percentage-point difference
An instrument for one disputed denominator
Three Rulers for an Empty Chair
A jury pool can be 1.28 percentage points short and 17.58 percent short at the same time. Measure the same shares three ways, then change only a hypothetical pool size and watch statistical surprise move while two disparities stand still. The arithmetic is kept separate from evidence of systematic exclusion.
This is a measurement model, not a universal legal test and not legal advice. It concerns the sources from which juries are drawn, not a promise that a seated jury mirrors community demographics.
Layer 1 · Put down the rulers
The same shortfall, three answers
Start with the Smith preset from Berghuis v. Smith. Move observed share from the eligible share down to the reported pool share. Then move only n. The first two rulers never look at it.
%
%
draws
shortfall relative to p
signed standard-deviation gap
A = p − q
C = (p − q) / p
z = (np − nq) / √(np(1 − p))
The instrument is loading.
Layer 2 · The statistic does not finish the proof
A small tail is not a selection mechanism
A significance calculation can ask whether a share would be surprising under a declared random-draw model. The constitutional pipeline asks separate questions. Numerical underrepresentation belongs to one stage. Evidence that a process produced it belongs to another.
Disparity
How far apart are the eligible and observed shares? The answer ignores sample size.
Random-sampling evidence
How surprising is an integer count under a chosen probability model? The answer depends on sample size and assumptions.
Selection mechanism
What feature of the jury-selection process accounts for persistent underrepresentation? A tail probability cannot name it.
Operate the reported Duren pipeline
Where the share fell
%
%
Jury-eligible population54.0%
↓
Summoned for service…
↓
Postsummons weekly venires…
The pipeline is loading.
Keep the ledger open
One denominator, two later stages
| Comparison to eligible share | Absolute | Comparative |
|---|---|---|
| summoned | … | … |
| weekly venire | … | … |
What the reported record added
- The underrepresentation occurred in every weekly venire for almost a year.
- Questionnaires conspicuously told women they could opt out.
- The summons stage supplied another opportunity to opt out, and an ignored summons was treated as opting out.
Those are factual features the Supreme Court recounted from Duren. They are not outputs of the sliders. Changing a share can change disparity, but it cannot create this causal record.
SHOWING 1Distinctive groupA legal classification, not a numeric output.
SHOWING 2Fair and reasonable representationThe three rulers inform this question, without a universal Supreme Court threshold.
SHOWING 3Systematic exclusionRequires proof about the selection process that accounts for the underrepresentation.
The check · live and independently toothed
What would turn this panel red?
The page’s decimal route is compared with an integer hundredths-of-a-percent route that shares none of its formula helpers. A published anchor and an exact binomial special case stand outside both. Finally, a deliberately corrupted Smith share must be rejected.
Free choices and uncertainties: the sample size is hypothetical; the binomial model assumes independent draws with replacement; the historical percentages are rounded; no underlying Smith or Duren counts are recovered; group and eligibility categories are jurisdiction-specific legal and historical constructions; the page does not choose a legal threshold or infer a cause from a p-value.
Primary source
The opinion that declined the ruler
Berghuis v. Smith, 559 U.S. 314 (2010), reports the Smith shares and recounts the Duren pipeline. At pages 319 and 330, the Court sets out the three showings and says that no Supreme Court decision specifies which underrepresentation measure courts must use. It calls absolute disparity, comparative disparity, and standard deviation imperfect.
Official United States Reports volume · accessed for this build on 2026-08-25