The space-or-newness trade-off that the budget invented

Dark navy estate-agent slide, masthead “Prairie & Vine Realty · Buyer briefing · Ames, Iowa · 2,930 recorded sales, 2006–2010”, headlined “The $200,000 law: every 1,000 square feet costs you twenty years”. A subhead says this is every residential sale in the assessor’s record that closed between $180,000 and $220,000 — 485 homes, all of them, nothing sampled and nothing smoothed; each dot is one house that actually changed hands, floor area across and year of construction up; both scales are linear and unbroken, neither truncated nor logged, no second axis, and every one of the 485 sales is on the panel and in the tables; the correlation is −0.34 and the least-squares line falls 20.1 years for every 1,000 square feet added, so in this bracket space and newness are not two things you shop for but one thing you divide. Down the left, under the label “Homes sold $180,000–$220,000 · floor area vs. year built · n = 485”, a scatter plot runs from 900 to 5,200 square feet across and 1870 to 2015 up, gridlines every 1,000 square feet and every 20 years, both labelled. Four hundred and eighty-five gold dots cluster densely between about 1,000 and 2,600 square feet, thickest near the top around the year 2000, with a long tail of older houses reaching down past 1900 and a handful of very large homes out to 5,095 square feet. A thick white least-squares line runs from the upper left to the lower right, marked at two points: “1,200 sq ft → built 1997” and “2,400 sq ft → built 1973”. In the lower right of the panel, in gold, “r = −0.34”, above the lines “485 sales · slope −20.1 yr per 1,000 sq ft” and “every sale in the bracket, 2006–2010”. A note beneath records the axis ranges and that the fit is ordinary least squares on all 485 points. The middle column holds three tables. “Your budget, cut five ways by size · 97 homes in each fifth · 485 divides exactly” gives floor-area range, homes, median year built and median price: 988–1,364 sq ft, 97, 2004, $193,879; 1,364–1,516, 97, 2001, $192,000; 1,522–1,674, 97, 1998, $195,800; 1,675–1,851, 97, 1998, $196,000; and 1,855–5,095, 97, 1978, $202,500 — captioned as twenty-six years between the smallest fifth and the largest for a median price difference of $8,621. “The same 485 homes, cut the other way · by decade built” gives before 1950, 35 homes, median 1,824 sq ft, $200,000; 1950s–60s, 48, 1,689, $191,000; 1970s–80s, 83, 1,621, $192,000; 1990s, 110, 1,658, $194,500; 2000s, 209, 1,504 sq ft, $199,000, captioned that cut by age the same law appears from the other side, the oldest homes running 320 sq ft larger than the newest. “The law, year by year · the bracket, split by year of sale” gives homes, correlation and slope for each year: 2006, 102, −0.27, −21.8; 2007, 123, −0.24, −11.3; 2008, 99, −0.46, −40.4; 2009, 101, −0.52, −30.1; and 2010 to July, 60, −0.45, −26.3 — five separate years, five negative numbers, no exceptions. Down the right-hand side, a gold-bordered card carries “−20.1 years” over “per 1,000 sq ft added · least squares, 485 sales · r = −0.34 · every sale in the bracket included”. A card headed “What the line prices” reads the fit at 1,200 sq ft (1997), 1,800 (1985) and 2,400 (1973), concluding that doubling your floor area buys a house a quarter of a century older. A card headed “We tried to break it” reports that dropping the five largest houses in the file, which the data’s own author suggests doing, gives −0.44, and keeping only ordinary arm’s-length sales gives −0.38, so the law survived everything they could think of. A card headed “Advice to buyers” tells readers to decide which kind of household they are before viewing, because the market has already decided they cannot be both. A card headed “Why we trust this number” stresses that these are recorded transactions from the city assessor, every residential sale in the file that landed in the bracket, nothing sampled, smoothed or trimmed. A final card, “Basis of measurement”, names the Ames City Assessor’s Office, the three fields used, and the 485 sales out of 2,930, or 16.6%, that closed in the bracket. Three ticked boxes run along the foot: “Nothing dropped”, “No axis tricks” and “One source, one definition — one assessor’s office, one continuous window from January 2006 to July 2010”. A footnote credits the Ames Housing data set of Dean De Cock, Journal of Statistics Education 19(3), 2011, gives the file and retrieval date, the bracket definition, r = −0.3354 and the OLS slope, records that the 2,930 are the residential sales De Cock retained from an original 3,970 with repeat sales de-duplicated, notes that sales run January 2006 to July 2010 and how medians of even-sized groups are rounded, and states that the slide is our own demonstration drawn in the manner of an estate agent’s buyer briefing, that the firm is invented, and that every figure on it is real. — Showing the misleading chart

The same data on a cream background, kicked “The same 2,930 sales · with the price filter taken off · Ames City Assessor · 2006–2010” and headlined “Across the market the two go together. The trade-off is the budget.” A subhead grants that every figure on the slide is correct — inside the $180,000–$220,000 bracket the 485 sales really do run −0.34, in all five years of sale — and notes that the slide also says plainly which homes it kept, six times over, in the headline, the panel label, both tables and two cards; what it never says is what keeping them does. Price is not a third thing to hold constant, it is what floor area and construction year jointly produce: fix it and you have not controlled for anything, you have required the two to compensate, so a house can only be big here by being old and new here by being small. Take the filter off and the same 2,930 sales run +0.24. The main panel, labelled “All 2,930 sales · floor area vs. year built · the $180,000–$220,000 bracket picked out in gold”, plots every sale from 300 to 5,700 square feet across and 1870 to 2015 up. Small navy dots fill the field and 485 gold dots pick out the bracket, tracing a diagonal band from small-and-new at the upper left down to large-and-old at the lower right. A navy least-squares line rises left to right, labelled “all 2,930 sales: +14.5 years per 1,000 sq ft”; a gold line falls, labelled “the bracket alone: −20.1”. Two dashed navy rectangles mark the corners the budget deletes: the upper right, “150 big and new · median $342,322”, and the lower left, “250 small and old · median $99,500”. In the lower right of the panel, “whole market r = +0.24” in navy above “inside the bracket r = −0.34” in gold, and the note “2,445 sales in navy · 485 in gold · nothing removed”. Beneath it a strip chart headed “Every tenth of the market, sliced by price · 293 sales in each · Pearson r between floor area and year built” shows ten gold dots hanging below a zero line at −0.16, −0.43, −0.41, −0.46, −0.26, −0.33, −0.29, −0.41, −0.34 and −0.39, labelled by decile top price from $105k to $755k, and one navy dot above the line at +0.24 marked “no filter · all 2,930”; its caption notes that all ten are negative while the unsliced figure is +0.24, so the slide’s bracket was no lucky pick and any band tight enough to be a budget does it. The middle column carries four tables. “Median sale price by size and age · all 2,930 sales · cells landing in the bracket shaded” is a five-by-five grid of median prices, floor-area bands down the side and decade built across the top, rising in both directions from $97,200 for small pre-war homes to $356,383 for large new ones, with the six cells whose medians fall between $180,000 and $220,000 shaded gold and forming a diagonal — captioned that a price bracket is therefore a diagonal, and a diagonal slice through two rising quantities slopes down before a single house is looked at. “Median floor area by decade built · the whole market against the bracket” sets the two side by side: before 1950, 1,345 sq ft against 1,824; 1950s–60s, 1,209 against 1,689; 1970s–80s, 1,257 against 1,621; 1990s, 1,680 against 1,658; 2000s, 1,636 against 1,504 — one filter apart, the newest homes 291 sq ft larger than the pre-war ones across the market and 320 sq ft smaller inside the bracket. “Cut by year of sale · five separate years · 2010 runs to July” gives all sales at +0.28, +0.22, +0.21, +0.27 and +0.22 against the bracket’s −0.27, −0.24, −0.46, −0.52 and −0.45. “Tighten the filter, deepen the trade-off · every band centred on $200,000” lists ± $5,000 keeping 112 sales at −0.56, ± $20,000 (the slide) keeping 485 at −0.34, ± $60,000 keeping 1,548 at −0.11, ± $100,000 keeping 2,463 at +0.09 and no filter at all keeping 2,930 at +0.24, captioned that this is not a feature of the houses but a dial, and the filter is the hand on it. Five cards run down the right. “The corners the budget deleted” gives the 150 big-and-new homes at a median $342,322 with four reaching the bracket and the 250 small-and-old homes at a median $99,500 with none reaching it, noting that one kind is priced out of the top of the budget and the other out of the bottom. “What a price band actually is” explains that price is a common effect of size and age, that selecting on a common effect is the one kind of filter that creates a relationship rather than revealing one, and that once the price is known, learning a house is large tells you it must be old. “Not Simpson’s paradox — its mirror image” sets the two apart: Simpson’s paradox as you usually meet it comes from a confounder, a cause the two variables share, and the fix is to break the comparison out by it; a collider is an effect they share and breaking it out by that is the cause; the test is direction. “Where you meet this in your own data” lists customers who converted, candidates hired, students admitted, start-ups funded, studies that reached significance, and the hospital patients Joseph Berkson worked through in 1946, where two unrelated diseases come out looking like alternatives. “What the slide got right” counts ten things — one assessor, one definition, one window, every sale in the bracket plotted and tabulated, both axes linear, unbroken and printed on the panel, nothing sampled, smoothed or trimmed, and the bracket named six times — and names the one missing sentence as what naming the bracket does to the two columns. A gold-ruled callout at the foot reads “A filter is a variable you have already used,” and says the fix is a sentence rather than a better model: say what a row had to do to be in the table, then ask whether the columns being compared helped it do that; where the filter cannot be lifted the direction is still knowable in advance, because selecting on a common effect pushes the association between its causes negative, which is why the tighter the slice the surer the trade-off; and where the slice really is the question, as it is for a buyer with $200,000 to spend, the finding is a fact about the price bracket rather than about houses. A footnote gives the source, the retrieval date, the fields used, the whole-market and bracket statistics, all ten decile correlations, the note that the 2,930 records are the residential sales De Cock retained from an original 3,970 and run to July 2010, how medians of even-sized groups are rounded, and two robustness checks: dropping the five sales over 4,000 sq ft as De Cock’s own documentation recommends gives +0.239 and −0.437, and restricting to normal sale conditions gives +0.213 and −0.384. — Showing the misleading chart

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Inside a $180,000–$220,000 house-hunting bracket, floor area and year built run −0.34: every extra 1,000 square feet costs you twenty years of building age. Take the price filter off and the same 2,930 sales run +0.24 the other way — bigger houses are newer. Price is what size and age jointly produce, so slicing by it forces the two to compensate.

Ames, Iowa assessor’s residential property sales, 2006–2010 (2,930 records) — the Ames Housing data set, Dean De Cock, “Ames, Iowa: Alternative to the Boston Housing Data as an End of Semester Regression Project”, Journal of Statistics Education 19(3), 2011Added August 28, 2026

01The claim

Here is every residential sale in the assessor’s record that closed between $180,000 and $220,000 — 485 homes, all of them, nothing sampled and nothing smoothed. Each dot is one house that actually changed hands: floor area across, year of construction up. Both scales are linear and unbroken, neither is truncated, neither is logged, there is no second axis, and every one of the 485 sales is on the panel and in the tables. The pattern is not subtle and it is not our opinion: the correlation is −0.34, and the least-squares line falls 20.1 years for every 1,000 square feet you add. Read it at three sizes and the exchange rate is plain — at 1,200 sq ft the line sits at 1997, at 1,800 sq ft at 1985, at 2,400 sq ft at 1973. Cut the same 485 homes into fifths by size — 97 in each, since 485 divides exactly — and the median year built runs 2004, 2001, 1998, 1998, 1978. Cut them the other way, by decade built, and the law appears from the other side: the pre-1950 homes in the bracket have a median of 1,824 square feet and the 2000s homes 1,504. Split the bracket by year of sale and it holds five times out of five: −0.27, −0.24, −0.46, −0.52, −0.45, with 2010 running to July where the record ends. We tried to break it and could not: drop the five largest houses in the file, which the data’s own author suggests doing, and it comes out at −0.44, stronger; keep only ordinary arm’s-length sales and it is −0.38. So decide which kind of household you are before you start viewing, because the market has already decided you cannot be both. In this bracket, space and newness are not two things you shop for. They are one thing you divide.

02The trick

Every number on that slide is right, and the crime is not on the panel — it is in the first line of the subhead, where the slide says which homes it kept. It filtered on price. Price is not a third variable to hold constant while you study the other two; price is what floor area and construction year jointly produce. Make a house bigger and it costs more; make it newer and it costs more. Fix the price and you have not controlled for anything — you have required the two to compensate for each other, because inside a $40,000 window a house can only be large by being old and only new by being small. That is why the bracket is a diagonal. Tabulate the median sale price of all 2,930 sales by size and by decade built and it climbs in both directions at once, from $97,200 for a small pre-war house to $356,383 for a large new one; the cells whose medians land between $180,000 and $220,000 form a stripe running from top-right to bottom-left, and a diagonal slice through two rising quantities slopes downward before a single house has been looked at. Take the filter off and the same 2,930 sales run +0.24 the other way: across the whole market bigger houses are newer, by 14.5 years per 1,000 square feet. The same five decades tell it plainly without a correlation anywhere — across the market the 2000s homes are 291 square feet larger than the pre-war ones; inside the bracket they are 320 square feet smaller. Two corners of the picture are what the budget actually removed. One hundred and fifty homes in the record are both big (2,000 sq ft and over) and new (built 2000 or later); their median price is $342,322 and exactly four of them reach the bracket. Two hundred and fifty are both small (under 1,200 sq ft) and old (built 1950 or earlier); their median is $99,500 and none reach it. Both kinds exist in numbers. One is priced out of the top of your budget and the other out of the bottom, and with those two corners gone the survivors have no choice but to trade off. Every check the slide offers as proof turns out to be a check on the filter rather than on the finding. Five years of agreement is what a filter looks like: it is applied every year. So is robustness to dropping outliers — the filter is still there afterwards, which is why the trade-off strengthens to −0.44 rather than weakening. Every one of the ten price deciles is negative — −0.16, −0.43, −0.41, −0.46, −0.26, −0.33, −0.29, −0.41, −0.34, −0.39 — so the bracket was not a lucky pick; any band tight enough to be a budget would have done it. And the strength is a dial with the filter’s hand on it: a band of ±$5,000 around $200,000 gives −0.56, ±$20,000 gives the slide’s −0.34, ±$60,000 gives −0.11, ±$100,000 gives +0.09, and no filter at all gives +0.24. This is Berkson’s paradox, named for the Mayo Clinic statistician who worked out in 1946 that two unrelated diseases come out looking like alternatives among hospital patients, because being admitted takes one or the other. It is worth keeping distinct from Simpson’s paradox, which it resembles exactly on the page and wants the opposite treatment. Simpson’s paradox as you usually meet it comes from a confounder — a cause the two variables share — and the fix there is to break the comparison out by it. This comes from a collider — an effect they share — and breaking the comparison out by that is what caused it. The test is a question about direction rather than about statistics: does the thing I am slicing on produce my two variables, or is it produced by them? (Both drawings are our own; the estate agency on the first is invented, and every figure on both is computed from the assessor’s file.)

03The fix

Take the filter off and draw it on. The honest chart here is all 2,930 sales with the bracket shaded — one picture in which the slice is visibly a diagonal and the two corners it deleted are sitting just outside it, still on the page, at median prices of $342,322 and $99,500. Everything else follows from that image: the whole-market line rising at +14.5 years per 1,000 square feet, the bracket line falling at −20.1, and the reader able to see that both are true and only one is about houses. Where the filter cannot be lifted — the customers who never converted were never recorded, the rejected applicants were never measured, the studies that missed significance were never published — the direction is still knowable in advance, and this is the most useful thing about the whole phenomenon: selecting on a common effect pushes the association between its causes negative, so you can say which way the number leans before you have seen it — and why the tighter the slice, the surer the trade-off looks. So say what a row had to do to be in the table, in a line on the chart, the way you would print a unit on an axis, and then ask whether the two columns you are about to compare helped it do that. If they did, some of the relationship you find between them is the entry rule. Two habits make it routine. Draw the two-by-two and look for the empty corner: a real trade-off thins out gradually at both extremes, while a filter deletes exactly the corner that could not clear the bar. And move the boundary: if narrowing the slice deepens the negative relationship and widening it brings the positive one back, you have found a dial rather than a finding. Keep the distinction from Simpson’s paradox sharp, because the two are identical on the page and opposite in treatment — adjust for a confounder, never for a collider, and settle which you have by asking whether the thing you sliced on causes your variables or is caused by them. And note what does not help: every robustness check on this exhibit — five separate years, outlier removal, arm’s-length sales only — leaves the filter in place, so all of them come back agreeing. And when the slice genuinely is the question, which it is for anyone with a fixed budget or a fixed shortlist, keep the finding attached to it: this is a true and useful fact about what $200,000 buys in Ames, and a false one about houses. The moment the budget moves, the law it looked like goes with it.