What Does “Overlap” Tell Us?
When two groups are displayed on the same numerical scale, their distributions may occupy some of the same horizontal or vertical territory. Describing that overlap can help answer a practical question: are the observed values mostly separated, or do the groups share some numerical span? The answer depends on what is being compared. The overall ranges, the interquartile intervals, and the individual plotted observations each describe overlap in a different way.
Overlap is a descriptive feature of the data, not a significance test. A graph can show where observed values fall, but visual overlap alone does not tell us whether a difference between groups is statistically significant. Nor does a lack of visible overlap, by itself, establish a particular inferential conclusion.
A range interval marks the span from the smallest to the largest observation. It is an envelope around the data, not a claim that every value inside the envelope was observed. The interquartile interval marks the span from the first quartile to the third quartile. It describes the central portion of the distribution, but the exact observations included at its boundaries can depend on ties and the method used to calculate quartiles.
Finding the Shared Span of Two Intervals
Suppose Group A has an interval \([L_A,U_A]\) and Group B has an interval \([L_B,U_B]\). The lower end of a possible shared span must be at least as large as both lower endpoints, so it is the larger of them. The upper end must be no larger than either upper endpoint, so it is the smaller of them.
This span exists when its lower endpoint is no greater than its upper endpoint. If the lower endpoint is greater, the intervals are separated. If the endpoints are equal, they touch at one point but have no positive-width overlap. This calculation can be applied to range intervals or, separately, to interquartile intervals.
Worked Examples
Worked Example: Overlapping Range Intervals Do Not Guarantee Observed Values There
Two hypothetical groups have the following observations, measured in minutes:
Group A: 0 and 100 minutes.
Group B: 40 and 60 minutes.
Group A’s range interval is \([0,100]\), and Group B’s range interval is \([40,60]\). Their shared range span is
This calculation says that the two range intervals cover a common numerical span from 40 to 60 minutes. But Group A has observations only at 0 and 100 minutes; neither observation falls in that shared span. Group B’s observations, 40 and 60 minutes, do fall there. Thus, overlapping range intervals do not guarantee that both groups have observed values in the shared span. In general, they do not guarantee that either group has an observation in every part of it.
Worked Example: Comparing Interquartile Intervals
Suppose hypothetical summaries, all in centimeters, give Group A’s first and third quartiles as 18 and 31, and Group B’s as 29 and 39. Compare their interquartile intervals and their IQRs.
Group A’s interquartile interval is \([18,31]\), so its IQR is
Group B’s interquartile interval is \([29,39]\), so its IQR is
The shared interquartile span is
Its width is \(31-29=2\) cm. The interquartile intervals therefore overlap over a 2 cm span. This indicates that the groups’ central quartile spans share some values, while most of each interval’s width lies outside the shared span. It does not tell us that the groups have identical distributions, nor does it establish whether any difference between them is statistically significant.
The IQR widths also give a separate descriptive comparison: Group A’s central quartile span is 13 cm wide, compared with 10 cm for Group B. Overlap and spread answer different questions. The overlap describes shared territory; the IQR widths describe the widths of the two central spans.
Worked Example: Using Plotted Values to Describe Observed Overlap
In a hypothetical exercise, two groups’ recorded completion times, in minutes, are:
Group A: 2, 3, 4, 5, 6.
Group B: 5, 6, 7, 8, 9.
Group A’s range interval is \([2,6]\), and Group B’s is \([5,9]\). Their range intervals share the span \([5,6]\), with width \(6-5=1\) minute. Unlike the previous example, the plotted data here also show observations from both groups in this span: each group has observations at 5 and 6 minutes.
On a dotplot with both groups on a common scale, the dots from Group A would occur at 2, 3, 4, 5, and 6; those from Group B would occur at 5, 6, 7, 8, and 9. The plot makes clear that the groups share the recorded values 5 and 6, while Group A also has lower recorded times and Group B also has higher recorded times. The same units and scale are essential: comparing positions on different scales would not show meaningful overlap.
Exact matching values are not required to describe distributions as occupying nearby territory. For example, if times were recorded to tenths of a minute, two groups could have nearby dots at 5.0 and 5.1 without sharing an exact recorded value. A plot helps show that distinction; the definition of “overlap” should match what is being described—shared exact values, shared interval coverage, or nearby plotted observations.
Reading Overlap From Summaries and Plots
A five-number summary gives the minimum, \(Q_1\), median, \(Q_3\), and maximum. Comparing the minimum and maximum gives the two range intervals. Comparing \(Q_1\) and \(Q_3\) gives the two interquartile intervals. These comparisons can be useful when a graph is unavailable, but they do not reveal every individual data value. A dotplot, when the individual observations are shown, gives more direct evidence about where the recorded values actually occur.
A boxplot displays the quartiles and usually the median in the box, with whiskers indicating further values according to the graph’s convention. Overlap between boxes concerns the interquartile intervals; overlap between whiskers or overall plotted extents concerns other spans. Check what each graphical element represents before describing it. In particular, do not treat a box as if it displayed every observation in the interval.
For a careful comparison, state what kind of overlap you mean and keep it distinct from the groups’ centers and spreads. For example, “The interquartile intervals overlap from 29 to 31 cm” is more precise than “the groups overlap.” If discussing the data points, say whether the actual plotted values appear in a shared region or whether there are exact shared recorded values.
Decide whether you are comparing range intervals, interquartile intervals, or individual plotted observations.
Check that both groups are displayed or summarized in comparable units on a common numerical scale.
Give the shared span when useful, and distinguish interval coverage from actual observed values.
Do not infer statistical significance or its absence from visual overlap alone.
Why Visual Overlap Is Not a Significance Test
A significance test addresses a specified question using a statistical model and the variability of the data. A visual display of two groups is useful for understanding their observed distributions, but overlap alone does not supply a test statistic, a \(p\)-value, or a decision at a significance level. In particular, the amount of visible overlap does not have a universal cutoff that separates “significant” from “not significant.”
Two groups can have overlapping ranges while their central portions are positioned differently; the reverse pattern is also possible. In addition, the range is determined by the extreme observations, while the IQR focuses on the central quartile span. A statement about one of these features should not be silently substituted for a statement about another. These descriptive comparisons are valuable, but inferential claims require an appropriate inferential method.
Common Mistake
Wrong reasoning: “The range intervals overlap from 40 to 60, so both groups must have observations between 40 and 60.” This treats an interval from a minimum to a maximum as though it contained every observed value in between.
Correction: The range intervals only show that the groups’ minimum-to-maximum envelopes cover a common numerical span. The data might be sparse within that span. Inspect the plotted values before claiming that each group has observations there, and identify whether your statement concerns interval coverage or actual observations.
Check Your Understanding
Use interval calculations where appropriate, and be precise about what the evidence does and does not establish.
- Group A has observations 1 and 20, while Group B has observations 8 and 12. Find the range intervals and their shared span. Does Group A have an observed value in that span?
- Two interquartile intervals are \([10,22]\) and \([18,27]\). Find their shared span and its width.
- In your own words, explain why overlapping range intervals do not guarantee that both groups have observations in the shared span.
- What extra information can a dotplot provide beyond two five-number summaries when describing observed overlap?
- Why can visual overlap not, by itself, establish whether a difference between groups is statistically significant?