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Lord's Paradox: Two Statisticians, One Dataset

By Alexis Diamond · Statistics · causal inference · Undergraduate · Computational / statistical · version 21

Two statisticians analyse the same six students' weights and reach opposite conclusions. Students compute both analyses in R and then work out what each one assumes about the missing counterfactual.

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Licence: CC BY 4.0

Topicslord's paradoxchange scorescovariate adjustmentpotential outcomescausal inference

Assumption ledger

  • dataSix invented students from the Holland and Rubin (1983) setting: women weigh 110, 130, 150 lb in September and 120, 130, 140 in June; men 140, 160, 180 and 150, 160, 170
  • parameterStatistician 1 models june - sept = a + b*female. Statistician 2 models june = a + c*sept + b*female. female is 1 for women and 0 for men. These models are described in the reveal; students do not fit them in code in this version
  • simplificationNo sampling noise and no randomization: the six points lie exactly on two parallel lines, so every fitted value is exact
  • distributionNo distributional model; lm is used only as least-squares fitting, and no standard errors or p-values are interpreted
  • interpretationReading follows Holland and Rubin (1983): the dining-hall diet is the treatment, sex is an attribute rather than a manipulable treatment, and every student's Y(0) is missing
  • interpretationThe key-elements summary frames the puzzle (data, treatment, question, two analyses, paradox, task) without stating the resolution; the reveal stays hidden until the button is pressed
  • interpretationBoth analyses target the same causal question, the difference between the diet's effect on women and on men; they differ only in the assumed Y(0). Statistician 2 assumes men and women with the same September weight would have had the same June weight without the diet, on one line of slope 0.5 (the within-sex slope, not the pooled slope of about 0.73)
  • interpretationThe opening prediction is a deliberate quick gut-check after Questions 1 and 2 and before Statistician 1's quote is read; Question 3 then asks the group to agree or disagree with that quote and point to supporting numbers. Question 1's group check asks only whether group members get the same numbers, because no prediction has been made at that point
  • interpretationQuestion 4 counts both gainers and losers (two of each, two unchanged), so students see that the group averages hide movement in both directions
  • interpretationQuestion 15 (fitting both models) and the explore cell on within-sex slope were removed at the author's request; the reveal is opened after Question 14. The exit question (formerly Question 16) was also removed at the author's request, so the page has no closing written question and ends with the reveal
  • interpretationStyle kit: EB Garamond substitutes for Chronicle Display, and a Helvetica Neue stack for Neue Haas Grotesk. Clay #905112 is used for buttons and links on light backgrounds only. Goldenrod and Tangerine are not used, because they fail contrast for text. White is used for the character guide on Charcoal sections for contrast. Light backgrounds carry the copy, and there is no dark colour scheme
  • interpretationThe logo is omitted, because the shared kit supplies no logo file

Breakout check

✓ Ready for breakouts: every control we tried showed teammates the same thing.

Code checks

✓ Code checked: the reference solutions pass every exercise's checks.

Built-in checks 4 of 4 pass

  • both groups have zero average change (passes)
  • parallel lines of slope 0.5 with men 15 lb above women (passes)
  • every code cell has a matching well-formed checks block (passes)
  • setState round-trips getState (passes)

Cite this

Alexis Diamond (2026). Lord's Paradox: Two Statisticians, One Dataset [Interactive model], version 21. Ideas at Play. https://ideasatplay.org/play/lords-paradox-two-statisticians-one-dataset-05d1b2?v=21 (CC BY 4.0)

History

  1. v21Removed the exit question (Question 16): its section, its response box and its state key response_q16; all other state keys and the window.artifact interface are unchanged. (generated, Oct 19, 2026)current
  2. v20Removed Question 15 and the 'What if June weight tracked September weight more or less closely?' explore cell, with their reference solution and ledger entry, and updated the reveal prompt, exit question and structure assertion to match. (generated, Oct 19, 2026)View
  3. v19Question 1's group check now asks whether everyone gets the same numbers, and the opening prediction is kept as an explicit quick gut-check before Statistician 1's quote, so its overlap with Question 3 is intentional. (generated, Oct 19, 2026)View
  4. v18No change to the page: the B4 and B5 FIND texts (a Y(0)-based exercise 4 with miss_s1/miss_s2 and an exercise 5 with effect_s3) do not exist in this version, so nothing was replaced and the ledger records this. (generated, Oct 19, 2026)View
  5. v17No edit was applied: the B2/B3 FIND text (Y(0) exercises with y0_s1, y0_s2, effect_s2) does not exist in this version, and the page already uses the within-sex slope of 0.5 in Questions 5 and 15, so the HTML and solutions are unchanged and only the ledger notes this. (generated, Oct 19, 2026)View
  6. v16Edited by hand. (edited, Oct 19, 2026)View
  7. v15Edited by hand. (edited, Oct 19, 2026)View
  8. v14Edited by hand. (edited, Oct 19, 2026)View
  9. v13No change made: the text 'He assumes no systematic change: Y(0) is just the September weight.' is not in this artifact, whose reveal already says 'Statistician 1 ... assumes', so the artifact is returned as it was. (generated, Oct 19, 2026)View
  10. v12Edited by hand. (edited, Oct 19, 2026)View
  11. v11Edited by hand. (edited, Oct 19, 2026)View
  12. v10Edited by hand. (edited, Oct 19, 2026)View
  13. v9Added colons after the statisticians' names, extended Question 4 to count losers as well as gainers (new n_lost check and solution), moved the opening prediction to just before Statistician 1 with its new wording, switched to American spelling, and made the character guide white on dark sections; state keys and the window.artifact interface are unchanged. (generated, Oct 19, 2026)View
  14. v8Edited by hand. (edited, Oct 19, 2026)View
  15. v7Applied the five requested edits: corrected Statistician 2's Y(0) assumption and the closing reveal paragraph, removed the unused seed from setup, and made the Question 5 vertical line dot-dash with matching text. (generated, Oct 19, 2026)View
  16. v6Edited by hand. (edited, Oct 19, 2026)View
  17. v5Added a key-elements summary after the Holland and Rubin quote, and replaced the explore cell's shift knob with a slope knob (with corrected hints and a ledger entry); everything else, including state keys and window.artifact, is unchanged. (generated, Oct 19, 2026)View
  18. v4Edited by hand. (edited, Oct 19, 2026)View
  19. v3Edited by hand. (edited, Oct 19, 2026)View
  20. v2Edited by hand. (edited, Oct 19, 2026)View
  21. v1New Coding artifact in R built from the Lord's Paradox workbook, with Minerva styling, written-response boxes for the free-response questions, a prediction anchor, a reveal and a break-it cell; the red/blue plot colours were replaced with palette colours plus shapes to meet the brand and accessibility rules. (generated, Oct 19, 2026)View

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