The 61.5% of schools that hold 32.7% of the children

Dark navy briefing slide, masthead “Facilities briefing · US public schools · Enrollment size of school · School year 2021–22 · Census · not a sample”, headlined “Six in ten American public schools enroll fewer than 500 students”. A standfirst says this is a census rather than a survey — every public elementary and secondary school in the fifty states and the District of Columbia files to the federal Common Core of Data, 99,239 of them in 2021–22 — and that the publisher’s size distribution runs over the 95,839 that reported an enrollment above zero, the other 3,400 having reported none, those 95,839 schools holding 49,089,640 students between them. It adds that the axis starts at zero, every band is plotted at the publisher’s own boundaries, nothing is smoothed, indexed, annualized or rebased and there is no second axis, and that on that count the commonest school in the country holds 300 to 399 students, 61.5% of schools enroll fewer than 500, 32.0% fewer than 300 and very nearly one school in ten — 9.96% — fewer than 100, while only 9.0% reach 1,000 students and 0.34% reach 3,000. The main panel, labelled “Share of public schools, by enrollment size · 2021–22 · axis 0 to 16% · every school counted once”, is a column chart of thirteen blue bars with gridlines at 0, 4, 8, 12 and 16 percent, each bar labelled with its value: Under 100 at 10.0, 100–199 at 9.4, 200–299 at 12.6, 300–399 at 15.3 picked out in gold as the tallest, 400–499 at 14.2, 500–599 at 10.9, 600–699 at 7.7, 700–799 at 5.2, 800–999 at 5.7, 1,000–1,499 at 5.1, 1,500–1,999 at 2.0, 2,000–2,999 at 1.5 and 3,000+ at 0.3. The x-axis is captioned “Enrollment size of school · percent of the 95,839 reporting schools in each band”. A table headed “The distribution as drawn” lists every band with its exact percentage of schools and the school count that percentage implies: 9.96 and 9,546; 9.41 and 9,018; 12.62 and 12,095; 15.35 and 14,711; 14.15 and 13,561; 10.88 and 10,427; 7.71 and 7,389; 5.18 and 4,964; 5.73 and 5,492; 5.11 and 4,897; 2.02 and 1,936; 1.53 and 1,466; 0.34 and 326. A second table, “Small schools dominate the count”, gives cumulative shares — fewer than 100 at 9.96%, fewer than 200 at 19.37%, fewer than 300 at 31.99%, fewer than 500 at 61.49%, fewer than 800 at 85.26%, 1,000 or more at 9.00%, 2,000 or more at 1.87% and 3,000 or more at 0.34% — and notes beneath that the median school, the fiftieth of every hundred counted from the smallest, falls in the 400–499 band, and that mean enrollment per reporting school is 512.2 students. Down the right-hand side a card carries the figure “61.5%” over “Public schools enrolling fewer than 500 · 2021–22” and “Modal band 300–399 · mean 512.2 students”. A card headed “Read-out for the facilities board” says America is a nation of small schools, that the median school sits in the 400–499 band and the average school holds 512.2 students, so a site of roughly 500 is the country’s normal case rather than its small case. A “Planning note” card says a staffing model, a counselling ratio and a per-site technology package specified for a school of about 500 will fit the majority of American sites without modification, and that large-campus provision is the exception since fewer than one site in ten reaches 1,000 students. A card headed “Why a census, and what its base is” says these are not estimates, that every public school files to the Common Core of Data so there is no sample and no margin of error and the percentages are counts, and that the publisher computes them over the 95,839 schools reporting enrollment above zero while the 3,400 reporting none are outside the distribution. A “What we did not do” card lists a zero-based axis, every band plotted at the publisher’s own boundaries, nothing smoothed, indexed, annualized, projected or rebased, no second axis, no log scale and no break, and “one year, one country, one unit”. A “Basis of measurement” card names NCES, Digest of Education Statistics 2022, Table 216.40, with underlying data from the NCES Common Core of Data, 2021–22. Five ticked boxes run along the foot: “Zero-based axis”, “Every reporting school counted”, “Publisher’s own size bands”, “Census, not a sample” and “Base stated”. A footnote gives the source, notes that percentages are of the 95,839 schools out of 99,239 reporting enrollment above zero and sum to 99.99 through rounding, that the school counts in the table are those percentages applied to that base, and says the slide is a demonstration drawn by us in the manner of a facilities briefing, that the board, the read-out and the planning note are invented, and that every figure behind it is real. — Showing the misleading chart

The same thirteen bands on a cream background, kicked “The same 95,839 schools · counted a second time, by student · US public schools, 2021–22 · Nothing added · nothing dropped” and headlined “Six in ten schools are under 500. Three in ten students are.” A standfirst says nothing on the briefing was cropped, truncated, smoothed or given a second axis, that it was a census, the bands and the base were the publisher’s own and every percentage on it is correct, and that what sets the picture is the unit: the bars counted schools, and a school enters the count once whether it holds 45 children or 4,007. It notes that the publisher prints a second column on the same table — the same thirteen bands weighted by the students inside them — and that it is a different distribution: schools under 500 students are 61.49% of schools and 32.71% of students, schools under 100 are 9.96% of schools and 0.87% of students, schools of 1,000 or more are 9.00% of schools and 28.47% of students, and the average school holds 512.2 students while the school the average student attends holds about 916. The main panel is a paired column chart on the same zero-based axis running to 16%, a blue bar for share of schools beside a gold bar for share of students in each of the thirteen bands, with a key reading “Share of schools · Share of students · Same thirteen bands · same base of 95,839 schools · same zero-based axis”. Every bar is labelled: Under 100, 10.0 against 0.9; 100–199, 9.4 against 2.8; 200–299, 12.6 against 6.2; 300–399, 15.3 against 10.5; 400–499, 14.2 against 12.4; 500–599, 10.9 against 11.6; 600–699, 7.7 against 9.7; 700–799, 5.2 against 7.6; 800–999, 5.7 against 9.9; 1,000–1,499, 5.1 against 12.0; 1,500–1,999, 2.0 against 6.8; 2,000–2,999, 1.5 against 7.0; and 3,000+, 0.3 against 2.7 — so the blue bars are taller in the first five bands and the gold bars in the last eight, the gold peak sits one band to the right of the blue one, and the gold bars stay high across the whole right-hand half of the chart where the blue ones have fallen away. The x-axis is captioned “Enrollment size of school · percent of schools and percent of students in each band · the bands are the publisher’s own and are not equal in width”. A table headed “Band by band · and the average school size the two columns imply” gives each band’s percent of schools, percent of students, the ratio between them and the mean school size that ratio implies: 9.96 and 0.87, 0.09×, 45 students; 9.41 and 2.77, 0.29×, 151; 12.62 and 6.20, 0.49×, 252; 15.35 and 10.49, 0.68×, 350; 14.15 and 12.38, 0.87×, 448; 10.88 and 11.61, 1.07×, 547; 7.71 and 9.73, 1.26×, 646; 5.18 and 7.56, 1.46×, 748; 5.73 and 9.91, 1.73×, 886; 5.11 and 11.96, 2.34×, 1,199; 2.02 and 6.82, 3.38×, 1,729; 1.53 and 7.03, 4.59×, 2,353; and 0.34 and 2.66, 7.82×, 4,007. Beside it a panel headed “Running share, from the smallest school up” draws two cumulative curves against a log axis of school size marked 100, 200, 300, 500, 700, 1,000, 2,000 and 3,000, with y-gridlines labelled 0%, 25%, 50%, 75% and 100% and a dashed rule at 50%: a blue curve for schools crossing it at a marked point labelled “Median school ≈ 420” and a gold curve for students crossing it further right at a point labelled “Median student’s school ≈ 660”. Both curves run dashed past the 3,000 tick under a note reading “dashed: top band open-ended”. The panel is captioned “Enrollment size of school (log scale) · running share at or below”, and beneath it a line says half of American public schools hold fewer than about 420 students, half of American public-school students are in schools of more than about 660, and the gap between the two curves is the whole of the effect. Down the right-hand side, a card headed “Two averages, one census” tabulates the average school at 512.2 students, the average student’s school at about 916, the median school at about 420 and the median student’s school at about 660, adding that neither number is wrong, that they answer different questions and that only one of them is about children. A card headed “The arithmetic behind the gap” says weighting each unit by its own size raises the mean by exactly the variance divided by the mean, that the two averages agree only when every school is the same size and diverge fastest where the spread is widest — which is why the eight bands from 500 up are 38.50% of schools and 67.28% of students — and that this is not survivorship bias, since nothing dropped out, every school in the base is in both columns, and the selection happens at the moment of counting. A card headed “The same shape elsewhere” says to ask a university its average class size and then ask a student theirs, and to ask a transit agency the average gap between buses and then a passenger their wait, which is longer than half the gap unless the buses run to the second, adding that whenever containers differ in size, counting containers and counting contents give two distributions and the contents-weighted one is the larger. A card headed “What both charts still owe you” says the publisher’s bands are not equal in width — eight of 100, then 200, 500, 500, 1,000 and an open-ended top — and that both charts draw them as equal bars, so “the modal band” is not a statistic worth quoting off either, the curves being dashed past 3,000 for the same reason. A “Basis of measurement” card notes that both columns are printed side by side in NCES Digest 2022, Table 216.40, over the 95,839 schools of 99,239 reporting enrollment above zero, and that the mean-size column is derived by dividing each band’s students by its schools. A gold-ruled callout at the foot reads “Decide who the sentence is about, then count that,” and says that “most schools are small” and “most students are in small schools” are different findings which one bar chart cannot carry both of, so draw the pair as the publisher’s own table already does and label each bar with the unit it counts; where only one can be shown, choose the unit the claim is about, since a staffing model is about sites and a class-size promise is about children; say which you used, print both means, and read a long right tail as a warning that the two will differ by a lot. A footnote gives the source, notes that both percentage columns are as published and were computed by NCES over the 95,839 schools — of 99,239 — reporting enrollment above zero, that they sum to 99.99 through rounding and that those schools hold 49,089,640 students, and that the band mean sizes, the ratio column, the cumulative curves, the two medians and the ≈916 figure are computed from those two columns and that base. It explains that because the underlying data is banded the ≈916 figure is a floor rather than an exact value, since within-band variation can only push it up, and that the two medians are linear interpolations within their bands. It says both drawings are ours, that the briefing, the board, the read-out and the planning note on the first are invented, and that every figure on both is real. — Showing the misleading chart

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A facilities briefing on the size of American public schools: a census rather than a survey, the publisher’s own thirteen enrollment bands, zero-based axis, nothing smoothed or dropped. On that count the country is unmistakably a nation of small schools — 61.5% enroll fewer than 500 students and very nearly one in ten enrolls fewer than 100. The bars count schools, and a school enters the count once whether it holds 45 children or 4,007. The same table’s second column weights the same thirteen bands by the students inside them, and it is a different distribution: schools under 500 hold 32.7% of students, and the 9.0% of schools with 1,000 or more hold 28.5%. The average school has 512.2 students; the school the average student attends has about 916.

Number and percentage distribution of public elementary and secondary schools and enrollment, by level, type, and enrollment size of school — National Center for Education Statistics, Digest of Education Statistics 2022, Table 216.40; school year 2021–22, from the NCES Common Core of Data. Both percentage distributions are computed over the 95,839 schools, of 99,239, that reported enrollment above zero; those schools hold 49,089,640 students, 2022Added September 3, 2026

01The claim

A facilities briefing on one question — how big is an American public school? — answered from a census rather than a survey. Every public elementary and secondary school in the fifty states and the District of Columbia files to the federal Common Core of Data, so there is no sample here and no margin of error: 99,239 schools in the 2021–22 school year, of which the 95,839 that reported an enrollment above zero make up the publisher’s size distribution, the other 3,400 having reported none. Those 95,839 schools hold 49,089,640 students between them. The axis starts at zero. All thirteen bands are plotted, at the boundaries the publisher drew them. Nothing is smoothed, indexed, annualized, projected or rebased; there is no second axis, no log scale and no break in the scale anywhere. On that count the shape of American schooling is unambiguous, and it is a nation of small schools. The commonest school in the country holds 300 to 399 students. 61.49% of schools enroll fewer than 500; 31.99% enroll fewer than 300; and very nearly one school in ten — 9.96% — enrolls fewer than 100. Only 9.00% reach 1,000 students, 1.87% reach 2,000 and 0.34% reach 3,000. The median school falls in the 400–499 band and mean enrollment per reporting school is 512.2 students. Read-out for the facilities board: a site of roughly 500 is the country’s normal case rather than its small case. Planning note: a staffing model, a counselling ratio and a per-site technology package specified for a school of about 500 will fit the majority of American sites without modification, and large-campus provision is the exception, since fewer than one site in ten reaches 1,000 students.

02The trick

Every count needs a unit, and this one counts schools. A school enters the tally once — the same once — whether it holds 45 children or 4,007, so the picture is a fair description of the country’s stock of buildings and not of anybody’s schooling. The reader is never a school. The reader is inside one. Count the children instead and the same thirteen bands, from the same census, on the same base, rearrange themselves: the schools under 500 students that are 61.49% of schools hold 32.71% of students; the schools under 100 that are 9.96% of schools hold 0.87% of students; and the schools of 1,000 or more that are 9.00% of schools hold 28.47% of students. Read it from the other end and it is starker still: the eight bands from 500 students up are 38.50% of the schools and 67.28% of the children. The median school holds about 420 students and the median student’s school about 660. The average school holds 512.2 students and the school the average student attends holds about 916. Nothing has been added, removed, rescaled or reweighted by us to produce that. The second column is printed on the publisher’s own table, one column to the right of the first, and the exhibit simply draws both. What is doing the work is that the units differ in size and the count does not know it. Weighting each unit by its own size raises the mean by exactly the variance divided by the mean, so the two answers coincide only when every school is the same size and separate fastest where the spread is widest — which is why the long right tail matters so much here. It is a sliver of the count and a slab of the population, and a chart of schools is obliged to draw it as negligible. This is not survivorship bias and it is not a self-selected sample: nothing dropped out, nothing walked in, and every school in the base is present in both columns. Nor is it a missing denominator, because both figures are already rates — the argument is over what they are rates of. The selection happens at the moment of counting, in the choice of what one bar represents. (Both drawings are ours; the briefing, the board, the read-out and the planning note on the first are invented, and every figure behind them is real.)

03The fix

Decide who the sentence is about, then count that. Here the honest chart is barely a redraw — it is the same thirteen bands on the same zero-based axis with a second series beside the first, because the publisher already did the work and printed the student column next to the school column. Drawn as a pair, the blue bars lead in the first five bands and the gold bars in the last eight, and the gold bars stay high across the whole right-hand half of the chart where the blue ones have fallen away. The gap between the two distributions is itself the finding: it costs one extra series and it is more informative than either bar alone. Print both means where the reader can see them together, since the container mean and the contents-weighted mean are a matched pair and the distance between them is the spread in plain sight — 512.2 against about 916 is a ratio of nearly 1.8, and a reader who has seen the two numbers side by side will not confuse them again. Say in the caption which unit the bars are, the way you print a unit on an axis. Where only one chart can be shown, choose the unit the decision actually runs on, because both are legitimate and the same census honestly supports two charts for two different meetings: a maintenance budget, a heating bill and a site inspection rota really are about buildings, and a class-size promise, a counselling ratio and a per-pupil funding formula really are about children. What is not legitimate is the swap — reading the school chart and writing a sentence about children — which is what the briefing’s planning note does when it infers from “fewer than one site in ten reaches 1,000 students” that large-campus provision is an edge case. Nearly three students in ten are in one. Two things both of these charts still owe the reader, and the honest one says so on its face. The publisher’s bands are not equal in width — eight of 100, then 200, 500, 500, 1,000 and an open-ended top — and both charts draw them as equal bars, which is the uneven-intervals trick riding along on the same picture: it is why “the modal band” is not a statistic worth quoting off either chart, and why the running-share curves are drawn dashed past 3,000, since nothing on the table says where that band ends. The shares themselves are unaffected, which is why the exhibit still works; the shape of the histogram is not, which is why it is worth saying. The same test catches this everywhere it lives, which is nearly everywhere: ask a university its average class size and then ask a student theirs; ask a transit agency the average gap between buses and then ask a passenger their wait, which is longer than half the gap unless the buses run to the second, because a long gap is a bigger target to land in; ask how many employees the average firm has and then how large a firm the average employee works for. Watch for the average of averages in particular, which quietly picks the container as its unit while looking like an average of everything, and treat a long right tail as advance warning that the two answers will be far apart. And keep the distinction from the neighbouring techniques sharp, because the remedies differ. Survivorship bias is about cases that left the dataset before you looked and self-selection about cases that stepped forward, while nothing here is missing at all. The missing denominator is the nearest miss of the three and the one worth holding on to: that is a count that should have been a rate, whereas both of these figures are already rates and the whole argument is over what they are rates of. Both counts here are complete. Only one of them is about children.