Margin of Error: Technical Details
How district-level MOEs are computed from ACS tract and block group data
What the ACS Publishes
For every estimate it releases, the ACS also publishes a margin of error (MOE) at the 90% confidence level. This applies to estimates at every geography the ACS reports — including census tracts and block groups. The MOE is a 90% confidence half-width: if the survey were repeated many times with independent samples, the true population value would fall within ±MOE of the published estimate approximately 90% of the time.
As with the main estimates, the ACS does not publish district-level margins of error directly. Computing them requires aggregating tract and block group estimates, and propagating their individual MOEs into a combined district-level MOE. Our methadology for doing so is explained below.
MOE Propagation — Count Variables
Interior tracts only
When all tracts in a district fall entirely within the district boundary, the district estimate is a direct sum of tract estimates:
\[\hat{Y}_\text{district} = \sum_i \hat{y}_i\]
The MOE propagates via quadrature — the square root of the sum of squared component MOEs:
\[\text{MOE}_\text{district} = \sqrt{\sum_i \text{MOE}_i^2}\]
This follows the Census Bureau’s standard formula for aggregating estimates across geographies (ACS Handbook, Chapter 8). The formula assumes independence across units, which is an approximation — nearby tracts draw from overlapping survey samples — but the Census Bureau treats it as the appropriate method for geographic aggregation. The district-level MOE grows with the number of tracts but slower than linearly: a district whose tracts each have MOE of ±1,000 will have a district-level MOE of roughly \(\pm 1{,}000\sqrt{n}\), where \(n\) is the number of tracts.
Split tracts
When a tract straddles the district boundary, we only want to count residents within the district. We scale the tract estimate by a population weight \(w_i\) — the fraction of the tract’s 2020 decennial block population residing inside the district boundary — giving a general estimate:
\[\hat{Y}_\text{district} = \sum_i w_i \cdot \hat{y}_i\]
The MOE scales proportionally with the weight:
\[\text{MOE}_\text{district} = \sqrt{\sum_i \left(w_i \cdot \text{MOE}_i\right)^2}\]
For interior tracts \(w_i = 1\) and this reduces to the simple case above. For split tracts \(w_i < 1\), the contribution to the sum is reduced: a split tract with weight 0.4 and MOE of ±2,000 contributes \((0.4 \times 2{,}000)^2 = 640{,}000\) to the sum, versus \(4{,}000{,}000\) for an interior tract with the same MOE.
Block group resolution. For split tracts where block group estimates are available and none are suppressed, we substitute block group-level estimates for the tract-level estimate. Block groups are smaller subdivisions of a tract (typically 600–3,000 people) that often align more cleanly with the district boundary, reducing extrapolation. Each block group enters the sum with its own population weight \(w_j\) and its own published MOE, using the same formula above. When any block group estimate within a split tract is suppressed, the full tract estimate and weight are used as a fallback.
MOE Propagation — Rate and Median Variables
For rate and median variables (e.g., median household income, percentage renter-occupied), the district estimate is a universe-weighted mean across tracts:
\[\hat{R}_\text{district} = \frac{N}{D} = \frac{\displaystyle\sum_i w_i \cdot u_i \cdot \hat{r}_i}{\displaystyle\sum_i w_i \cdot u_i}\]
where \(\hat{r}_i\) is the rate or median for unit \(i\), and \(u_i\) is its universe (e.g., number of households for median income; number of renter-occupied units for median gross rent).
The MOE propagation uses the Census Bureau ratio formula. First, the numerator and denominator MOEs are accumulated separately via quadrature:
\[\text{MOE}_N = \sqrt{\sum_i \left(w_i \cdot u_i \cdot \text{MOE}_{r,i}\right)^2}\]
\[\text{MOE}_D = \sqrt{\sum_i \left(w_i \cdot \text{MOE}_{u,i}\right)^2}\]
The combined MOE of the ratio is then:
\[\text{MOE}_{\hat{R}} = \frac{1}{D}\sqrt{\text{MOE}_N^2 + \hat{R}^2 \cdot \text{MOE}_D^2}\]
If the expression inside the square root is negative (which can occur at extreme proportions), the Census Bureau prescribes a conservative fallback using addition rather than subtraction under the radical.
Percentage-Point Conversion
For variables displayed as population shares, the raw count MOE is converted to percentage points by dividing by the universe estimate:
\[\text{MOE}_\text{pp} = \frac{\text{MOE}_\text{count}}{\hat{U}} \times 100\]
where \(\hat{U}\) is the published district-level estimate of the relevant universe (total population, housing units, or occupied units, depending on the variable).
Note that this uses the universe estimate as the denominator, not the universe MOE-corrected value. The resulting MOE_pp therefore understates the true uncertainty slightly, since the universe estimate is itself uncertain. For most cases the universe CVs are small enough that this simplification is inconsequential; for small subgroups it may modestly understate the true percentage-point MOE.