Unit 1 · Topic 1.8 · about 20 minutes

Graphical Representations of Summary Statistics for One Quantitative Variable

Draw a boxplot from data, outliers included, read one correctly, and use the mean and median to judge whether a distribution is likely skewed.

Predict first

In a boxplot, the whisker on the left is three times as long as the whisker on the right. Which part of the plot holds more of the data?

The five-number summary and the boxplot

Five numbers sketch a whole distribution: the minimum, $Q_{1}$ , the median, $Q_{3}$ and the maximum. Together they are the five-number summary, and 1.7 Summary Statistics for One Quantitative Variable showed how to find each one. A boxplot draws them.

The box runs from $Q_{1}$ to $Q_{3}$ , so it holds the middle 50% of the data. A line inside the box marks the median.
The whiskers run from the box out toward the smallest and largest values. Each one covers about 25% of the data.
When there are outliers, each whisker stops at the most extreme value that is not an outlier, and every outlier is drawn on its own with a dot, an asterisk or some other symbol.

Outliers are usually decided by the $1.5 \times IQR$ rule from 1.7 Summary Statistics for One Quantitative Variable, so a boxplot that shows outliers takes one extra step: find the fences before you draw the whiskers.

Worked exampleDrawing a boxplot with an outlier

Fifteen students timed how long they took to solve a scrambled cube puzzle, in seconds: 45, 52, 58, 61, 63, 70, 74, 78, 81, 88, 95, 102, 110, 125, 240. Draw a boxplot that shows outliers.

Five-number summary. The data are already sorted and $n = 15$ , so the median is the 8th value, 78 seconds. The seven values below it give $Q_{1} = 61$ and the seven above it give $Q_{3} = 102$ . The minimum is 45 and the maximum is 240.
Fences. $IQR = 102 - 61 = 41$ , and $1.5 \times 41 = 61.5$ . The fences are $61 - 61.5 = - 0.5$ and $102 + 61.5 = 163.5$ seconds.
Outliers. No time is below $- 0.5$ . Only 240 is above 163.5, so the 240-second solve is an outlier.
Draw. A box from 61 to 102 with a line at 78. The left whisker runs to 45. The right whisker runs to 125, the largest value that is not an outlier. The 240 gets its own dot.

Answer.

The boxplot below. The right whisker stops at 125 seconds, not at 240.

Puzzle solve times for 15 students

The box holds the middle 50% of the times, from 61 to 102 seconds. The separate dot at 240 seconds is the outlier.

Reading a boxplot

The median line shows the center. The length of the box, the IQR, shows the spread of the middle half of the data. Comparing the two halves of the box, and the two whiskers, hints at the shape. In the solve times, the right half of the box (78 to 102) is longer than the left half (61 to 78), and the right whisker is longer than the left one. The slower solvers are more spread out than the faster ones, which is a sign of right skew.

What a boxplot cannot show is anything between its five numbers. Remember the birthday-party ages from 1.6 Descriptions for One Quantitative Variable Distributions: a cluster of children, a cluster of adults and a gap of almost 20 years between them. Here are those ages again, first as a dotplot and then as a boxplot.

Birthday-party ages as a dotplot

The same ages as a boxplot

The boxplot shows a median of 8.5 years and a long box reaching up to 36, but the two clusters and the gap between them have disappeared.

Mean, median and shape

When all you have is a table of summary statistics, comparing the mean with the median is a quick check on shape.

Roughly symmetric: the mean and the median are close together.
Skewed right: the mean is usually larger than the median, because the long right tail pulls the mean up.
Skewed left: the mean is usually smaller than the median.

For the solve times, the mean is about 89.5 seconds and the median is 78, which fits the right skew in the boxplot. Notice the word usually. A mean above the median hints at right skew but does not prove it: the birthday-party ages have a mean of about 19.7 and a median of 8.5, and they are two clusters. So when the evidence is only a pair of numbers, say that the distribution is likely skewed.

Check your understanding

A boxplot of the lengths of the 200 songs on a playlist has $Q_{1} = 3.1$ minutes, median 3.6 minutes and $Q_{3} = 4.2$ minutes. About how many of the songs are longer than 3.1 minutes?

The five-number summary of the goals a hockey team scored in each of its 41 games is 0, 2, 3, 5, 11. The two highest totals were 8 goals and 11 goals. In a boxplot that shows outliers, where does the right whisker end?

For the 120 employees of a company, the mean commute distance is 18.4 miles and the median is 11.2 miles. What is the most likely shape of the distribution of commute distances?

A dotplot of quiz scores shows two clusters, one around 60 and one around 90, with no scores from 70 to 80. A student makes a boxplot of the same scores. What will the boxplot fail to show?

Which five-number summary most likely comes from a distribution that is skewed left?

Practice

Practice until it is automatic

Every problem here is one step of drawing a boxplot: quartiles, the IQR, the fences or the outliers.

Quartiles, IQR and outliers practice page

Course alignment, for teachers

AP Statistics topic 1.8, Unit 1: Exploring One-Variable Data and Collecting Data.

Skill 3.A: Construct tabular and graphical representations of data and distributions.
Skill 4.A: Describe and compare tabular and graphical representations of data, as well as summary statistics.