Side-by-Side Boxplots Compared

What Side-by-Side Boxplots Make Easy to Compare

In Comparing Two Histograms on the Same Scale, you practiced checking whether displays use matching scales before comparing distributions. Side-by-side boxplots also make comparisons easier by placing groups on a shared numerical axis. They emphasize a distribution’s center and spread rather than showing how many observations fall in each interval.

A boxplot displays the median and quartiles. The line inside the box marks the median. The box extends from the first quartile, \(Q_1\), to the third quartile, \(Q_3\); its length represents the interquartile range, or IQR. The whiskers and any plotted points show additional information about the data’s endpoints or unusual observations, depending on the boxplot convention.

As in Comparing Centers Using Medians and Means and Comparing Spread Using IQR and Standard Deviation, compare like with like: median with median, IQR with IQR, and range with range. State the size and direction of a difference in the original units. For instance, if one group’s median is 5 minutes greater, say which group has the greater median and what variable those minutes measure.

The IQR focuses on the middle half of a group’s observations. The range includes the two extremes, so a single unusually high or low value can affect it substantially. Tutorial 89, Choosing IQR or Standard Deviation to Describe Spread, explains why the IQR is often more useful for skewed data or data with outliers. When a question asks you to compare range as well, report it, but distinguish it from the IQR rather than treating the two measures as interchangeable.

A Careful Reading Routine

Before comparing features, check that both plots display the same quantitative variable in the same units and use a common axis. A horizontal distance should represent the same amount in both plots. If the scales differ, apparent distances between medians or box widths are not directly comparable. Read the axis labels and tick marks rather than estimating from the physical size of the plot alone.

Then read each group’s median and quartiles. The median is the line inside the box; the box edges are \(Q_1\) and \(Q_3\). Subtract \(Q_1\) from \(Q_3\) to obtain the IQR. To compare ranges, identify the actual minimum and maximum for each group and subtract. In a standard boxplot the whiskers may mark those extremes. In a modified boxplot, whiskers may stop at the most extreme values that are not flagged as outliers, with outliers plotted separately. Check the display’s key or description: if outlier points are present, include them when finding the true minimum or maximum.

A boxplot summarizes a distribution; it does not show every observation. If the plots are drawn from samples, describe the groups represented by those samples. Do not claim that the displays prove a difference for all members of a larger population.

Worked Example: Comparing Two Groups’ Centers and Spreads

Worked Example: When IQR and Range Tell Different Stories

Worked Example: Same Median, Different Spreads

What Boxplot Overlap Can—and Cannot—Tell You

The box is the interval from \(Q_1\) to \(Q_3\), so comparing box locations and widths gives a quick visual comparison of the middle halves. If two boxes overlap, both groups’ middle-half intervals include some of the same numerical values. This does not mean the same people, objects, or observations occur in both groups. Nor does it reveal exactly how the observations are distributed inside the boxes.

Likewise, overlapping whiskers or overall spans do not mean the distributions are identical. Groups can have similar ranges but different medians or IQRs. They can also have similar IQRs but different ranges, as in the first worked example. Report the specific features the plot supports instead of saying only that the groups are “similar” or “different.”

Boxplots emphasize selected summaries, so they do not show all distribution features equally well. They can suggest differences in center and spread, but they may hide clusters or gaps that would be visible in a dotplot. As in Comparing Two Dotplots, use a display that reveals the feature relevant to the question, and avoid claiming a boxplot shows details it does not display.

Common Mistakes and AP Exam Tips

• Comparing the wrong parts. Compare median with median, IQR with IQR, and range with range. Do not compare one group’s median to another group’s \(Q_1\).
• Calling the box width the range. The box runs from \(Q_1\) to \(Q_3\), so its width is the IQR. The range uses the actual minimum and maximum.
• Ignoring plot conventions. In a modified boxplot, whisker ends may not be the data minimum and maximum. Check for plotted outlier points and the display’s definition before calculating a range.
• Using a vague comparison. “Group A is more spread out” is incomplete if the question asks for numerical evidence. Give the IQRs or ranges and state their difference in context.
• Forgetting units and direction. Say which group has the greater median or spread, and include the variable’s units. A difference in travel-time medians should be reported in minutes, not as an unexplained number.
• Overstating what overlap means. Overlapping boxes do not establish identical data, and separated medians do not by themselves establish a population-level difference.

A full-credit comparison names the variable and groups, reports the relevant summaries accurately, and interprets their differences with units. If the IQR and range point in different directions, state that clearly instead of forcing them into one broad conclusion.