Distributions, and the Summaries That Hide
Them
Four hundred House districts drawn four ways, and
an average that describes almost nothing
Democracy’s Data
Last updated September 15, 2026 at 4:42p EDT
The American House district, you will read, is a fifty-fifty
proposition: the average district splits its vote almost evenly between
the parties. The average is real. In the 2024 election, the Democratic
share of the two-party vote across the 397 contested races had a mean —
the ordinary average, every share added up and divided by 397 — of
49.8% and a median — the middle value, the 199th share
once all 397 are sorted, with as many districts above it as below — of
49.6%. The two agree to within 0.20 of a point. So is
the typical district a coin flip?
The question matters because summaries are how citizens meet data.
Nobody hands you 397 numbers; they hand you one, and the one number
decides what you believe about whether your own vote is contested. This
brief draws the same column four ways: a box, a histogram, a cumulative
curve, and a stack of years. The same mean, it turns out, can sit on top
of opposite electorates.
The test
The data is the Democratic share of the two-party vote — the votes
cast for the Democrat and the Republican, with any other candidate left
out — in every contested U.S. House race from 1946 to 2024. The series
is two sources joined end to end at a year: the political scientist Gary
Jacobson’s file of House results from 1946 to 2014, which circulates
among researchers and has no public address, and for 2016 onward the
Clerk of the House’s Statistics of the Congressional Election,
the official tally the Clerk publishes after every election from what
each state certifies. Both are read through this book’s house-competition
chapter, which assembles them and documents its repairs; the figures
here read that assembled file rather than rebuilding it.
The question needs this source because it is about every district at
once. A summary hides its distribution only when you hold the whole
distribution. House returns are the one record with a number for every
district in every election: complete and official rather than sampled,
for the reasons this cluster’s
chapter on election returns lays out. The test itself is simple:
compute the standard summaries, then draw the actual 397 numbers several
ways and see which claims survive.
The data
One row of districts.csv is one House race in one
election year. A second table, by_year.csv, carries one row
per election with the summaries already computed. The columns of the
district file, by name:
| year |
the election year |
discrete, 1946 to 2024 |
| stcd |
a state-district code — the race’s identifier |
categorical, one per race |
| dv |
the Democratic share of the two-party vote, 0 to
100 |
continuous |
| uncontested |
whether only one candidate stood |
dichotomous |
dv is a share of the two-party vote,
not a share of all votes and not a margin. A district at 50 split evenly
between the two major parties; a district at 70 gave the Democrat seven
of every ten major-party votes. Four rows of 2024, from the two ends of
the column:
| 2004 |
88.7% |
the most Democratic contested seat |
| 1005 |
85.7% |
the second most |
| 4201 |
19.9% |
the second least |
| 2703 |
19.6% |
the least Democratic contested seat |
The cleaning decision that matters most is an absence. 38 of the 435
seats in 2024 were uncontested — only one major-party candidate stood —
so there is no second party and no share to compute. In those rows
dv is missing, and those seats are not among the 397 drawn
below. They are also, by definition, the least competitive seats in the
country. Every figure in this brief is therefore drawn on a set with the
extreme already removed, a deletion that is patterned rather than random
— the ordinary condition of a computed variable.
Here are the summaries of the 397 that remain:
| Mean |
49.8% |
| Median |
49.6% |
| Standard deviation |
15.7 |
| 25th percentile |
36.4% |
| 75th percentile |
62.0% |
| Within 5 points of even |
16.9% of races |
| Within 10 points of even |
37.3% of races |
Two of those rows need a word. The standard deviation is a measure of
spread: roughly, how far a typical district sits from the mean, so a
bigger number means the shares are more scattered. The 25th and 75th
percentiles are the values a quarter and three quarters of the way up
the sorted list, so half of all districts fall between them. Read the
table on its own and a picture assembles itself: a distribution — the
whole set of 397 shares and how they are spread along the line from 0 to
100 — centred a shade under even, spread about fifteen points either
way, with half the districts between 36.4 and 62.0. It is the picture a
well-behaved bell would produce — one hump, symmetric, most values
crowded near the middle — and every number in the table is correct.
Before you read on, commit to a number. On the
evidence above, how many of the 397 districts do you expect to
sit within five points of even? Write it down. The table gives
you everything you need to guess, and the guess is the point.
What it says about democracy
A box says one thing. Start with the form the
summary table implies.
Figure 1. The same 397 numbers, twice. A boxplot
above — a box spanning the middle half of the districts, from the 25th
to the 75th percentile, a bar at the median, and whiskers reaching out
to the extremes — and below it the districts themselves, one dot each,
jittered — nudged a little up or down at random — so they do not stack
on one line. This paired form fits the question because the argument is
that the second row contradicts the first, and only one shared axis
makes that visible. Hover any point for its district; the dashed line
carries the median down from the box.
The boxplot is not wrong: every mark on it is a correct summary of
the numbers below it. But the dashed line falls through a thin
patch in the strip. The box has quietly asserted that the
middle of the data is where the data is densest, and in this variable
that is false.
A histogram says another.
Figure 2. The same numbers again, as a histogram.
The axis is cut into equal stretches called bins, and each bar counts
the districts whose share falls inside one; the control sets the bin
width, and nothing else changes. At five points the shape is
unmistakable and it is not a bell — it is two hills with a
valley between them, and the median sits in the valley. There
are 22 districts in the five points below even, against 40 in the five
points below that and 45 in the five points above. Those two middle bins
are the answer to the prompt: 22 + 45 = 67 districts within five points
of even, which out of 397 is 16.9%.
Now widen the bins to twelve or fifteen. The valley closes, the hills
merge, and the picture becomes the single well-behaved hill the summary
table implied. No number changed. The only thing that
changed is a parameter nobody is obliged to report, and a histogram
published without its bin width is an argument with a step missing. The
two peaks sit near 35.6% and 55.5%, and a district at either peak is a
seat one party wins comfortably. The modal American House district — the
most common kind — is safe, and there are two kinds of it.
A cumulative curve answers the question you asked. A
histogram is a good way to see a shape and a bad way to answer how
many. The opening question, how many districts are close to even,
is a counting question, and there is a form that answers it exactly.
Figure 3. The same numbers as a cumulative curve.
For any share on the horizontal axis, the curve gives the percentage of
districts at or below it. Set the band and the figure reads the count
off the curve directly: at ±5 points it is 16.9% of
contested districts, and at ±10 points 37.3%.
Compare that with the number you wrote down. The summary table said
the middle half of districts ran from 36.4 to 62.0, a spread of 25.7
points, and it is easy to read that as the middle is crowded.
It is not. The curve climbs steeply on both flanks and flattens
through the middle, and a flat stretch on a cumulative curve is
exactly what an empty region looks like.
Five elections, stacked. The valley is often
explained as something recent, a product of the last two redistricting
cycles. One year cannot test that; a stack of years can — five elections
drawn the same way on the same axis, one above another, so that the eye
compares shapes rather than reading numbers.
Figure 4. Five elections, same bandwidth, same axis.
Each year is drawn as a density curve, a smoothed histogram, and the
bandwidth is how much smoothing was applied — the curve’s equivalent of
a bin width, held the same for every year. The dashed line is an even
split and the red dot is that year’s median. Hover a band for its
counts.
The valley is not new. It is there in 1976, when only 17.2% of
contested races fell within five points of even — a lower share
than in 2024. It was already visible to David Mayhew in the returns of
the early 1970s, who called the disappearance of close seats “the
vanishing marginals.” What has moved is not the hole but the
position of the whole distribution: in the 1970s and 1980s it sat well
to the Democratic side of even, and it has slid since. A reader shown
only 2024 would reasonably conclude the two-hill shape is a modern
artifact of redistricting. Five elections say otherwise, and the only
reason to believe the stack over the single year is that it contains
more of the record.
What this brief cannot tell you
It cannot show you the least competitive seats at
all. The 38 uncontested seats of 2024 have no two-party share,
so they are absent from every figure above. The valley would be no
shallower with them; the flanks would be taller.
It cannot name an honest form. The boxplot’s
quartiles — the 25th and 75th percentiles that bound its box — are
correct. The histogram is correct at every bin width, and the density
curves in Figure 4 use one bandwidth chosen once and applied everywhere,
which is a defensible choice and still a choice. The argument is not
that some charts deceive and others do not. It is that every
summary is a choice about what to discard, and the only defence
is to draw more than one and say which you chose.
A share is not a seat. Nothing here counts who won.
A district at 49% elected a Republican and a district at 51% elected a
Democrat, and this brief has treated them as neighbouring numbers, which
they are, and as similar outcomes, which they are not.
What you should have learned
- A mean of 49.8% and a median of 49.6% agree to within 0.20 of a
point, and both sit in the emptiest part of the distribution.
- The 2024 House is two hills, not a bell: peaks near 35.6% and 55.5%,
with only 16.9% of contested districts within five points of even.
- A histogram’s finding can appear and disappear with its bin width
while no number changes, which is why the bin width has to travel with
the chart.
- The valley is not a product of recent redistricting: 1976 put 17.2%
of contested races within five points of even, less than 2024.
- The 38 uncontested seats of 2024 are missing from every figure, and
their absence is patterned, not random.
Extensions
None of these is answered above, and every one can be answered from
the tables the Sources section hands over.
by_year.csv has mean and
median for every year alongside sd. Find the
years where mean and median are furthest apart. What is the distribution
doing there?by_year.csv also counts uncontested seats.
Follow that column across the series. When was the ballot emptiest, and
does it move with the within5 column or against it?by_year.csv carries within5 and
within10. Follow them from 1946 to 2024. Has the number of
genuinely close districts fallen, and by how much?districts.csv is one row per district per year, with
the share in dv. Pick a year this brief does not draw and
build its histogram in a spreadsheet. Then say what that year’s mean was
hiding.
Two stretch ideas point past the spreadsheet. Senate races have the
same two-party share and only 33-odd contests per cycle; pull one year’s
results from the Clerk’s Statistics and see whether so few
races can even have a valley. And find a published news chart of
“average district partisanship” and work out which of the four forms
above it silently chose.
Sources
Data
Academic literature
- Mayhew, David R. (1974). “Congressional Elections: The Case of the
Vanishing Marginals.” Polity 6(3): 295–317. — the first account
of the valley Figure 4 shows in 1976. https://www.jstor.org/stable/3233931
The data itself
Every figure above is drawn from these tables. They are plain CSV,
they open in a spreadsheet, and they are yours to take further.
by_year.csv, districts.csv
One more file is not data about the subject – it is how this chapter
checked its own numbers: facts.csv, the individual
numbers the text above quotes.
