Chapter 10.4 · Sorting instead of ranking

Families of places

K-means sorts 75,717 census tracts into 4 families on 15 deprivation measures, and life expectancy for deaths in 2010 to 2015, which the sorting never used, runs from 74.7 years in the high-poverty family to 81.7 years in the high-income family.

Author: Vishal Singh
Affiliation: NYU Stern
Published: 9 October 2026

K-means sorts the 75,717 census tracts of the frame into 4 families on 15 of the measures that the Area Deprivation Index (ADI) combines, and the families differ by 6.9 (95% CI 6.7 to 7.2) years of life expectancy at birth between the longest-lived and the shortest-lived, for deaths in 2010 to 2015. The sort never used life expectancy. Two of the families show no detectable difference in life expectancy, 78.6 years for the crowded renters and 78.6 years for the mid-range family, and the crowded renters do no better on adult income rank or on modeled estimates of diabetes, smoking and self-rated health, although the ADI scores them as less deprived than the mid-range family by 16.3 (95% CI 8.4 to 22.6) points on its national rank.

The families are regions of a continuum and not separate clusters. The silhouette, a score of how well tracts fit their own group against the nearest other, is 0.21 for the four families, against 0.12 for simulated data with no clusters, and a second arrangement of four groups has only 1.1% more within-group variance. This chapter chooses the number of families, describes them, compares them with outcomes held out of the sort and with deciles of the ADI, and labels counties by the family they resemble.

Related material is in other chapters. Chapter 10.2, The anatomy of a deprivation index rebuilds the ADI from these measures and shows how much of its weight falls on home value. Chapter 9.1, Eight types of neighborhood sorts tracts on measures that include life expectancy, mobility, diabetes and smoking, so its types describe bundles of conditions and outcomes together, while the families here use deprivation measures alone. Chapter 10.1, Gradients, slopes and variance covers the slopes of outcomes on the ADI and the split of variance among states, counties and tracts, and Chapter 10.3, A deprivation index at two zooms repeats the index for counties.

The measures and how they are sorted

The frame is the one that Chapter 10.2, The anatomy of a deprivation index uses: the 75,717 tracts of the 2020 census that have a published ADI rank, residents and all 15 measures. They hold 307.1 million residents, 93.0% of the residents of tracts with a rank in the states and the District of Columbia. Puerto Rico's 981 tracts are outside the frame, as they are in that chapter. The measures come from the American Community Survey (ACS) for 2019 to 2023. The measures of advantage are median family income, median home value, median gross rent, the monthly mortgage cost, the white-collar share of workers and the homeownership rate. The measures of hardship are the poverty rate, the near-poverty share, income disparity (the log ratio of low-income to high-income households), unemployment, the single-parent share, the share of households without a vehicle, crowding, homes without complete plumbing and the share of adults with under 9 years of schooling.

Each measure is standardized, which means subtracting its mean across tracts and dividing by its standard deviation, so that a dollar amount and a proportion count equally. K-means then places 4 centers in the space of the standardized measures so that the total squared distance from each tract to its nearest center is as small as it can be made,[1]

\min_{c,\ \mu}\ \sum_{i} \lVert z_i - \mu_{c(i)} \rVert^{2},

where $z_i$ is the vector of standardized measures for tract $i$ , $\mu_j$ is the center of family $j$ and $c(i)$ is the family that tract $i$ is assigned to. The fit starts from many random sets of centers and keeps the best. Each tract counts once, so a family is a family of places and not of people. Residents return as weights in every profile and outcome mean below.

The ADI is not an input, but it is built from the same ACS measures, so the sort and the index share their raw material. Life expectancy, adult income rank and the health measures from PLACES are also outside the sort. They come back below as held-out outcomes.

Four families depart from a cloud with no clusters more than any other number does

The data do not say how many groups there are, and k-means returns a partition for any number. Figure 1 compares fits with 2 to 10 groups on the silhouette, computed on a seeded sample of tracts.[2] The silhouette of the tract data is 0.22 for two groups, 0.21 for four and 0.15 for five, so it is no higher for four groups than for two and drops after four. Taken alone it favors two groups.

A positive silhouette does not show that groups exist, because k-means divides any cloud into parts and some of them look separate. The build therefore fits the same k-means to simulated data as well. The simulated clouds keep the distribution of each measure, including its skew and its top codes, and the correlations among measures, but their dependence is a single multivariate normal cloud with one dense core. At two groups they score 0.21, nearly the same as the tract data, which means the high silhouette at two groups comes from the shape of the cloud and not from groups in it. At four groups the simulated score is 0.12, and the tract data exceed it by 0.09, against at most 0.03 at any other number of groups. The sample silhouette has a margin of error of at most 0.003 either way, and the 3 simulated clouds differ from one another by at most 0.007. This chapter uses four families because that is where the data depart most from a cloud without clusters. The choice is a judgment about where to draw lines on a continuum, and it does not count the kinds of neighborhood that exist.

Figure 1

The tract data beat a cloud with no clusters by 0.09 in silhouette at four families, more than at any other number of families

Mean silhouette of k-means fits to the 15 standardized deprivation measures, tract data and simulated data with no clusters, and agreement of refits on resamples of counties

The tract data beat simulated data with no clusters by 0.09 at four families and by at most 0.03 at any other number

Mean silhouette by number of families, tract data and simulated data with no clusters, with 95% intervals for the sample

Mean silhouette (higher = better separated)

The gap between the lines is the evidence for clusters. It is widest at four families.

Show data

Data for The tract data beat a cloud with no clusters by 0.09 in silhouette at four families, more than at any other number of families, Silhouette
conf_high	conf_low	k	n	series	silhouette
0.221	0.216	2	10,000	Tract data	0.218
0.208	0.205	2	10,000	Simulated, no clusters	0.207
0.214	0.207	3	10,000	Tract data	0.210
0.182	0.179	3	10,000	Simulated, no clusters	0.180
0.212	0.206	4	10,000	Tract data	0.209
0.122	0.120	4	10,000	Simulated, no clusters	0.121
0.153	0.148	5	10,000	Tract data	0.151
0.126	0.123	5	10,000	Simulated, no clusters	0.124
0.156	0.151	6	10,000	Tract data	0.153
0.127	0.124	6	10,000	Simulated, no clusters	0.125
0.131	0.126	7	10,000	Tract data	0.128
0.125	0.123	7	10,000	Simulated, no clusters	0.124
0.134	0.129	8	10,000	Tract data	0.132
0.122	0.120	8	10,000	Simulated, no clusters	0.121
0.140	0.135	9	10,000	Tract data	0.137
0.113	0.110	9	10,000	Simulated, no clusters	0.111
0.142	0.137	10	10,000	Tract data	0.140
0.112	0.109	10	10,000	Simulated, no clusters	0.110

In the first tab, the tract line sits above the simulated line most clearly at four families, where the two are farthest apart; at two families the lines nearly meet. The tract line plots the mean silhouette of a seeded sample of tracts for each number of families, and the simulated line plots the same fit to clouds that keep each measure's distribution and the correlations among measures but have no clusters. In the second tab, agreement of refits with the fit on all tracts is high up to five families, lower at six and lowest at seven.Note: The silhouette uses a seeded random sample of tracts; the intervals show its sampling error, at most 0.003 in either direction. The simulated line is the mean of 3 draws that differ by at most 0.007 at any number of families.Note: Simulated data are drawn from a Gaussian copula: each measure keeps its distribution in the tract data, and the dependence among measures is a single multivariate normal cloud, so k-means has no clusters to find.Note: The band in the second tab runs from the tenth to the ninetieth percentile of the refits, each a k-means fit on a resample that draws whole counties with replacement and assigns every tract to its nearest center (replicates and seed in the methods note).Source: ACS 2019-2023 deprivation measures, k-means fits to all tracts, silhouette on a sample of tracts; unweighted (each tract counts once)

The departure is modest. The four families hold 41.8% of the variance of the standardized measures between them, against 38.3% for four groups cut from the simulated cloud, and 13.4% of the sampled tracts have a negative silhouette, which means they sit closer to another family than to their own.

The partition changes with weights, scale and the sample of counties

A fit can be reproducible and still have soft borders. The adjusted Rand index compares two partitions of the same tracts, with the maximum score for identical partitions and a score of zero for the agreement of random ones.[3] Refits from 6 other starting seeds agree with the families at 0.99 or higher. The result depends more on the choices in the sort. A refit that weights each tract by its population agrees at 0.74, so whether tracts or residents count changes the partition. A refit with the four dollar amounts replaced by their logs agrees at only 0.62, so the families depend on how the dollar measures are scaled, as the ADI does (Chapter 10.2, The anatomy of a deprivation index).

Resampling whole counties tests stability with the county as the unit of sampling, because tracts in a county resemble one another.[4] The second tab of Figure 1 refits k-means on 200 resamples that draw counties with replacement, at each number of groups and with as many starts as the main fit, and assigns every tract to the nearest refit center. The median agreement with the families at four groups is 0.88, but 17.5% (95% CI 12.9% to 23.4%) of the refits find a different arrangement, which this chapter defines as an agreement below 0.5. That is more often than at three groups (0.0% (95% CI 0.0% to 1.9%)) or at five (2.0% (95% CI 0.8% to 5.0%)). The different arrangement is in the full data too. Of 30 fits that start from a single random set of centers, 13 end there, and its within-group variance is only 1.1% higher than that of the families. It puts 76.9% of the crowded-renter tracts in one group with 47.6% of the high-poverty tracts, so a separate crowded-renter family is the least secure part of the partition. Median agreement is between 0.88 and 0.92 up to five groups, falls to 0.80 at six and 0.54 at seven, and is 0.78 at eight. Reproducibility therefore does not favor four groups, and the benchmark above is the reason for choosing them.

What defines each family

Figure 2 shows the profile of each family on the 15 measures, and Figure 3 gives the same families in the units of the measures. The four families are the high-income family, the crowded renters, the mid-range family and the high-poverty family, numbered by the mean ADI rank of their tracts from the least to the most deprived. The names describe the measures that set each family apart and are the book's labels. They say nothing about who lives in the tracts, and no measure in the sort is a race, an ethnicity or a place of birth.

Figure 2

Crowded renters average 2.2 standard deviations above the mean tract on crowding, and the high-poverty family 1.4 above on poverty

Population-weighted mean z-score of each of the 15 measures by family, tracts, ACS 2019 to 2023; zero is the mean tract

−2.50.0+2.5

A grid of fifteen measures by four families, shaded from negative to positive z-scores. The high-income column is high on income, home value, rent, mortgage cost and white-collar work and low on poverty. The crowded-renter column is high on crowding and on the share of adults with little schooling, and low on homeownership. The mid-range column stays within 0.34 standard deviations of zero throughout. The high-poverty column is high on poverty, near-poverty, unemployment and income disparity and low on income and housing costs.
Measure	High income	Crowded renters	Mid-range	High poverty
Family income	+1.4, interval +1.3 to +1.4, n = 17,284	−0.6, interval −0.7 to −0.6, n = 6,870	−0.1, interval −0.1 to −0.1, n = 37,746	−1.0, interval −1.0 to −0.9, n = 13,817
Home value	+1.1, interval +0.9 to +1.3, n = 17,284	+0.3, interval +0.1 to +0.6, n = 6,870	−0.3, interval −0.3 to −0.3, n = 37,746	−0.7, interval −0.7 to −0.6, n = 13,817
Gross rent	+1.3, interval +1.1 to +1.4, n = 17,284	+0.2, interval +0.1 to +0.3, n = 6,870	−0.3, interval −0.3 to −0.2, n = 37,746	−0.7, interval −0.7 to −0.6, n = 13,817
Mortgage cost	+1.3, interval +1.2 to +1.4, n = 17,284	+0.4, interval +0.2 to +0.6, n = 6,870	−0.3, interval −0.4 to −0.3, n = 37,746	−0.8, interval −0.8 to −0.7, n = 13,817
White-collar jobs	+1.2, interval +1.1 to +1.2, n = 17,284	−0.9, interval −1.0 to −0.8, n = 6,870	−0.1, interval −0.1 to −0.1, n = 37,746	−0.8, interval −0.8 to −0.8, n = 13,817
Homeownership	+0.3, interval +0.2 to +0.4, n = 17,284	−1.0, interval −1.2 to −0.9, n = 6,870	+0.3, interval +0.3 to +0.4, n = 37,746	−0.7, interval −0.7 to −0.7, n = 13,817
Poverty	−0.8, interval −0.8 to −0.7, n = 17,284	+0.6, interval +0.5 to +0.7, n = 6,870	−0.3, interval −0.3 to −0.3, n = 37,746	+1.4, interval +1.4 to +1.4, n = 13,817
Below 150% of poverty	−0.9, interval −0.9 to −0.9, n = 17,284	+0.7, interval +0.6 to +0.8, n = 6,870	−0.3, interval −0.3 to −0.3, n = 37,746	+1.4, interval +1.4 to +1.4, n = 13,817
Income disparity	−0.7, interval −0.8 to −0.7, n = 17,284	+0.2, interval +0.2 to +0.3, n = 6,870	−0.1, interval −0.1 to −0.1, n = 37,746	+0.9, interval +0.9 to +0.9, n = 13,817
Unemployment	−0.3, interval −0.3 to −0.2, n = 17,284	+0.4, interval +0.4 to +0.5, n = 6,870	−0.3, interval −0.3 to −0.2, n = 37,746	+0.8, interval +0.7 to +0.8, n = 13,817
Single parents	−0.6, interval −0.6 to −0.5, n = 17,284	+0.7, interval +0.6 to +0.9, n = 6,870	−0.1, interval −0.2 to −0.1, n = 37,746	+0.9, interval +0.9 to +1.0, n = 13,817
No vehicle	−0.2, interval −0.3 to 0.0, n = 17,284	+0.8, interval +0.3 to +1.3, n = 6,870	−0.3, interval −0.3 to −0.3, n = 37,746	+0.5, interval +0.4 to +0.6, n = 13,817
Crowding	−0.2, interval −0.3 to −0.2, n = 17,284	+2.2, interval +1.9 to +2.5, n = 6,870	−0.3, interval −0.3 to −0.2, n = 37,746	+0.1, interval 0.0 to +0.1, n = 13,817
No plumbing	−0.4, interval −0.4 to −0.4, n = 17,284	−0.3, interval −0.3 to −0.2, n = 6,870	−0.1, interval −0.1 to −0.1, n = 37,746	+0.6, interval +0.5 to +0.7, n = 13,817
Under 9 years of school	−0.4, interval −0.5 to −0.4, n = 17,284	+2.0, interval +1.8 to +2.2, n = 6,870	−0.3, interval −0.3 to −0.2, n = 37,746	+0.3, interval +0.2 to +0.3, n = 13,817

Show data

Data for Crowded renters average 2.2 standard deviations above the mean tract on crowding, and the high-poverty family 1.4 above on poverty
conf_high	conf_low	family	measure	n	z
1.44	1.29	High income	Family income	17,284	1.36
−0.551	−0.654	Crowded renters	Family income	6,870	−0.599
−0.121	−0.148	Mid-range	Family income	37,746	−0.134
−0.942	−0.966	High poverty	Family income	13,817	−0.954
1.32	0.927	High income	Home value	17,284	1.12
0.553	0.058	Crowded renters	Home value	6,870	0.329
−0.307	−0.349	Mid-range	Home value	37,746	−0.328
−0.644	−0.686	High poverty	Home value	13,817	−0.666
1.36	1.14	High income	Gross rent	17,284	1.25
0.333	0.096	Crowded renters	Gross rent	6,870	0.225
−0.231	−0.315	Mid-range	Gross rent	37,746	−0.273
−0.644	−0.707	High poverty	Gross rent	13,817	−0.674
1.38	1.17	High income	Mortgage cost	17,284	1.28
0.587	0.166	Crowded renters	Mortgage cost	6,870	0.399
−0.316	−0.367	Mid-range	Mortgage cost	37,746	−0.342
−0.748	−0.814	High poverty	Mortgage cost	13,817	−0.782
1.23	1.10	High income	White-collar jobs	17,284	1.17
−0.779	−0.970	Crowded renters	White-collar jobs	6,870	−0.877
−0.061	−0.102	Mid-range	White-collar jobs	37,746	−0.082
−0.776	−0.827	High poverty	White-collar jobs	13,817	−0.802
0.400	0.222	High income	Homeownership	17,284	0.309
−0.917	−1.15	Crowded renters	Homeownership	6,870	−1.05
0.353	0.304	Mid-range	Homeownership	37,746	0.328
−0.651	−0.733	High poverty	Homeownership	13,817	−0.692
−0.722	−0.779	High income	Poverty	17,284	−0.750
0.669	0.492	Crowded renters	Poverty	6,870	0.572
−0.307	−0.327	Mid-range	Poverty	37,746	−0.316
1.42	1.35	High poverty	Poverty	13,817	1.38
−0.864	−0.925	High income	Below 150% of poverty	17,284	−0.895
0.821	0.645	Crowded renters	Below 150% of poverty	6,870	0.726
−0.271	−0.296	Mid-range	Below 150% of poverty	37,746	−0.283
1.41	1.36	High poverty	Below 150% of poverty	13,817	1.39
−0.683	−0.806	High income	Income disparity	17,284	−0.743
0.271	0.162	Crowded renters	Income disparity	6,870	0.217
−0.086	−0.119	Mid-range	Income disparity	37,746	−0.102
0.925	0.890	High poverty	Income disparity	13,817	0.907
−0.226	−0.331	High income	Unemployment	17,284	−0.282
0.522	0.366	Crowded renters	Unemployment	6,870	0.449
−0.231	−0.273	Mid-range	Unemployment	37,746	−0.252
0.844	0.692	High poverty	Unemployment	13,817	0.764
−0.538	−0.581	High income	Single parents	17,284	−0.559
0.853	0.631	Crowded renters	Single parents	6,870	0.735
−0.120	−0.156	Mid-range	Single parents	37,746	−0.137
0.987	0.894	High poverty	Single parents	13,817	0.940
0.020	−0.322	High income	No vehicle	17,284	−0.174
1.29	0.331	Crowded renters	No vehicle	6,870	0.757
−0.301	−0.331	Mid-range	No vehicle	37,746	−0.317
0.575	0.400	High poverty	No vehicle	13,817	0.480
−0.180	−0.304	High income	Crowding	17,284	−0.241
2.49	1.94	Crowded renters	Crowding	6,870	2.23
−0.244	−0.284	Mid-range	Crowding	37,746	−0.264
0.094	0.018	High poverty	Crowding	13,817	0.054
−0.409	−0.425	High income	No plumbing	17,284	−0.417
−0.227	−0.290	Crowded renters	No plumbing	6,870	−0.261
−0.103	−0.145	Mid-range	No plumbing	37,746	−0.125
0.655	0.512	High poverty	No plumbing	13,817	0.580
−0.382	−0.462	High income	Under 9 years of school	17,284	−0.421
2.24	1.83	Crowded renters	Under 9 years of school	6,870	2.05
−0.235	−0.279	Mid-range	Under 9 years of school	37,746	−0.256
0.310	0.215	High poverty	Under 9 years of school	13,817	0.262

Read down a column for one family. The high-income column is positive on income and housing costs and negative on poverty, the high-poverty column is the reverse on income and poverty, and the crowded-renter column stands out on crowding and schooling. The mid-range column stays within 0.34 standard deviations of zero on every measure. Each cell is the mean of tract z-scores weighted by residents, with the 95% interval in the tooltip.Note: Values are in standard deviations of the tract distribution of each measure, with each tract counted once, so zero is the mean tract and not the mean resident. Weighting by residents moves a cell by at most 0.2 from the k-means center.Note: Income, home value, rent, mortgage cost, white-collar work and homeownership are measures of advantage, and the rest are measures of hardship. Intervals are 95% intervals from a bootstrap over counties, given the families.Note: The share of single-parent households is defined differently in the tract file and in the county tables, and the tracts of Utah and the District of Columbia, which came from the Census API, follow the county definition. They are 819 of the tracts in the frame.Source: ACS 2019-2023 deprivation measures (15) and Area Deprivation Index 2023; k-means families; population-weighted (z-scores use the unweighted mean and standard deviation across tracts)

The high-income family holds 76.6 million residents (24.9% of the frame) in 17,284 tracts. The mean tract median family income is 162 (95% CI 159 to 166) thousand dollars and the mean median home value 714 (95% CI 654 to 774) thousand dollars, with a poverty rate of 5.7% (95% CI 5.4% to 5.9%). Its tracts sit above the mean tract on income, housing costs and white-collar work and below it on every hardship measure, although the interval for households without a vehicle includes zero.

The crowded-renter family is the smallest, with 29.9 million residents in 6,870 tracts. Its tracts average 14.9% (95% CI 13.4% to 16.2%) of homes with more than one person per room, 43.3% (95% CI 41.0% to 46.1%) homeownership and 17.1% (95% CI 15.8% to 18.2%) of adults with under 9 years of schooling, which puts crowding at 2.2 (95% CI 1.9 to 2.5) standard deviations above the mean tract. Housing costs are above the mean tract (a home value of 472 (95% CI 389 to 540) thousand dollars) while family incomes are below it (74 (95% CI 71 to 76) thousand dollars). The family is concentrated: 42.6% of its residents live in California, 13.8% in New York and 11.8% in Texas.

The mid-range family is the largest, with 153.1 million residents in 37,746 tracts. Its column in Figure 2 stays within 0.34 standard deviations of zero on every measure, with a family income of 95 (95% CI 94 to 95) thousand dollars and a poverty rate of 9.9% (95% CI 9.8% to 10.0%). The high-poverty family holds 47.4 million residents in 13,817 tracts, with a poverty rate of 26.5% (95% CI 26.2% to 26.8%), a family income of 58 (95% CI 57 to 58) thousand dollars and a home value of 168 (95% CI 162 to 175) thousand dollars, so low income and low housing cost go together.

Figure 3

Mean tract family income runs from 58 thousand dollars in the high-poverty family to 162 thousand dollars in the high-income family

Population-weighted means of tract values by family, in the units of each column, tracts and residents of the frame, ACS 2019 to 2023

Family	Tracts	Residents (millions)	ADI national rank (mean)	Family income (thousands of dollars)	Home value (thousands of dollars)	Poverty (%)	Crowded homes (%)	Homeownership (%)	Adults with under 9 years of school (%)
High income	17,284	76.6	16.9	162	714	5.7	2.3	72.4	2.3
Crowded renters	6,870	29.9	38.5	74	472	18.6	14.9	43.3	17.1
Mid-range	37,746	153.1	54.7	95	271	9.9	2.2	72.8	3.3
High poverty	13,817	47.4	79.3	58	168	26.5	3.8	50.9	6.4

Show data

Data for Mean tract family income runs from 58 thousand dollars in the high-poverty family to 162 thousand dollars in the high-income family
adi	crowd	edu	family	home	income	own	pov	residents	tracts
16.91	2.31	2.27	High income	714.5	162.2	72.42	5.67	76.58	17284
38.48	14.85	17.06	Crowded renters	471.7	73.80	43.35	18.56	29.92	6870
54.75	2.20	3.26	Mid-range	271.4	94.73	72.82	9.90	153.1	37746
79.28	3.81	6.36	High poverty	168.3	57.86	50.93	26.46	47.42	13817

Read across a row for one family. The mid-range family is the largest by residents, the crowded-renter family the smallest, and the Area Deprivation Index (ADI) puts the crowded renters between the high-income and mid-range families. Each cell is the mean of tract values weighted by residents; income and home value are means of tract medians.Note: The ADI is not an input to the families. Families are numbered from least to most deprived by the population-weighted mean ADI of their tracts. The names describe the measures that define each family and are the book's labels, not categories in the data.Note: Dollar amounts are 2019-2023 ACS medians of tracts, averaged over tracts with residents as weights; they are not medians of the family.Source: ACS 2019-2023 deprivation measures (15) and Area Deprivation Index 2023; k-means families; population-weighted

Counting residents instead of tracts changes the profile little. Every cell of Figure 2 differs from the center that k-means found, which counts each tract once, by at most 0.2 standard deviations.

Outcomes the sort did not use

Figure 4 compares the families on measures that were not inputs. Life expectancy runs from 81.7 (95% CI 81.5 to 81.8) years in the high-income family to 74.7 (95% CI 74.5 to 74.9) years in the high-poverty family, a difference of 6.9 (95% CI 6.7 to 7.2) years. The adult income rank of children raised in low-income families, which the Opportunity Atlas reports as a percentile, runs from 48.1 (95% CI 47.7 to 48.6) to 37.2 (95% CI 36.9 to 37.6). Both outcomes are older than the measures: life expectancy is for deaths in 2010 to 2015, mobility is for children born in 1978 to 1983, and the families are drawn on conditions in 2019 to 2023. The comparison shows that present conditions line up with past outcomes, and it does not show that the families produce them.

Figure 4

Life expectancy runs from 81.7 years in the high-income family to 74.7 years in the high-poverty family, and the crowded-renter and mid-range families do not differ detectably

Population-weighted mean of tract values by family with 95% intervals from a bootstrap over counties; none of these measures was used to form the families

Life expectancy and adult income rank are highest in the high-income family and lowest in the high-poverty family

Life expectancy at birth for deaths in 2010 to 2015 and adult income rank of children born 1978 to 1983, population-weighted family means with 95% intervals

Life expectancy at birth (years)

Adult income rank, children from low-income families (percentile)

Population-weighted mean of tract values (units in each panel title)

The order of the families is the same on both measures except that the crowded renters are below the mid-range family on income rank and not detectably different on life expectancy.

Show data

Data for Life expectancy runs from 81.7 years in the high-income family to 74.7 years in the high-poverty family, and the crowded-renter and mid-range families do not differ detectably, Measured outcomes
conf_high	conf_low	estimate	family	label	n	panel
81.81	81.50	81.66	High income	High income	16,221	Life expectancy at birth (years)
78.99	78.09	78.58	Crowded renters	Crowded renters	6,587	Life expectancy at birth (years)
78.67	78.50	78.59	Mid-range	Mid-range	36,163	Life expectancy at birth (years)
74.94	74.55	74.74	High poverty	High poverty	12,903	Life expectancy at birth (years)
48.57	47.71	48.13	High income	High income	17,256	Adult income rank, children from low-income families (percentile)
42.07	40.51	41.30	Crowded renters	Crowded renters	6,869	Adult income rank, children from low-income families (percentile)
43.41	42.98	43.20	Mid-range	Mid-range	37,728	Adult income rank, children from low-income families (percentile)
37.58	36.90	37.23	High poverty	High poverty	13,814	Adult income rank, children from low-income families (percentile)

In both tabs the high-income family has the most favorable mean and the high-poverty family the least favorable. The crowded-renter and mid-range families show no detectable difference in life expectancy, and the crowded renters are no better on any other measure, although the ADI ranks them as less deprived. Dots are family means and bars are 95% intervals; rows follow the family numbering, not the size of the value.Note: Life expectancy is for deaths in 2010 to 2015 on tracts of 2020 (area-weighted from 2010 tracts), with institutional tracts left out. The adult income rank is for children born in 1978 to 1983, interpolated by land area from 2010 tracts to 2020 tracts, and tract population stands in for the number of children. Both describe the past, while the families are built on measures for 2019 to 2023.Note: PLACES values are modeled estimates. The model combines each survey respondent's age, sex, race and education with the poverty rate of the respondent's county and applies the result to the age, race and sex counts of each tract, so differences among the families in the second tab reflect county effects and the demographic make-up of tracts, not tract-level inputs of the families. Intervals describe how the means would vary with the sample of counties, given the families, and not the choice of families.Source: USALEEP 2010-2015, Opportunity Atlas (children born 1978 to 1983) and PLACES 2024 release; families from ACS 2019-2023; population-weighted, modeled estimates for PLACES

The two middle families do not follow the ADI. The ADI scores the crowded renters at a mean rank of 38.5 (95% CI 32.0 to 46.2) and the mid-range family at 54.7 (95% CI 53.8 to 55.7), a difference of 16.3 (95% CI 8.4 to 22.6) points, but their life expectancy differs by 0.0 (95% CI −0.5 to 0.4) years, an interval that includes zero. The adult income rank of the crowded renters is 1.9 (95% CI 1.1 to 2.7) percentiles below that of the mid-range family, and their rate of fair or poor health is 8.2 (95% CI 7.2 to 9.2) points above it. The ADI follows home value closely, as Chapter 10.2, The anatomy of a deprivation index shows, and the crowded renters live in tracts with higher home values than the mid-range family does, 472 thousand dollars against 271 thousand dollars.

The health measures in the second tab are modeled. The PLACES model combines each survey respondent's age, sex, race and education with the poverty rate of the respondent's county and applies the result to the age, race and sex counts of each tract, so differences among the families in that tab reflect county effects and the demographic make-up of tracts, and no tract-level input of the families enters the model. Life expectancy and adult income rank are not modeled in that way; they come from death records and from tax records, interpolated from 2010 tracts to 2020 tracts by area. The intervals in the figure are bootstrap intervals over counties given the families, and they do not carry the uncertainty about the families themselves, which the refits above address.

Deciles of the ADI mix the families

A ranking puts tracts in order, and a decile of the ranking holds whatever kinds of tract land there. Figure 5 splits each decile of the ADI national rank by family. The deciles are close to equal in size, from 7,430 to 7,764 tracts, because ties in the published rank put tracts of equal rank in one decile. In the most deprived decile, 79.2% of residents live in tracts of the high-poverty family and 16.1% in tracts of the mid-range family, and the mid-range residents live 2.4 (95% CI 2.1 to 2.7) years longer than the high-poverty residents of the same decile. No decile is a single family. The most common family holds between 50.7% and 88.3% of a decile's residents, a second family holds at least 8.2% in every decile, and 72.0% of residents across the ten deciles live in a tract of their decile's most common family. The crowded renters appear in every decile, from 4.7% of the most deprived decile's residents to 19.3% of the second.

Figure 5

In the most deprived decile of tracts, 79.2% of residents live in high-poverty tracts and 16.1% in mid-range tracts

Share of residents in each decile of the ADI national rank by family of their tract, percent of residents, tracts of the frame, ACS 2019 to 2023

High income
Crowded renters
Mid-range
High poverty

Decile of the ADI national rank (1 = least deprived, 10 = most deprived)

Show data

Data for In the most deprived decile of tracts, 79.2% of residents live in high-poverty tracts and 16.1% in mid-range tracts
decile	family	share	tracts
1	High income	88.30	6606
1	Crowded renters	11.09	901
1	Mid-range	0.378	39
1	High poverty	0.234	15
2	High income	70.94	5256
2	Crowded renters	19.28	1508
2	Mid-range	9.17	770
2	High poverty	0.617	54
3	High income	50.70	3698
3	Crowded renters	12.78	960
3	Mid-range	35.16	2708
3	High poverty	1.36	123
4	High income	18.56	1418
4	Crowded renters	10.33	795
4	Mid-range	68.03	5269
4	High poverty	3.07	282
5	High income	3.32	256
5	Crowded renters	8.23	589
5	Mid-range	82.74	6171
5	High poverty	5.72	477
6	High income	0.440	42
6	Crowded renters	8.14	559
6	Mid-range	82.69	6223
6	High poverty	8.73	699
7	High income	0.048	6
7	Crowded renters	7.61	516
7	Mid-range	77.17	5966
7	High poverty	15.18	1177
8	High income	0.015	1
8	Crowded renters	6.78	430
8	Mid-range	68.51	5193
8	High poverty	24.70	1806
9	High income	0.006	1
9	Crowded renters	5.13	331
9	Mid-range	52.11	4127
9	High poverty	42.76	3176
10	High income	0	0
10	Crowded renters	4.71	281
10	Mid-range	16.14	1280
10	High poverty	79.16	6008

Look for the mix inside each bar: every decile holds residents of at least two families, and the crowded renters appear in every decile. Each bar is one decile of tracts, ranked from least to most deprived on the published ADI, and the segments are the families, as shares of the decile's residents.Note: Shares are of residents (ACS 2019-2023 tract population) of the tracts in the frame, not of tracts. Deciles of the frame's tracts are close to equal in size, from 7,430 to 7,764 tracts, because ties in the published tract ADI national rank put tracts of equal rank in one decile. The shares are counts of ACS estimates and carry no sampling interval here.Source: ACS 2019-2023 tract population by family; Area Deprivation Index 2023; full counts of ACS estimates

The mix matters for how much of the variation in outcomes a grouping carries. Ten deciles of the ADI account for 36.6% (95% CI 34.3% to 39.0%) of the variance of tract life expectancy and the four families for 33.3% (95% CI 31.4% to 35.2%), so the deciles carry 3.3 (95% CI 2.0 to 4.5) points more, and four groups formed from the measures come close to ten groups formed from the rank. For adult income rank the families carry more: 27.7% (95% CI 25.6% to 30.0%) against 23.3% (95% CI 21.0% to 26.2%) for the deciles, a difference of 4.4 (95% CI 2.5 to 6.2) points.

The cells formed by decile and family account for 41.3% (95% CI 39.2% to 43.3%) of the variance of life expectancy and 30.9% (95% CI 28.7% to 33.4%) of adult income rank. The families add 4.7 (95% CI 4.1 to 5.4) points for life expectancy and 7.6 (95% CI 6.3 to 9.1) points for adult income rank to what the deciles carry, and the deciles add 8.0 (95% CI 7.3 to 8.8) and 3.2 (95% CI 2.6 to 4.0) points to what the families carry. A grouping with more cells accounts for more variance even when it is unrelated to the outcome. If 40 cells were unrelated to life expectancy and each county counted as a single observation, which gives the most that chance could carry, they would account for 1.3%, below each of the gains above.

At county scale the families blur

A county has one set of measures and many kinds of tract. Labeling each of the 3,127 counties with the family whose center lies nearest the county's own measures, standardized as the tracts are, gives 96 high-income counties, 29 crowded-renter counties, 2,261 mid-range counties and 741 high-poverty counties (Figure 6). The distance uses 14 of the measures, without the single-parent share, because the county tables define it differently from the tract file. Each tract then takes its county's label, and the agreement between the county labels and the tract families is 0.22 on the adjusted Rand index. Across the country, 62.5% of residents live in a tract of the most common family of their county's label, and the share runs from 50.7% to 70.6% across the four labels.

Figure 6

Crowded-renter tracts hold 50.7% of the residents of counties that resemble the crowded-renter family, and high-income tracts hold 34.6%

Share of residents by family of their tract, in counties labeled by the nearest family center, percent of residents, ACS 2019 to 2023

County label (nearest family center)

High income
Crowded renters
Mid-range
High poverty

Show data

Data for Crowded-renter tracts hold 50.7% of the residents of counties that resemble the crowded-renter family, and high-income tracts hold 34.6%
county_label	family	share
High-income counties	High income	70.56
High-income counties	Crowded renters	14.05
High-income counties	Mid-range	14.06
High-income counties	High poverty	1.33
Crowded-renter counties	High income	34.63
Crowded-renter counties	Crowded renters	50.73
Crowded-renter counties	Mid-range	9.97
Crowded-renter counties	High poverty	4.67
Mid-range counties	High income	16.29
Mid-range counties	Crowded renters	5.20
Mid-range counties	Mid-range	63.01
Mid-range counties	High poverty	15.49
High-poverty counties	High income	2.91
High-poverty counties	Crowded renters	4.43
High-poverty counties	Mid-range	39.72
High-poverty counties	High poverty	52.93

Each bar is a set of counties labeled by the family whose center lies nearest the county's measures, and the segments split its residents by the family of the tract they live in. No bar is a single family: the share in tracts of the family that the label names runs from 50.7% to 70.6%.Note: A county takes the label of the family whose center lies nearest the county's own 14 measures, standardized with the mean and standard deviation of the tract distribution, without the single-parent share, which is defined differently for counties. No clustering is run on counties. Counties include county equivalents and Connecticut's planning regions, as the ACS tables report them.Note: Shares are of residents of the tracts in the frame and are counts of ACS estimates without a sampling interval. The county measures are medians and rates for the whole county, not averages of its tracts.Source: ACS 2019-2023 county measures and tract population by family; labels from the nearest tract-family center; full counts of ACS estimates

The crowded-renter and high-poverty labels are the weakest. The 29 counties with the crowded-renter label hold 21.8 million residents, and 50.7% of them live in crowded-renter tracts while 34.6% live in high-income tracts. In the 741 counties labeled high-poverty, 52.9% of residents live in high-poverty tracts. The crowded-renter family is also not contained in its counties: 37.0% of its residents live in counties with that label, 37.2% in counties labeled mid-range and 22.3% in counties labeled high-income. A county average combines crowding, renting and costly housing with the tracts around them, so a county label does not recover the family. Chapter 10.3, A deprivation index at two zooms makes the same comparison for the index itself.

What these data cannot show

The families describe tracts, and every comparison in this chapter is ecological: the life expectancy of a family is an average over tracts, and it says little about any one resident of one tract (ecological inference). Nothing here shows that living in a family's tracts changes a person's outcome. The families are arrangements of census measures, and the outcomes that line up with them reflect selection, history, prices and policy that these data do not separate.

The number and the names of the families are choices. The silhouette favors no number of groups on its own, the benchmark picks four by a stated rule, refits find another arrangement more often at four groups than at three or five, a second arrangement of four groups has only 1.1% more within-group variance than the first, and weighting by residents or rescaling the dollar measures changes the partition. The crowded-renter family depends most on those choices. The frame leaves out tracts that lack a median rent, home value or mortgage cost, which the ACS withholds for tracts with almost no renters or almost no owners, and it leaves out Puerto Rico, so none of these results describes the island.

The measures are tract estimates for 2019 to 2023 without margins of error, and the tract file defines the share of single-parent households differently from the county tables, while the 819 tracts of Utah and the District of Columbia carry the county definition. Life expectancy is for deaths in 2010 to 2015 and was interpolated from 2010 to 2020 tracts by area, and the USALEEP estimates suit maps and gradients and not single tracts. Adult income rank describes children born in 1978 to 1983 and was interpolated to 2020 tracts by area, PLACES values are modeled, and the ADI averages block-group ranks over tracts without population weights. The county labels use county estimates of the same measures and treat Connecticut's planning regions as counties.

Data sources

American Community SurveyACS 5-year estimates 2019-2023 for 2020 tracts and counties: the 15 deprivation measures, from the staged tract file and the Census API pull of housing, occupation, income-distribution and schooling tablesYears: 2019-2023 (5-year) · Weights: ACS 2019-2023 tract population · License: Public domain (U.S. Census Bureau)
Area Deprivation IndexArea Deprivation Index 2023 (v4.0.1) national rank, tract values averaged from block groups, used to number the families and to form decilesYears: 2023 release (version 4.0.1; built on ACS 2019-2023) · Weights: ACS 2019-2023 tract population · License: Free for non-commercial use; cite the Neighborhood Atlas and Kind and Buckingham
USALEEP life expectancyNCHS small-area life expectancy at birth, deaths in 2010 to 2015, 2020 tracts (area-weighted), institutional tracts left outYears: 2010-2015 · Weights: ACS 2019-2023 tract population · License: Public domain (NCHS, CDC)
Opportunity AtlasAdult household income rank of children born 1978 to 1983 raised in families at the 25th income percentile, 2020 tractsYears: 1978-1983 birth cohorts (2018 release) · Weights: ACS 2019-2023 tract population · License: Public use (Opportunity Insights and the U.S. Census Bureau)
CDC PLACESCDC PLACES 2024 release (BRFSS 2022): diabetes, current smoking and fair or poor health among adults, modeled prevalenceYears: 2024 release (tracts); 2025 release (ZCTAs, 2023 data year) · Weights: ACS 2019-2023 tract population · License: Public domain (CDC, U.S. Department of Health and Human Services)

All data sources

Methods

This chapter sorts the 75,717 census tracts of the frame into 4 families with k-means on 15 ACS 2019-2023 measures, and describes each family with outcomes that were not inputs. The fit counts each tract once and is unweighted by design, family means use tract population as weights, and the intervals resample whole counties and do not carry uncertainty about the fit. The held-out outcomes come from other periods and models: life expectancy for deaths in 2010 to 2015, adult income rank for children born 1978 to 1983, and CDC PLACES health measures that are modeled estimates. Puerto Rico's tracts are outside the frame, and county labels come from each county's nearest center, since no clustering is run on counties.

Which tracts and measures are used

The frame is the 2020 tracts that have a published ADI rank, residents and all 15 measures (the frame of the chapter on the anatomy of a deprivation index, loaded by its loader). The 15 measures are a subset of the 17 that the Area Deprivation Index combines. Puerto Rico's tracts are not in the frame, as in that chapter.

The tracts of Utah and the District of Columbia carry the county definition of the single-parent share, which differs from the tract file's definition. The measures are in their own units: dollars for the four dollar amounts, proportions for the other ten, and the log ratio of low-income to high-income households.

How the families are fitted

Each measure is standardized across tracts with each tract counted once, and k-means (scikit-learn, Lloyd's algorithm, k-means++ starts, 4 groups, 20 starts, seed 12345, one thread) places the centers. The fit is unweighted by design. Families are numbered by the population-weighted mean of the published ADI, least deprived first.

How the number of families is chosen

For 2 to 10 groups the build fits k-means with 10 starts and computes the mean silhouette on a seeded random sample of 10,000 tracts (seed 12345). The same fits are run on 3 simulated clouds of the same size that keep each measure's distribution and the correlations of normal scores among measures (a Gaussian copula, seeds 12345 to 12347) and have no clusters.

Figure explained here: Figure 1

Held-out outcomes, deciles and county labels

Life expectancy leaves out institutional tracts (25 percent or more of residents in group quarters). Adult income rank, life expectancy and PLACES measures are not inputs. PLACES values are modeled from individual age, sex, race and education and county poverty, post-stratified to tract counts.

Deciles of the published ADI national rank are formed on the tracts of the frame and are close to equal in size, because tied ranks fall in one decile.

Each county takes the label of the family whose center lies nearest its own 14 measures (all but the single-parent share, which is defined differently for tracts and counties), standardized with the tract mean and standard deviation; no clustering is run on counties, and each tract takes its county's label.

Figures explained here: Figure 4, Figure 5 and Figure 6

Means and intervals

Family means use ACS 2019-2023 tract population as weights. Intervals are 95% percentile intervals from 2,000 draws of whole counties with replacement (seed 12345); differences between two families and the variance explained by groups use the same draws. Intervals condition on the families and do not carry uncertainty about the fit.

Figures explained here: Figure 2 and Figure 3

Stability refits

Stability is tested with refits using 6 other seeds; a refit weighted by tract population; a refit with the four dollar amounts logged; 30 single-start fits to find other arrangements; and 200 refits at each number of groups from 2 to 8 (20 starts each, as many as the main fit) on resamples that draw whole counties with replacement (Connecticut tracts are assigned to its eight legacy counties; the generator of resample b is seeded with 12345 and b, and the same resamples serve every number of groups), each assigning every tract to its nearest center.

Agreement is the adjusted Rand index. The share of refits with an agreement below 0.5 has a Wilson interval that treats the refits as independent.

The refits run in worker processes; each depends only on its seed, so the result does not depend on the number of workers.

References

MacQueen, J. (1967). Some methods for classification and analysis of multivariate observations. Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, Volume 1: Statistics, 281-297.
Rousseeuw, Peter J. (1987). Silhouettes: a graphical aid to the interpretation and validation of cluster analysis. Journal of Computational and Applied Mathematics 20: 53-65. doi:10.1016/0377-0427(87)90125-7
Hubert, Lawrence; Arabie, Phipps (1985). Comparing partitions. Journal of Classification 2(1): 193-218. doi:10.1007/BF01908075
Hennig, Christian (2007). Cluster-wise assessment of cluster stability. Computational Statistics & Data Analysis 52(1): 258-271. doi:10.1016/j.csda.2006.11.025

How to cite

@misc{singh2026familiesofplaces,
  author = {Singh, Vishal},
  title = {Families of places},
  year = {2026},
  url = {https://vishalsingh.org/zipcode/families-of-places},
  note = {In ZIP Code Destiny. Published 9 October 2026.}
}

Reading paths

Methods first