of 15 observations and use them to draw a boxplot by hand.

The calculation depends on the positions of values after they have been arranged from least to greatest. Keep every observation in the list, including repeated values. A repeated value occupies a position just like any other observation; removing a duplicate would change the data set and could change the summary.

Definition: The five-number summary consists of the minimum, first quartile (\(Q_1\)), median, third quartile (\(Q_3\)), and maximum, in that order. A boxplot represents these values on a numerical scale: the box extends from \(Q_1\) to \(Q_3\), a line inside the box marks the median, and whiskers extend to the minimum and maximum.

Find the Five Values for 15 Observations

First, sort all 15 observations from least to greatest. With 15 values, the median is the 8th value. To find the quartiles, divide the ordered data into a lower half and an upper half, leaving out the overall median. Each half then contains seven observations. The median of the lower half is \(Q_1\), and the median of the upper half is \(Q_3\).

$$ \begin{aligned} \text{Lower half: positions }&1\text{ through }7 &\quad Q_1&=\text{4th value}\\ \text{Overall median: }&\text{position }8 &\quad \text{Median}&=\text{8th value}\\ \text{Upper half: positions }&9\text{ through }15 &\quad Q_3&=\text{12th value} \end{aligned} $$

The quartiles are medians of their respective halves, not averages of the smallest and largest values. For 15 observations, each half has seven values, so its middle value is its fourth. Since the upper half begins at overall position 9, its fourth value is the 12th value in the complete ordered list.

Position guide for 15 observations: In the ordered list, take the 1st value as the minimum, the 4th as \(Q_1\), the 8th as the median, the 12th as \(Q_3\), and the 15th as the maximum.

This position guide is specific to a data set of 15 values. The key idea is to count positions in the complete ordered list carefully. A useful practice is to number the observations from 1 to 15 before selecting the five values, especially when values repeat.

Draw the Boxplot by Hand

Once you have the five-number summary, choose a horizontal number line that covers the minimum and maximum. Use evenly spaced tick marks and a scale that makes the five values easy to locate. Then draw the box from \(Q_1\) to \(Q_3\), mark the median inside the box, and draw whiskers from the box edges to the minimum and maximum.

1
Sort and identify positions.
Arrange all observations from least to greatest and locate positions 1, 4, 8, 12, and 15.
2
Write the five-number summary.
Record the minimum, \(Q_1\), median, \(Q_3\), and maximum in that order.
3
Set up the number line.
Choose a consistent scale that includes both endpoints and allows the five values to be located accurately.
4
Draw and label the boxplot.
Draw the box from \(Q_1\) to \(Q_3\), add a line at the median, and extend whiskers to the minimum and maximum. Label the scale and identify the variable and units.

The boxplot is a summary, not a picture of each individual observation. Its five marked values do not show every value in the data set. For instance, observations between \(Q_1\) and the median may be spread unevenly even though they lie within the same half of the box. For this basic boxplot, the whiskers reach the minimum and maximum.

Worked Example: Reading Minutes

Worked Example: Reading Minutes

A fictional set of 15 students reports the number of minutes they spent reading on one evening. Find the five-number summary and describe how to draw a boxplot.

Order the observations. The values, in minutes, are already listed from least to greatest:

\(12, 14, 15, 16, 18, 19, 21, 22, 23, 24, 26, 27, 29, 31, 34\)

Locate the five positions. The minimum is the 1st value, 12. The lower half contains positions 1 through 7, so \(Q_1\) is the 4th value, 16. The median is the 8th value, 22. The upper half contains positions 9 through 15, so \(Q_3\) is its 4th value, which is overall position 12, or 27. The maximum is the 15th value, 34.

$$ \text{Five-number summary: }(12,\ 16,\ 22,\ 27,\ 34)\text{ minutes} $$

Draw the boxplot. Draw a horizontal scale that includes 12 through 34 minutes; for example, label evenly spaced marks from 10 to 35. Place the left edge of the box at 16 and its right edge at 27. Draw a line inside the box at 22. Draw the left whisker from 16 back to 12 and the right whisker from 27 out to 34. Label the axis “Reading time (minutes).”

The boxplot therefore shows the middle half of the reported reading times between 16 and 27 minutes, with a median of 22 minutes. The whisker endpoints show that the smallest reported time was 12 minutes and the largest was 34 minutes.

Worked Example: Device Setup Times

Worked Example: Device Setup Times

A fictional class records how many minutes it takes to set up a tablet for a project. Find the five-number summary and give the drawing instructions for a boxplot.

Sort and number the observations. The ordered setup times are:

\(8, 9, 11, 12, 13, 15, 15, 16, 18, 20, 20, 23, 24, 26, 29\)

There are two observations equal to 15 and two equal to 20. Keep both copies of each value when counting positions. The minimum is 8. The 4th value is 12, so \(Q_1=12\). The 8th value is 16, so the median is 16. The 12th value is 23, so \(Q_3=23\). The maximum is 29.

$$ \text{Five-number summary: }(8,\ 12,\ 16,\ 23,\ 29)\text{ minutes} $$

Construct the display. Use a horizontal scale that covers 8 to 29 minutes, such as evenly spaced marks from 5 to 30. Draw the box from 12 to 23 and place the median line at 16. Extend one whisker from 12 to 8 and the other from 23 to 29. Label the number line “Tablet setup time (minutes).”

Notice that the median line does not have to divide the box into two equal lengths. Here, the median is 4 minutes to the right of \(Q_1\) and 7 minutes to the left of \(Q_3\). The drawing should place the line at its actual value on the scale, rather than automatically centering it in the box.

Worked Example: Daily Rainfall

Worked Example: Daily Rainfall

A fictional weather log gives the daily rainfall, in millimeters, for 15 days. Calculate the five-number summary and explain how to sketch the boxplot.

Use the ordered list. The rainfall amounts are:

\(3, 4, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 18\)

The minimum is 3 mm. The lower half is \(3,4,4,5,6,7,8\), whose middle value is 5 mm, so \(Q_1=5\) mm. The overall median is the 8th value, 9 mm. The upper half is \(10,11,12,13,14,16,18\), whose middle value is 13 mm, so \(Q_3=13\) mm. The maximum is 18 mm.

$$ \text{Five-number summary: }(3,\ 5,\ 9,\ 13,\ 18)\text{ millimeters} $$

Make the boxplot. Draw a horizontal scale from 0 to 20 millimeters with evenly spaced marks. Draw the box from 5 to 13, mark the median at 9, and extend the whiskers to 3 and 18. Label the axis “Daily rainfall (millimeters).” The box should have its median line halfway between its edges in this example because 9 is equally distant from 5 and 13.

Check Your Construction

A careful check catches many errors before they affect the drawing. Confirm that the ordered list still has 15 observations, then check the five selected positions: 1, 4, 8, 12, and 15. Finally, compare each part of the boxplot with the summary: the whisker endpoints must match the minimum and maximum, the box edges must match the quartiles, and the line inside the box must match the median.

A boxplot should also be readable on its own. Give the number line a scale with evenly spaced tick marks, and name the variable and units. If the scale begins at a convenient value below the minimum or ends above the maximum, that is fine as long as the plotted values are placed accurately.

Common Mistakes and AP Exam Tips

  • Finding the median in an unsorted list. Sort the data first. The middle position only identifies the median after the observations are ordered.
  • Including the overall median in both halves. For 15 values, leave out the 8th value when finding \(Q_1\) and \(Q_3\). Each half then has seven observations.
  • Using the wrong position for \(Q_3\). The 4th value of the upper half is overall position 12, not position 4 in the complete list.
  • Removing repeated observations. Count every data value, including duplicates. Removing a repeat changes the number of observations and can shift the positions.
  • Drawing the median at the center of the box by default. Place it at its numerical value on the scale. It may or may not be centered between the quartiles.
  • Making the whiskers stop at the quartiles. The box ends at \(Q_1\) and \(Q_3\); the whiskers continue from those edges to the minimum and maximum in the boxplot constructed here.
  • Leaving out context and units. A full-credit display identifies the quantitative variable, includes its units, and shows a numerical scale that makes the five values clear.

For a full-credit response, show the ordered data or explain how you identified the five positions, state the five-number summary in order, and make the boxplot match that summary. Label the variable and units. Do not rely on a sketch alone to communicate the values when the question asks you to calculate the summary.

Key takeaway: For 15 ordered observations, the five-number summary uses positions 1, 4, 8, 12, and 15. Draw the box from \(Q_1\) to \(Q_3\), mark the median, extend whiskers to the minimum and maximum, and label a clear scale in context.

Check Your Understanding

Use the position guide for 15 observations and explain your choices clearly.

  1. For 15 ordered observations, which positions give the minimum, \(Q_1\), median, \(Q_3\), and maximum?
  2. Why is the overall median left out when dividing 15 observations into a lower half and an upper half?
  3. For the ordered list \(2,4,5,6,7,8,9,10,12,13,14,15,17,18,20\), find the five-number summary.
  4. A boxplot has \(Q_1=6\), median \(=9\), \(Q_3=14\), minimum \(=3\), and maximum \(=19\). Where should the box edges, median line, and whiskers go?
  5. What should you do with repeated values when sorting and numbering the observations?

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