Chapter 10.1 · The factory floor

Gradients, slopes and variance

For deaths in 2010 to 2015, tract life expectancy is 1.12 years lower per ten-point step of the ADI within counties, and lower in more deprived tracts in all 267 metros.

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

Across the 77,936 census tracts that have a life expectancy estimate for deaths in 2010 to 2015 and a rank on the Area Deprivation Index (ADI), life expectancy at birth is lower in more deprived tracts. It is 0.84 (95% CI 0.75 to 0.92) years lower for each ten-point step up the index. The ADI gives each tract a national percentile, and a higher percentile means a more deprived tract. That one number is a slope, and this essay follows it through the checks the book uses to decide what a slope of this kind can and cannot say. The gradient itself, tenth by tenth of deprivation, is the subject of Chapter 8.4, Place and the length of a life.

Comparing tracts in the same county gives a steeper slope

The slope comes from a straight line fitted through the tracts, each weighted by its population. It is reported as the years of life expectancy lower for a ten-point step of the ADI, so a slope above zero means life expectancy is lower where deprivation is higher. A national slope could come from composition alone, with the least deprived tracts concentrated in some states and the most deprived in others. Fixed effects test for that by giving each state, and then each county, its own baseline, so the slope compares only tracts in the same state or the same county.

Within counties, life expectancy is 1.12 (95% CI 1.07 to 1.16) years lower per step, against 0.84 years across all tracts and 0.92 within states. A composition story predicts the opposite order. A slope that survives these comparisons shows that rich and poor states at opposite ends of one line do not produce the pattern.

Figure 1

Life expectancy is 1.12 years lower per ten-point step of the ADI within counties, against 0.84 years across all tracts

Slope of tract life expectancy at birth on the Area Deprivation Index national percentile, years lower per ten-point step, three specifications, deaths in 2010 to 2015

Population-weighted
Each tract once

Show data

Data for Life expectancy is 1.12 years lower per ten-point step of the ADI within counties, against 0.84 years across all tracts
Specification	Weights	Years lower per ten points	Lower limit of the 95% interval	Upper limit of the 95% interval	Clusters	Tracts
All tracts	Population-weighted	0.84	0.75	0.92	51	77,936
All tracts	Each tract once	0.84	0.75	0.93	51	77,936
Within states	Population-weighted	0.92	0.78	1.05	51	77,936
Within states	Each tract once	0.91	0.77	1.05	51	77,936
Within counties	Population-weighted	1.12	1.07	1.16	2,890	77,722
Within counties	Each tract once	1.13	1.09	1.17	2,890	77,722

The within-county dots lie right of the all-tracts dots, with the within-state dots between them. Each dot is a slope of tract life expectancy on the ADI with a 95% interval, and the two dots in a row weight tracts by population or count each tract once. Values in Table 1.

Note: Intervals use cluster-robust errors, clustered on states for the first two rows and on counties for the third, with t intervals on one fewer degree of freedom than there are clusters. The county fit leaves out tracts that are alone in their county, so it uses 77,722 of 77,936 tracts.Note: Institutional tracts, such as prisons and dormitories, are dropped. USALEEP covers deaths in 2010 to 2015 and the 2020-tract file is area-weighted from 2010 tracts.Source: USALEEP life expectancy, 2010-2015; Area Deprivation Index, 2023; ACS 2019-2023; population-weighted and unweighted regression slopes, modeled estimates

The book also fits the same regression separately in each metro, a core-based statistical area with at least 40 tracts that have a life expectancy and an ADI rank. All 267 metros show lower life expectancy in more deprived tracts. The metro slopes spread out, and estimation error accounts for 48% of the variance of the estimated slopes. The intervals in most metros use errors that may be too narrow, which would make that share larger. Sorting the estimates puts the metros with the least precise slopes at both ends, so the chapter names no metro as the steepest or the flattest.

How we know

Life expectancy comes from USALEEP, a National Center for Health Statistics release of life expectancy at birth by census tract for deaths in 2010 to 2015.[1] The values are modeled estimates, and the book uses a file area-weighted from 2010 tracts onto 2020 tracts, so the estimate for a single tract is not reliable. The ADI is the 2023 national percentile, so the outcome and the deprivation measure come from different periods.[2] The weights are tract populations from the American Community Survey for 2019 to 2023. Institutional tracts, such as prisons and dormitories, are dropped, and the sample covers the states and the District of Columbia. The intervals are clustered on states for the national and within-state slopes and on counties for the within-county slope, which lets the errors of tracts in the same area be related. The report gives every estimate with its interval and sample size.

Part of the variation within counties is estimation error

A slope says how much life expectancy differs per step of the ADI. A split of the variance says where the differences lie. Variance measures how far tract values spread around their average. The chapter splits the population-weighted variance of tract life expectancy into a part between states, a part between counties in the same state and a part between tracts in the same county. The three shares sum to the whole. For life expectancy, 70.3% of the variance lies between tracts in the same county. A county map of life expectancy shows county averages and so hides the largest share of its variance.

That share needs a caveat. The tract estimates are modeled, and the average of their squared standard errors equals 26.7% of the total variance, which sits inside the within-county share. Taking the published errors at face value, 60% of the variance that is not estimation error lies within counties. The published errors of the 2020-tract file understate the noise, so the share of the true differences that lies within counties is lower than that. The ADI is the exception among the measures in the figure. Only 34% of its variance lies within counties, and the rest lies between states and between counties in a state.

Figure 2

70.3 percent of the variance of tract life expectancy lies between tracts in the same county, 26.7 points of it estimation error, against 34% percent for the ADI

Share of the population-weighted variance of three tract measures between states, between counties in a state and between tracts in a county, percent

Between states
Between counties in a state
Between tracts in a county

Show data

Data for 70.3 percent of the variance of tract life expectancy lies between tracts in the same county, 26.7 points of it estimation error, against 34% percent for the ADI
level	measure	share
Between states	Tract life expectancy	13.75
Between counties in a state	Tract life expectancy	15.92
Between tracts in a county	Tract life expectancy	70.33
Between states	ADI national percentile	36.64
Between counties in a state	ADI national percentile	29.38
Between tracts in a county	ADI national percentile	33.98
Between states	Nine-measure index percentile	5.13
Between counties in a state	Nine-measure index percentile	23.29
Between tracts in a county	Nine-measure index percentile	71.59

The segment between tracts in the same county is the largest in the life expectancy bar and in the index bar, and the segment between states is the largest in the ADI bar. Each bar sums to the whole variance of its measure, and the shares come from every tract in the sample and have no sampling interval. The third bar is an index the chapter builds from nine census measures. Values in Table 7.

Note: Tract life expectancy is modeled. The mean squared standard error of the tract estimates equals 26.7 percent of the total variance and sits inside the between-tract segment; the 2020-tract standard errors are mapping grade and understate the noise, so the share of the true differences that lies between tracts is lower than the net share in the text.Note: The life expectancy bar uses 78,008 tracts in 3,112 counties and Connecticut planning regions; the ADI and index bars use the 82,545 tracts of the index. Connecticut's planning regions stand in for its counties.Source: USALEEP life expectancy, 2010-2015; ACS 2019-2023; Area Deprivation Index, 2023; population-weighted decomposition of the variance, full counts of tracts

Across states, the weights decide which way a correlation moves

Counting each tract once instead of weighting by population moves none of the three slopes by more than 0.015 years, so the weights do not produce the result. At a coarser grain they can decide it. The same data can give different correlations across tracts, counties and states, a case of the modifiable areal unit problem.[3] The chapter computes 6 pairs of tract measures at all three grains on the same tracts, with a county or state value the population-weighted mean of its tracts.

The correlation of the college share with life expectancy rises from +0.56 across tracts to +0.71 (95% CI 0.55 to 0.83) across states when each unit is weighted by its population. Counting each unit once, it falls from +0.55 to +0.48 (95% CI 0.13 to 0.81). A weighted correlation across states gives populous states most of the weight, so the two weightings answer different questions, and neither is the correct one.

Figure 3

Weighted by population, the same pair of measures has a different correlation at each grain, and homeownership and poverty go from −0.53 across tracts to −0.05 across states

Correlation of six pairs of tract measures across tracts, across counties and across states, weighted by population or counting each unit once, with 95% intervals from a bootstrap over counties or states, 2019-2023

Census measures only
Includes life expectancy

Show data

Data for Weighted by population, the same pair of measures has a different correlation at each grain, and homeownership and poverty go from −0.53 across tracts to −0.05 across states, Weighted, counties to states
Pair of measures	Tracts, weighted	Counties, weighted	States, weighted	Tracts, each once	Counties, each once	States, each once	States, weighted, lower limit	States, weighted, upper limit	States, each once, lower limit	States, each once, upper limit
College share and income	+0.73	+0.79	+0.83	+0.72	+0.71	+0.81	+0.73	+0.94	+0.73	+0.91
Poverty rate and college share	−0.50	−0.56	−0.69	−0.49	−0.50	−0.52	−0.84	−0.50	−0.81	−0.20
Homeownership and poverty rate	−0.53	−0.40	−0.05	−0.52	−0.36	−0.11	−0.30	+0.17	−0.32	+0.12
Unemployment and poverty rate	+0.43	+0.58	+0.33	+0.45	+0.53	+0.50	+0.08	+0.58	+0.32	+0.65
College share and life expectancy	+0.56	+0.64	+0.70	+0.55	+0.51	+0.48	+0.55	+0.83	+0.13	+0.81
Poverty rate and life expectancy	−0.50	−0.60	−0.66	−0.52	−0.57	−0.78	−0.88	−0.45	−0.89	−0.65

In the views that join counties to states, the line for the college share and life expectancy changes direction between the weighted view and the view that counts each unit once. Each line joins the correlation of one pair of measures at two grains, and the four views pair the grains under each weighting. Values in Table 6.

Note: Every correlation uses the same 77,775 tracts. County and state values are population-weighted means of the tract values; the weighted views weight each correlation by the population of its unit and the others count each unit once. The county correlations rest on 3,112 counties and the state correlations on 51 states, so the state intervals are wide.Note: The correlation with life expectancy is between tract life expectancy, a modeled estimate, and census measures.Note: ACS tract values are survey estimates with sampling error, and the staged file has no margins of error, so they are treated as exact.Source: ACS 2019-2023 tracts and USALEEP life expectancy, 2010-2015; correlations of tract values and of population-weighted county and state means, population-weighted and unweighted

Averaging within an area removes the variation inside it, so a correlation of averages describes how the areas differ and says nothing about how the tracts or people inside them do. The slope also describes places. It tracks how life expectancy goes with deprivation across tracts, and families sort themselves among tracts for reasons the regression does not observe. It does not say what would happen to a family that moved, and the steeper slope within counties is compatible with more than one explanation that the data cannot separate.

What these estimates cannot tell you

Every estimate here compares tracts, counties, metros or states, and a gradient of tract life expectancy on the ADI says nothing about individuals, which is the ecological inference problem. Life expectancy is a modeled estimate for deaths in 2010 to 2015 that has not been updated. The ADI is a national percentile and understates deprivation in high-cost cities (Chapter 1.1, What the deprivation index measures), which bears on the metro slopes. The census measures are survey estimates with sampling error, and the file carries no margins of error, so the chapter treats them as exact.

Go deeper

The report: every estimate with its interval and sample size.
Tables: every figure of the report as a table.

Reading paths that include this chapter:

Data sources

USALEEP life expectancyUSALEEP life expectancy at birth, 2010-2015, on 2020 tracts (area-weighted from 2010 tracts), institutional tracts droppedYears: 2010-2015 · Weights: ACS 2019-2023 tract population · License: Public domain (NCHS, CDC)
Area Deprivation IndexNeighborhood Atlas Area Deprivation Index 2023, national percentile, tractsYears: 2023 release (version 4.0.1; built on ACS 2019-2023) · Weights: ACS 2019-2023 tract population (each tract once for the index split by county) · License: Free for non-commercial use; cite the Neighborhood Atlas and Kind and Buckingham
American Community SurveyACS 2019-2023 tracts: population, the nine measures of the index and the measures of the correlationsYears: 2019-2023 (5-year) · Weights: tract population; each tract once in the index · License: Public domain (U.S. Census Bureau)
Census TIGER/Line tractsCensus 2020 county file: the core-based statistical area code of each county (CBSAFP)Years: 2020 · Weights: none (assignment of counties to metros) · License: Public domain (U.S. Census Bureau)
CDC PLACESCDC PLACES 2024 release, tracts: six measures of the coefficient zooYears: 2024 release (tracts); 2025 release (ZCTAs, 2023 data year) · Weights: adult population of each tract (PLACES) · License: Public domain (CDC, U.S. Department of Health and Human Services)
CDC WONDER natalityCDC WONDER natality 2018-2024, county by birthweight class: counts of withheld cellsYears: 2018-2024 · Weights: none (full counts of cells) · License: Public domain (NCHS, CDC); WONDER data-use restrictions apply

All data sources

References

Arias, Elizabeth; Escobedo, Luis A.; Kennedy, Jiaquan; Fu, Chenxi; Cisewski, Jodi (2018). U.S. Small-area Life Expectancy Estimates Project: Methodology and Results Summary. Vital and Health Statistics, Series 2, Number 181, National Center for Health Statistics. www.cdc.gov/nchs/data/series/sr_02/sr02_181.pdf
Kind, Amy J. H.; Buckingham, William R. (2018). Making Neighborhood-Disadvantage Metrics Accessible: The Neighborhood Atlas. New England Journal of Medicine 378(26): 2456-2458. doi:10.1056/NEJMp1802313
Openshaw, Stan (1984). The Modifiable Areal Unit Problem. CATMOG 38, Concepts and Techniques in Modern Geography, Geo Books, Norwich.

How to cite

@misc{singh2026gradientsslopesandvariance,
  author = {Singh, Vishal},
  title = {Gradients, slopes and variance},
  year = {2026},
  url = {https://vishalsingh.org/zipcode/gradients-slopes-and-variance},
  note = {In ZIP Code Destiny. Published 9 October 2026.}
}