Part I Lab 3

Lab: How Much of a State Ranking Is Supported?

A league table of state estimates looks more definitive than the data allow. Replicate weights show by how much.

DataACS 2024 · 80 SDR replicate weights

Is our state worse than the one next door?

A reporter calls, or a dean does, with the same question: is our state worse than the one next door? The state uninsured rates came out this morning, your state sits a few places below its neighbor in the table, and you are the person on campus who works with the American Community Survey (ACS). You know that both rates are estimates. What you have to decide before you answer is whether the survey can tell the two places apart, and the answer is on the public-use file, in 80 weight columns most users never open.

Each September the Census Bureau releases the previous year's ACS, and within a day the state uninsured rates are a league table. The 2024 table for working-age adults reads: Texas first, Georgia second, Oklahoma third, Arizona tenth, Massachusetts last. Texas has the highest share of adults aged 19 to 64 without health insurance, 21.6%, and Massachusetts the lowest, 3.7%, against a national rate of 11.3% (95% CI 11.2%–11.4%). A governor's office in a state that dropped three places wants to know why. An advocacy group in a state that climbed two has the press release drafted.

Every step down the table is a claim about neighbors: this state's rate is higher than the next one's. There are 50 such neighboring pairs. With textbook standard errors, 7 of them differ at the 90 percent level. With the standard errors the ACS was built to support, 2 do. The estimated rates have not moved. The evidence for separating neighboring places has, and a story about who moved up or down is mostly a story about pairs the data cannot tell apart.

Predict first

Only a handful of neighboring pairs survive the design-based test. Before reading on, guess where in the table they sit: at the two ends, in the crowded middle, or scattered throughout. And guess whether a state's sample size or its rate decides the matter.

The rates describe adults aged 19 to 64 in the civilian noninstitutionalized population: everyone except residents of institutions such as prisons and nursing homes, and members of the Armed Forces. This is the Medicaid-relevant age group. Children have their own Medicaid and CHIP eligibility rules, nearly everyone 65 and older has Medicare, and the adults in between are the ones whose coverage depends most on each state's Medicaid choices. The data are the ACS 2024 1-year public-use sample from IPUMS (Ruggles et al. 2025), restricted to the Census Bureau's universe for health insurance: 1,871,404 adults aged 19 to 64, from 3,250 in Wyoming to 226,532 in California. Every test in this lab is a per-comparison test at the 90 percent level, the Census Bureau's convention for ACS comparisons. A set of such tests does not certify a whole ranking. It says which places the data can tell apart.

The naive test

The quickest way to attach uncertainty to each rate is the textbook standard error for a proportion from a simple random sample, $\SE_{\text{iid}} = \sqrt{\hat p(1-\hat p)/n}$ , followed by a test for every pair of states. At the 90 percent level, that analysis finds 1,174 of the 1,275 pairs significantly different, including 7 of the 50 pairs that sit next to each other in the ranking.

The analysis looks defensible and is wrong. It treats the ACS as a simple random sample of independent people. The ACS samples addresses and records everyone living at each one, so members of a household, who often share an insurance status, enter the sample together. Its weights also vary widely across respondents, and unequal weights add variance that the iid formula never sees.

Replicate weights: repeated sampling, pre-packaged

Predict first

The naive standard error for Texas is 0.10 percentage points. Is the design-based standard error about the same, a little larger, or more than half again as large?

A standard error answers a repeated-sampling question: draw another sample under the same design, weight it the same way, and how far does the estimate move? With the public-use file come 80 alternative sets of person weights (REPWTP1 to REPWTP80 in IPUMS). Each starts from a systematic perturbation of the sample's base weights and then goes through the full weighting again, with the nonresponse adjustments and the calibration to population controls redone for every replicate. Re-estimating a statistic with each set shows how far it moves.

That is why replicates travel with the data. The final weight is itself an estimate, and only the producer can repeat its own weighting on a perturbed sample. The spread of the replicate estimates therefore includes the variance that weight construction adds, and the variance that calibration removes. An ordinary standard-error formula applied to the final weight column cannot recover either, because the column alone does not reveal how it was built.A calibration-aware linearization could, but it needs the calibration's control totals and the weighting areas they apply to, which are not on the public file.

The ACS uses successive difference replication (SDR), developed at the Census Bureau for state and county income and poverty estimates (Fay and Train 1995). It is one of several replication procedures, alongside balanced repeated replication (BRR), Fay's variant of BRR, the jackknife, and the bootstrap. They form replicates differently and convert the spread into a variance with different constants; calling them all "the bootstrap" blurs procedures that are not interchangeable. For the ACS (U.S. Census Bureau 2024):

\widehat{\Var}(\hat\theta) = \frac{4}{80}\sum_{r=1}^{80}\bigl(\hat\theta_r - \hat\theta\bigr)^2, \qquad \text{df} = 79,

where $\hat\theta$ is computed with the final weight $w_i$ (PERWT) and $\hat\theta_r$ with the replicate weight $w_i^{(r)}$ (REPWTP $r$ ). The ACS publishes 90 percent margins of error (MOE), $1.645 \times \SE$ .

For Texas the replicate standard error is 0.17 percentage points, more than half again the naive 0.10. Across states the ratio of the two runs from 1.29× in Oklahoma to 2.05× in Minnesota, with a median of 1.59×.The ratio belongs to the estimate, not to the survey, as Chapter 3 argues for the design effect. For the national rate it is 2.57×, larger than for any state. Two states' replicate deviations are only slightly correlated, 0.04 on average, but the national estimate adds up every pair: those covariances supply 62% of its variance and make its standard error 1.62× what independent state estimates would give. A standard error that uses the weights but still treats people as independent closes much of that gap, yet it misses in both directions: depending on the state, the replicate standard error is between 0.84× and 1.26× the weights-only one. Figure 3L.1 shows each state's rate with its 90 percent margin of error.

State uninsured rates with 90% margins of error

Adults 19–64, 2024. Bars are ±1.645 SDR standard errors, the ACS convention.

Show data

Estimate for	Estimate	95% CI	n
Texas	22%	21%–22%	161,135
Georgia	16%	16%–17%	61,745
Oklahoma	16%	16%–17%	20,824
Nevada	15%	15%–16%	17,864
Florida	15%	15%–16%	112,312
New Mexico	15%	14%–16%	10,736
Alaska	14%	13%–16%	3,543
Mississippi	14%	13%–15%	15,542
Wyoming	14%	12%–15%	3,250
Arizona	14%	13%–14%	39,422
Tennessee	14%	13%–14%	41,223
South Carolina	13%	13%–14%	30,118
Arkansas	13%	12%–14%	16,841
Alabama	12%	12%–13%	28,110
Montana	12%	11%–13%	5,695
North Carolina	12%	12%–12%	61,230
Idaho	12%	11%–13%	10,907
Kansas	12%	11%–12%	15,843
Louisiana	11%	11%–12%	24,025
South Dakota	11%	10%–12%	4,801
New Jersey	11%	10%–11%	53,215
Nebraska	11%	10%–12%	10,556
Utah	11%	10%–11%	20,219
Missouri	10%	10%–11%	34,762
Colorado	10%	10%–11%	35,502
Indiana	10%	10%–10%	38,866
Illinois	10%	10%–10%	72,142
Kentucky	10%	9%–10%	25,565
Virginia	9%	9%–10%	48,887
Delaware	9%	8%–10%	4,942
Ohio	9%	9%–9%	66,038
Washington	9%	9%–9%	46,221
West Virginia	9%	8%–10%	9,582
Connecticut	8%	8%–9%	20,919
Maryland	8%	8%–9%	34,942
California	8%	8%–8%	226,532
Iowa	8%	8%–9%	17,637
Maine	8%	7%–9%	7,645
Pennsylvania	8%	8%–8%	71,843
North Dakota	8%	6%–9%	4,240
Oregon	8%	7%–8%	24,472
Wisconsin	7%	7%–8%	32,996
Minnesota	7%	7%–8%	31,455
Michigan	7%	7%–7%	54,833
New York	7%	7%–7%	119,434
New Hampshire	7%	6%–7%	7,750
Rhode Island	6%	6%–7%	5,696
Vermont	6%	5%–7%	3,631
District of Columbia	6%	5%–7%	4,365
Hawaii	5%	4%–5%	7,827
Massachusetts	4%	3%–4%	43,524

Figure 3L.1.Through most of the ranking, neighboring states' 90 percent margins of error overlap; Texas at the top and Massachusetts at the bottom stand clear.Source: IPUMS USA, ACS 2024 1-year; PERWT and REPWTP1-80; SDR variance (4/80, df 79).

Which differences are distinguishable

Predict first

Two states' 90 percent margins of error overlap. Can the difference between them still be statistically significant at the 90 percent level?

It can. A difference between two states is itself an estimate, and it gets the same treatment: compute it on every replicate and take the SDR spread of the replicate differences.

\widehat{\Var}(\hat\theta_a-\hat\theta_b) = \frac{4}{80}\sum_{r=1}^{80}\Bigl[\bigl(\hat\theta_{a,r}-\hat\theta_{b,r}\bigr)-\bigl(\hat\theta_a-\hat\theta_b\bigr)\Bigr]^2

For independent estimates with equal standard errors, the standard error of the difference is $\sqrt{2}$ times one of them, not twice, so overlapping intervals are not a test. The replicate formula also carries the covariance between the two estimates without being asked. For two different states the covariance is small: swapping in the independence formula $\sqrt{\SE_a^2 + \SE_b^2}$ changes the verdict for 11 of the 1,275 pairs. It matters when one estimate contains the other. California minus the nation has a replicate standard error of 0.089 percentage points; treating the two as independent gives 0.103.

With the replicate standard error of each difference, 1,102 pairs are distinguishable, against 1,174 in the naive analysis. The totals hide where the change is. Most pairs sit far apart in the ranking and differ under any method, while a league table's claims are about neighbors. Of the 50 adjacent pairs, 2 are distinguishable, not 7 (Figure 3L.2). One place apart, 4% of pairs are distinguishable (14% under the naive test), and every pair 11 or more places apart is distinguishable (under the naive test, every pair 6 or more apart).

How far apart must two states be in the ranking to be distinguishable?

Share of state pairs a given number of places apart whose difference is significant at the 90% level.

Replicate SE of the difference (SDR)
Naive iid SEs, independence assumed

Show data

Series	Distance in rank between the two states	Pairs distinguishable
Replicate SE of the difference (SDR)	1	4%
Replicate SE of the difference (SDR)	2	20%
Replicate SE of the difference (SDR)	3	33%
Replicate SE of the difference (SDR)	4	57%
Replicate SE of the difference (SDR)	5	67%
Replicate SE of the difference (SDR)	6	78%
Replicate SE of the difference (SDR)	7	91%
Replicate SE of the difference (SDR)	8	91%
Replicate SE of the difference (SDR)	9	100%
Replicate SE of the difference (SDR)	10	98%
Replicate SE of the difference (SDR)	11	100%
Replicate SE of the difference (SDR)	12	100%
Naive iid SEs, independence assumed	1	14%
Naive iid SEs, independence assumed	2	39%
Naive iid SEs, independence assumed	3	69%
Naive iid SEs, independence assumed	4	77%
Naive iid SEs, independence assumed	5	96%
Naive iid SEs, independence assumed	6	100%
Naive iid SEs, independence assumed	7	100%
Naive iid SEs, independence assumed	8	100%
Naive iid SEs, independence assumed	9	100%
Naive iid SEs, independence assumed	10	100%
Naive iid SEs, independence assumed	11	100%
Naive iid SEs, independence assumed	12	100%

Figure 3L.2.Neighbors in the ranking are rarely distinguishable; naive standard errors make many more of them look different.Source: IPUMS USA, ACS 2024 1-year; PERWT and REPWTP1-80; SDR variance (4/80, df 79).

The pairwise tests also give each state a range of ranks compatible with them (Figure 3L.3). The highest compatible rank is one plus the number of states whose rate is significantly higher; the lowest is 51 minus the number whose rate is significantly lower. Texas is first and Massachusetts last under every test. That answers the opening prediction: the distinguishable neighbors are at the ends, where the rates are far from the pack. Indiana, the median state, is compatible with any rank in 21–30 and is distinguishable from 41 of the other 50 (the naive analysis says 46). Only 6 states remain in the top ten under every ordering compatible with the tests; 13 are in it under at least one. These ranges summarize a set of per-comparison tests. They are not simultaneous confidence intervals for ranks, which would need a joint method.

Uninsured rate among adults 19–64 by state, 2024, with replicate-based uncertainty

Rank 1 = highest rate. MOE is the ACS 90% margin of error (1.645 × SDR SE). “Ranks” are the positions consistent with every pairwise 90% test.

Table of 51 states ranked by uninsured rate among adults 19 to 64, with 90% margins of error, naive and replicate standard errors, their ratio, the number of other states each is statistically distinguishable from, and the range of ranks consistent with the pairwise tests.
Rank	State	Uninsured	90% MOE (±)	SE, naive iid	SE, SDR	SDR ÷ iid	Distinguishable from (of 50)	Ranks	n
1	Texas	21.6%	0.3%	0.10%	0.17%	1.64×	50	1	161,135
2	Georgia	16.5%	0.4%	0.15%	0.22%	1.49×	49	2–3	61,745
3	Oklahoma	16.1%	0.5%	0.25%	0.33%	1.29×	47	2–5	20,824
4	Nevada	15.5%	0.7%	0.27%	0.42%	1.55×	45	3–8	17,864
5	Florida	15.4%	0.3%	0.11%	0.16%	1.45×	47	4–7	112,312
6	New Mexico	15.0%	0.9%	0.34%	0.57%	1.66×	44	3–9	10,736
7	Alaska	14.2%	1.5%	0.59%	0.91%	1.55×	41	4–13	3,543
8	Mississippi	14.2%	0.7%	0.28%	0.41%	1.48×	45	6–11	15,542
9	Wyoming	13.8%	1.5%	0.61%	0.92%	1.52×	40	5–15	3,250
10	Arizona	13.8%	0.5%	0.17%	0.29%	1.69×	45	7–12	39,422
11	Tennessee	13.6%	0.4%	0.17%	0.24%	1.40×	44	7–13	41,223
12	South Carolina	13.1%	0.5%	0.19%	0.29%	1.50×	46	9–13	30,118
13	Arkansas	13.0%	0.7%	0.26%	0.45%	1.73×	42	8–16	16,841
14	Alabama	12.4%	0.5%	0.20%	0.31%	1.60×	44	12–18	28,110
15	Montana	12.1%	1.1%	0.43%	0.65%	1.50×	42	12–20	5,695
16	North Carolina	12.1%	0.4%	0.13%	0.23%	1.75×	45	14–19	61,230
17	Idaho	12.0%	0.8%	0.31%	0.49%	1.57×	43	13–20	10,907
18	Kansas	11.7%	0.7%	0.26%	0.45%	1.75×	43	14–21	15,843
19	Louisiana	11.4%	0.5%	0.21%	0.28%	1.38×	45	16–21	24,025
20	South Dakota	10.9%	1.3%	0.45%	0.79%	1.76×	36	15–29	4,801
21	New Jersey	10.9%	0.4%	0.13%	0.24%	1.79×	46	20–24	53,215
22	Nebraska	10.7%	0.9%	0.30%	0.53%	1.77×	40	18–28	10,556
23	Utah	10.6%	0.6%	0.22%	0.39%	1.80×	44	20–26	20,219
24	Missouri	10.4%	0.4%	0.16%	0.23%	1.41×	43	20–27	34,762
25	Colorado	10.2%	0.5%	0.16%	0.27%	1.71×	42	21–29	35,502
26	Indiana	9.9%	0.4%	0.15%	0.23%	1.53×	41	21–30	38,866
27	Illinois	9.8%	0.3%	0.11%	0.19%	1.70×	43	23–30	72,142
28	Kentucky	9.7%	0.5%	0.18%	0.28%	1.51×	43	24–31	25,565
29	Virginia	9.3%	0.4%	0.13%	0.23%	1.75×	45	28–33	48,887
30	Delaware	9.3%	1.1%	0.41%	0.69%	1.67×	34	22–38	4,942
31	Ohio	9.2%	0.3%	0.11%	0.18%	1.64×	46	29–33	66,038
32	Washington	9.1%	0.3%	0.13%	0.19%	1.40×	46	29–33	46,221
33	West Virginia	9.1%	0.8%	0.29%	0.47%	1.59×	39	26–37	9,582
34	Connecticut	8.5%	0.5%	0.19%	0.32%	1.64×	43	32–39	20,919
35	Maryland	8.4%	0.4%	0.15%	0.25%	1.70×	43	32–39	34,942
36	California	8.3%	0.1%	0.06%	0.08%	1.44×	43	32–39	226,532
37	Iowa	8.1%	0.5%	0.21%	0.30%	1.46×	41	33–42	17,637
38	Maine	8.0%	0.8%	0.31%	0.47%	1.53×	38	32–44	7,645
39	Pennsylvania	7.9%	0.3%	0.10%	0.17%	1.74×	45	37–42	71,843
40	North Dakota	7.7%	1.3%	0.41%	0.77%	1.87×	36	34–48	4,240
41	Oregon	7.7%	0.4%	0.17%	0.23%	1.33×	43	37–44	24,472
42	Wisconsin	7.5%	0.5%	0.14%	0.28%	1.97×	40	37–47	32,996
43	Minnesota	7.3%	0.5%	0.15%	0.30%	2.05×	41	39–48	31,455
44	Michigan	7.2%	0.3%	0.11%	0.17%	1.58×	41	39–48	54,833
45	New York	7.1%	0.2%	0.07%	0.12%	1.59×	43	41–48	119,434
46	New Hampshire	6.7%	0.8%	0.28%	0.46%	1.61×	42	41–49	7,750
47	Rhode Island	6.5%	0.8%	0.33%	0.48%	1.48×	43	42–49	5,696
48	Vermont	6.4%	1.0%	0.41%	0.63%	1.54×	42	41–49	3,631
49	District of Columbia	5.8%	1.0%	0.35%	0.61%	1.72×	46	46–50	4,365
50	Hawaii	4.7%	0.7%	0.24%	0.40%	1.68×	49	49–50	7,827
51	Massachusetts	3.7%	0.2%	0.09%	0.14%	1.53×	50	51	43,524

Figure 3L.3.Only the ends of the league table hold their places; in the middle, a state's rank is a range.Note: Civilian noninstitutionalized adults 19–64 (IPUMS approximation). Pairwise tests use the SDR standard error of each difference (replicate covariance), |difference|/SE > 1.645, per comparison.Source: IPUMS USA, ACS 2024 1-year; PERWT and REPWTP1-80; SDR variance (4/80, df 79).

These are per-comparison tests, the ACS convention, and across 1,275 comparisons some false positives are expected. Holding the chance of any false positive to 10 percent with a Bonferroni correction leaves 848 distinguishable pairs.

Worse at what? The same states, a second table

Predict first

Restrict the same file to adults aged 55 to 64, the group closest to Medicare, and rank the states again. Will the order stay roughly the same, and will more or fewer neighboring pairs be distinguishable?

The dean's question has a hidden clause: worse at what, for whom? A ranking belongs to an estimand. Change the population and the table changes, and it changes by more than the uncertainty in either table would suggest. Among adults 55 to 64 the national uninsured rate is 7.4% (95% CI 7.3%–7.5%), and the same code, universe, and replicate variance give a second table from 460,897 respondents. Texas is still first and Massachusetts still last. In between, 18 states move five or more places and 7 move ten or more (Figure 3L.4). Delaware and New Jersey move the most, 14 places each. Delaware is 30th among all working-age adults, with an uninsured rate of 9.3% (95% CI 7.9%–10.6%), and 44th among adults 55 to 64, at 4.8% (95% CI 3.4%–6.2%). Its rank ranges in the two tables, 22–38 and 33–48, barely touch. That is not sampling noise moving one number. It is two different questions with two different answers, and a state can be middling at one and near the bottom at the other.

The same states, two tables: rank among adults 19–64 and among adults 55–64

Rank 1 = highest uninsured rate. Highlighted states move ten or more places.

Show data

Row	Rank, adults 19–64	Rank, adults 55–64	Change
Texas	1	1	0
Georgia	2	7	+5
Oklahoma	3	4	+1
Nevada	4	5	+1
Florida	5	6	+1
New Mexico	6	2	−4
Alaska	7	3	−4
Mississippi	8	13	+5
Wyoming	9	10	+1
Arizona	10	8	−2
Tennessee	11	11	0
South Carolina	12	14	+2
Arkansas	13	17	+4
Alabama	14	15	+1
Montana	15	12	−3
North Carolina	16	19	+3
Idaho	17	9	−8
Kansas	18	20	+2
Louisiana	19	18	−1
South Dakota	20	26	+6
New Jersey	21	35	+14
Nebraska	22	22	0
Utah	23	16	−7
Missouri	24	21	−3
Colorado	25	23	−2
Indiana	26	32	+6
Illinois	27	33	+6
Kentucky	28	24	−4
Virginia	29	28	−1
Delaware	30	44	+14
Ohio	31	29	−2
Washington	32	31	−1
West Virginia	33	36	+3
Connecticut	34	45	+11
Maryland	35	42	+7
California	36	34	−2
Iowa	37	39	+2
Maine	38	25	−13
Pennsylvania	39	41	+2
North Dakota	40	27	−13
Oregon	41	30	−11
Wisconsin	42	48	+6
Minnesota	43	43	0
Michigan	44	38	−6
New York	45	46	+1
New Hampshire	46	47	+1
Rhode Island	47	49	+2
Vermont	48	40	−8
District of Columbia	49	37	−12
Hawaii	50	50	0
Massachusetts	51	51	0

Figure 3L.4.Texas and Massachusetts hold the ends of both tables, but seven states move ten or more places between the working-age ranking and the ranking for adults 55 to 64: a rank is a property of an estimand, not of a state.Note: Both tables use the same universe (civilian noninstitutionalized), estimator, and replicate variance; only the age band changes. The 55–64 samples are smaller (from 593 in District of Columbia), so its intervals are wider and fewer pairs are distinguishable.Source: IPUMS USA, ACS 2024 1-year; PERWT and REPWTP1-80; SDR variance (4/80, df 79).

The second table is also less able to separate its own neighbors. The smallest state sample falls to 593, in the District of Columbia, and the largest coefficient of variation among the 51 rates is 0.26, close to the level at which producers suppress estimates. The replicate standard error is a median 1.40× the naive one, and it distinguishes 858 of the 1,275 pairs, down from 1,102 in the working-age table, and 2 of the 50 adjacent pairs. The naive test still claims 975 pairs and 2 adjacent ones; here its error is spread over the whole table rather than concentrated among neighbors. So the honest answer to the dean has two parts: state which population the ranking describes, and then say whether the two places differ under the replicate test in that table, which for neighbors they usually do not.

Re-rank on every replicate

Predict first

Wyoming ranks ninth and Arizona tenth among working-age adults. If you recompute the ranking with each set of replicate weights, which of the two stays in the top ten more often?

Re-rank the states on each replicate

Uninsured rate, civilian noninstitutionalized adults 19–64, 2024

View

RankUncertaintyTop-k frequency

A league table of point estimates. It looks definitive; the other two views ask how much of the order the data support.

What is held fixed. The published estimates and their 80 replicate estimates. The views change only how uncertainty is summarized; nothing is re-estimated.

Each replicate yields all 51 state rates, so it yields a complete ranking. The widget's top-k view re-ranks the states on every replicate and counts how often each lands in the top ten; its uncertainty view marks which states differ from any state you pick. One adjustment comes first. A single SDR replicate deviation $\hat\theta_r-\hat\theta$ carries only a quarter of the sampling variance (the formula multiplies the sum of 80 squared deviations by $4/80$ ), so the widget stretches every deviation by $\sqrt{0.05 \times 80} = 2$ before ranking.

Arizona stays in the top ten on 74 of 80 rescaled replicates; Wyoming on only 50 of 80. The two rates are equal to one decimal place, but Wyoming's rests on 3,250 respondents and Arizona's on 39,422: the sample size, not the rate, decides. Tennessee, eleventh, makes the top ten on 47 of 80. Across all replicates, 6 states are in the top ten every time and 13 at least once: the same states the pairwise tests identify.

Caution

Three cautions on replicate counts

Replicates approximate the spread of estimates across repeated samples; they are not posterior draws, and a top-ten frequency is not the probability that a state belongs in the top ten. Stretching each deviation by two matches the variance of each state's rate; it does not make the re-ranked replicates a validated sampling distribution for a rank, which is not a smooth function of the rates. And 80 replicates make a coarse distribution, whose frequencies move in steps of one-eightieth. Read the counts as a sensitivity summary under structured perturbations, not as probabilities of membership.

Doing it yourself

With the replicate weights joined on SAMPLE, SERIAL, and PERNUM, the hand computation takes three lines:

python

# one state's adults 19-64: y = uninsured (0/1), w = PERWT, W = REPWTP1-80 (n x 80)
theta = (w * y).sum() / w.sum()
theta_r = (W * y[:, None]).sum(axis=0) / W.sum(axis=0)
se = np.sqrt(4 / 80 * ((theta_r - theta) ** 2).sum())  # MOE = 1.645 * se

In R, the survey package supplies the SDR constant (Lumley 2010), and svyby with covmat = TRUE keeps the replicate covariance for contrasts:

des <- svrepdesign(data = d, weights = ~PERWT, repweights = "REPWTP[0-9]+",
                   type = "successive-difference", mse = TRUE)
est <- svyby(~uninsured, ~STATEFIP, des, svymean, covmat = TRUE)
svycontrast(est, c("48" = 1, "13" = -1))  # Texas minus Georgia

Setting mse = TRUE centers the squared deviations on the full-sample estimate, as the Census formula does. The second table is subset(des, AGE >= 55 & AGE <= 64) followed by the same two lines; subsetting a replicate design keeps all 80 weight columns, so the domain's variance is right without further thought.

Caution

svy ignores declared replicate weights unless told otherwise

With replicate weights declared, the default mean() call in svy 0.28 silently ignores them. For Texas it returns a standard error of 0.14 percentage points on 161,134 degrees of freedom, a linearization that treats every person as an independent draw. Passing method="replication" and variance_center="estimate" returns 0.17 percentage points on 79 degrees of freedom, identical to the hand computation. Assert the degrees of freedom:

python

s = svy.Sample(d, design=svy.Design(
    wgt="PERWT", rep_wgts=svy.SdrWgts(prefix="REPWTP", n_reps=80)))
est = s.estimation.mean("uninsured", method="replication",
                        variance_center="estimate")
assert est.to_polars()["df"][0] == 79  # the default call reports n - 1

An incorrect shortcut also circulates: clustering standard errors on PUMA, the public-use geography, instead of using the replicates. PUMAs are not units of the ACS sample design, and a cluster-robust standard error computed on them reproduces neither the address-level sampling nor the weighting.

Box

Validation

Three checks stand behind these numbers. R's survey package, reading the parquet files and joining the replicate weights on its own, reproduces the estimates, their standard errors, and the standard errors of the differences for eight states and three pairs to within 10⁻¹⁵, and does the same for the second table's estimates, standard errors, and the Delaware–New Jersey difference. The svy replication path agrees as well. The same code also reproduces an external benchmark, the Census Bureau's 2024 PUMS verification file: summing PERWT gives the published U.S. population of 340,110,990, 116 of 116 published totals (by sex, age group, and group-quarters type, for the nation and six states) match exactly, and 116 of their standard errors match to the published integer. Those PUMS standard errors, not the full-sample margins on data.census.gov, are the right target: the public-use file is a subsample of the ACS, so its standard errors are larger.

Analysis Ledger

Which state differences in the uninsured rate are distinguishable? (ACS 2024)

Target population: Civilian noninstitutionalized adults aged 19 to 64 in each of the 50 states and DC in 2024. IPUMS approximation: residents of institutional group quarters and members of the Armed Forces are dropped; noninstitutional group quarters (college dorms, shelters) stay in.
Estimand: For each state, the share of that population with no health insurance coverage at the time of interview; for every pair of states, the difference between their shares.
Estimator: Weighted share within each state (Hajek ratio): sum of PERWT x uninsured over sum of PERWT; pairwise differences of those shares.
Explicit weights: PERWT, the ACS final person weight: the inverse of the selection probability, adjusted for nonresponse and calibrated to population controls by the Census Bureau.
Implicit weights: None beyond PERWT: each estimate is a weighted mean, so a respondent's leverage is his or her share of the state's total weight.
Source of randomness: Repeated sampling of addresses and group-quarters residents under the ACS design, with the Census Bureau's weighting repeated on each hypothetical sample.
Variance estimator: Successive difference replication with the 80 person replicate weights REPWTP1-80: Var = (4/80) x sum over r of (theta_r - theta)^2, df = 79. Differences use the replicate covariance. 90% margin of error = 1.645 x SE (ACS convention); pairwise tests at the 90% level.
Assumptions: The replicates reproduce the sampling and weighting variability of the design. PUMS is a subsample of the full ACS, so these SEs are larger than those on data.census.gov. Coverage is point-in-time and self-reported. Pairwise tests are per comparison; with 1,275 of them some false positives are expected, and a Bonferroni family-wise version is stricter.
Numbers reported: 11.3%proportion
21.6%proportion
3.7%proportion
1.59×ratio of standard errors
1,102count
1,174count
2count

What we learned about the question

Among adults 19 to 64, Texas has a higher uninsured rate than every other state and Massachusetts a lower one, and every pairwise test distinguishes both from each of the others. Between the two ends a state's place is a range, not a number: Indiana, in the middle, is compatible with any rank in 21–30, and only 6 states remain in the top ten under every ordering the tests allow. The league table's first and last lines are findings; most of the lines between them are not. And the table is one estimand's table: among adults 55 to 64, 7 states sit ten or more places away from where the working-age ranking puts them.

What the method changed

Replacing the iid standard error with the replicate standard error made the typical state's interval 1.59× as wide. Testing each difference with its own replicate standard error cut the adjacent pairs that differ from 7 to 2, and turned single ranks into ranges. Changing the population from all working-age adults to adults 55 to 64 changed the ranking itself, and cut the distinguishable pairs from 1,102 to 858.

What this analysis does not establish

Why the states differ. The rates are descriptive: Medicaid expansion, income, immigration, and industry mix all vary across states, and none is separated here. The estimates describe coverage on the interview date during 2024, from the public-use subsample, so full-sample ACS tables will differ slightly and carry smaller margins. The tests are per comparison, and neither the rank ranges nor the replicate counts are simultaneous confidence statements about the ranking. The rank changes between the two tables are differences between two estimands, not evidence that either table is wrong. And the replicates measure sampling and weighting variability, not error in self-reported coverage.