Small-sample quartiles are unstable; box width is uninterpretable

What it is

Small-sample quartile instability arises because Q1, Q3, and the IQR are estimated from few order statistics; changing a few observations or the sample-quantile algorithm may move them visibly. The title combines two cautions: with a small sample, box length along the quantitative axis—the IQR—is not a stable estimate of population spread; ordinary physical box width along the categorical axis is styling and has no numeric interpretation unless a variable-width mapping is explicitly declared. A box plot remains computable, but precise geometry does not imply precise estimation.

Why it happens

Sample quantiles depend on rank positions and interpolation between nearby observations. With small n, each observation occupies a large share of the ranks, so one value near the target rank can move a box edge. Differences among software quantile definitions also become more visible. Stability further depends on population density near each quartile, discreteness, dependence, and sampling design, not on one universal n cutoff. Variation in the IQR moves Tukey fences as well, changing which observations appear separately. Those changes express sampling and algorithmic uncertainty, not proof of bad data.

Studying it

Repeatedly sample from the target population or a credible generating model and inspect the distributions of Q1, Q3, IQR, whisker ends, and outside-point membership. Bootstrap can aid exploration, but with extremely small, dependent, or unrepresentative samples it inherits the original sample's limitations. Compare common quantile algorithms and contrast boxes with raw points, ECDFs, and interval summaries. Report n, independent units, sampling design, and repetitions, and judge instability against task-specific tolerance rather than a fixed cutoff.

Where it stops holding

There is no n=10, 20, or 30 boundary valid for every distribution and task. A small census covering the full target population is not a sampling-estimation problem, while a large biased, clustered, or drifting sample may still miss its target. A small-sample box can describe the observed batch, but should not alone support subtle population comparisons. Plotting every point reveals the observations honestly but does not remove sampling uncertainty. A notch supplies a median-comparison cue under a particular approximation; it does not guarantee stability of the whole box.

Applying it

• Show all raw observations, n, and the box summary together for small samples, keeping overlaps countable and outside observations visible.
• Declare the quantile algorithm and pin the tool version. If reasonable algorithms change the substantive conclusion, report sensitivity instead of choosing the neatest box.
• Estimate uncertainty in box edges and target contrasts through repeated sampling, bootstrap, or a model appropriate to the sampling design, and disclose method limits.
• Keep ordinary categorical-axis widths constant. If width encodes n, label the mapping and provide text values so it is not confused with IQR span.
• Validate against the decision tolerance. When resampling or algorithm changes repeatedly reverse a group conclusion, show observations and uncertainty rather than a stable ranking.

Related

• Same group: U2.09.1 Box plots compress a distribution into five statistics · U2.09.2 Whisker and outlier rules are not unique; state them with the chart · U2.09.3 The box plot's strength is comparing many distributions side by side · U2.09.4 Sample size does not show in the box; annotate it separately
• Nearby: U2.08.3 Showing the raw point distribution is more reliable · U2.08.2 Box plots hide multimodal structure
• Search terms: small-sample box plot · quantile estimator · IQR uncertainty · bootstrap sensitivity