Comparing Five-Number Summaries Across Groups

Why Compare Five-Number Summaries?

When two groups have many observations, a compact summary can help describe how their values differ. A five-number summary gives a useful set of landmarks for each group: the smallest and largest observed values, the middle value, and two quartiles. Putting the summaries side by side makes it possible to compare corresponding landmarks in the same units.

These landmarks are informative, but they are not a complete description of either group. They do not show every observation, and similar summaries can conceal different patterns in the data. A careful comparison therefore describes what the summaries support and avoids treating them as if they were the original distributions.

Definition: A five-number summary consists of the minimum, first quartile \(Q_1\), median, third quartile \(Q_3\), and maximum of a data set, in that order. The minimum and maximum are the smallest and largest observed values. The median divides the ordered observations into lower and upper parts. The first quartile marks the lower quarter of the data, and the third quartile marks the lower three quarters. Quartile calculation conventions can differ, so comparisons should use summaries calculated with a consistent method.

Read Corresponding Values Across Groups

A side-by-side comparison should match the same summary position across groups. Compare the medians to describe their typical middle values; compare \(Q_1\) with \(Q_1\) and \(Q_3\) with \(Q_3\) to see how the middle portions are positioned. Then compare the minimums and maximums to describe the observed endpoints. Always include the variable and its units.

The difference between two medians is a difference in the groups' middle values, not a difference between every individual in one group and every individual in the other. Similarly, a larger maximum means that the largest recorded observation is larger; it does not, by itself, describe how common large values are.

Worked Example: Comparing Quiz Scores

Suppose two hypothetical groups of eight students have these quiz scores, already ordered from smallest to largest:

Group A: 10, 12, 13, 15, 17, 18, 20, 22 points
Group B: 14, 15, 16, 17, 19, 20, 23, 27 points

For this example, use the nearest-rank rule for the quartiles: \(Q_1\) is the observation at rank \(\lceil 0.25n\rceil\), and \(Q_3\) is at rank \(\lceil 0.75n\rceil\). The median for an even number of observations is the average of the two middle observations. Here \(n=8\), so the quartile ranks are \(\lceil 2\rceil=2\) and \(\lceil 6\rceil=6\).

For Group A, the minimum is 10 points and the maximum is 22 points. The quartiles are the second and sixth values, so \(Q_1=12\) and \(Q_3=18\). The median is \((15+17)/2=16\) points. Its summary is therefore \((10,12,16,18,22)\) points.

For Group B, the minimum is 14 points and the maximum is 27 points. The quartiles are \(Q_1=15\) and \(Q_3=20\). The median is \((17+19)/2=18\) points. Its summary is \((14,15,18,20,27)\) points.

The Group B median is \(18-16=2\) points higher. Its first and third quartiles are also higher, by 3 and 2 points respectively. This indicates that the corresponding middle landmarks are higher for Group B. It does not mean every Group B score exceeds every Group A score.

Compare Spread With Quartiles and Endpoints

The distance from \(Q_1\) to \(Q_3\) is the interquartile range, or IQR. It measures the width of the interval from the first to the third quartile, which contains the middle half of the observations as marked by those quartiles. The range is the distance from the minimum to the maximum and describes the span of all observed values.

\mathrm{IQR}=Q_3-Q_1 \qquad\text{and}\qquad \mathrm{Range}=\mathrm{maximum}-\mathrm{minimum}

These measures summarize different spans. The IQR focuses on the central portion, while the range includes the two endpoints. To compare group spreads, report which measure is being used: groups can have similar IQRs but different ranges, or the reverse. All differences and widths retain the original variable's units.

Justification: Since the first and third quartiles mark the lower and upper boundaries of the middle half, subtracting the lower boundary from the upper boundary gives that interval's width: \(Q_3-Q_1\). Subtracting the minimum from the maximum instead gives the width from the smallest to the largest observation. The two calculations describe different parts of the data.

Worked Example: Comparing Quiz-Score Spreads

Use the hypothetical quiz-score summaries from the previous example to compare the groups' IQRs and ranges.

For Group A, \(\mathrm{IQR}=18-12=6\) points, and \(\mathrm{Range}=22-10=12\) points.

For Group B, \(\mathrm{IQR}=20-15=5\) points, and \(\mathrm{Range}=27-14=13\) points.

Group A has an IQR 1 point larger, so its middle-half interval is wider by 1 point under these quartiles. Group B has a range 1 point larger, so its full observed span is wider by 1 point. There is no contradiction: the IQR describes the middle portion, while the range depends on the endpoints. These summaries alone do not explain why the spans differ.

Spacing Can Suggest Patterns, but Does Not Show the Shape

The gaps between consecutive entries in a five-number summary offer another way to describe a group. For instance, compare the distance from the minimum to \(Q_1\), from \(Q_1\) to the median, from the median to \(Q_3\), and from \(Q_3\) to the maximum. A notably longer gap can indicate that values in that portion cover a wider interval. It is a clue about spread across that interval, not a record of the individual values within it.

Worked Example: Interpreting the Gaps in Commute Times

Two hypothetical groups report commute times in minutes. Group A has the summary \((8,12,18,25,46)\); Group B has \((10,14,20,28,34)\). Compare the gaps between consecutive summary values.

For Group A, the four gaps are \(12-8=4\), \(18-12=6\), \(25-18=7\), and \(46-25=21\) minutes. For Group B, they are \(14-10=4\), \(20-14=6\), \(28-20=8\), and \(34-28=6\) minutes.

The uppermost interval from \(Q_3\) to the maximum is much wider for Group A (21 minutes) than for Group B (6 minutes). This supports saying that Group A's maximum is farther above its third quartile. It does not establish exactly how commute times are distributed within that interval or how many commuters have long trips. A graph or the underlying observations would be needed to see those details.

What the Summaries Leave Out

A five-number summary compresses many observations into five values. That compression makes comparison manageable, but it also discards information. In particular, the summary does not report the sample size, the number of repeated values, gaps between observations, or how observations are arranged within each quartile interval. It cannot identify the full distribution shape or show the exact location of every data point.

Even identical summaries do not imply identical data. The following example shows two small hypothetical data sets with matching five-number summaries but different observations.

Worked Example: Same Summary, Different Data

Consider these two hypothetical ordered data sets, measured in minutes:

Set A: 0, 1, 2, 3, 4, 5, 6
Set B: 0, 1, 1, 3, 5, 5, 6

Each set has seven observations. Using the median of the lower and upper halves to find the quartiles, the overall median is the fourth value: 3 minutes in both sets. For Set A, the lower half is 0, 1, 2, so \(Q_1=1\); the upper half is 4, 5, 6, so \(Q_3=5\). For Set B, the lower half is 0, 1, 1, so \(Q_1=1\); the upper half is 5, 5, 6, so \(Q_3=5\). Both sets also have minimum 0 and maximum 6 minutes.

Thus both summaries are \((0,1,3,5,6)\) minutes. Yet Set A has a different arrangement and frequency of values from Set B: for example, Set B repeats 1 and 5, whereas Set A does not. The five-number summary does not preserve that information. When a comparison depends on details the summary omits, consult the data or an appropriate graph rather than guessing from the five values.

Common Mistakes

Comparing different landmarks. Comparing one group's median with another group's \(Q_3\) does not compare the same feature. Match medians to medians, quartiles to the same quartiles, and endpoints to endpoints.
Calling the IQR the full spread. The IQR describes the distance from \(Q_1\) to \(Q_3\), not the distance between the minimum and maximum. Name the measure when describing spread.
Interpreting a higher median as universal superiority. A higher median means a higher middle value, not that every observation in that group is higher.
Inferring a complete shape from five values. Unequal gaps may suggest where values span wider intervals, but they do not reveal every observation or prove a particular distribution shape.
Ignoring the quartile convention. Different accepted methods can give different quartiles for the same data. State or check the method when exact values matter, and use a consistent method for the groups being compared.

Key takeaway: A five-number summary gives a compact comparison of group locations, middle-half widths, and observed endpoints. Describe the specific values and units, distinguish the IQR from the range, and remember that five landmarks cannot reproduce the full distributions.

Check Your Understanding

Use the definitions and comparisons in this tutorial to answer the questions.

A group has summary \((4,7,10,13,19)\) kilograms. What are its IQR and range, and what does each measure describe?
Group X has a median of 24 minutes and Group Y has a median of 20 minutes. State a correct comparison and one conclusion that the medians alone do not justify.
Two groups have the same \(Q_1\) and \(Q_3\), but different maximums. Which measure of spread is guaranteed to be the same, and which may differ?
Why should the quartile calculation method be consistent when comparing five-number summaries?
Can identical five-number summaries establish that two groups have identical observations? Explain why or why not.