On (Modest) Differences In Racial Distribution of Voting Eligible Population and Registered Voters in California

Each election cycle brings renewed concern about differential voter registration rates across demographic groups. In California, for instance, only 62.8% of eligible Latinos are registered to vote, compared to 72.9% of eligible Whites—a gap of over 10 percentage points. Such disparities rightfully worry those committed to democratic representation. Yet the relationship between registration rate gaps and the actual composition of the registered voter pool is more nuanced than raw percentages suggest.

This essay presents a mathematical framework for understanding how registration gaps translate into compositional shifts. The key insight is that two levers work together to determine representation: the initial population mix and the rate gaps in registration. Even substantial rate differences need not produce equally dramatic shifts in who ends up on the voter rolls—though the shifts remain real and policy-relevant.

The Mathematical Framework

Two-Group Model

Consider a population divided into two groups: group $a$ with population share $p$ and group $b$ with share $1-p$. Their registration rates are $q_a$ and $q_b$ respectively. We define the rate ratio as $r \equiv q_a/q_b$.

After registration, the shares become:

$P'_a = \frac{p \cdot q_a}{p \cdot q_a + (1-p) \cdot q_b}$

$P'_b = 1 - P'_a$

This formula reveals a fundamental principle: registration acts as a reweighting mechanism. Groups with higher registration rates gain proportionally larger shares among registrants.

The Odds Perspective

A particularly intuitive way to understand this reweighting is through odds. The odds of being in group $a$ versus group $b$ update according to:

$\frac{P'_a}{P'_b} = \frac{p}{1-p} \times \frac{q_a}{q_b}$

We can write this as:

• Initial odds: $p/(1-p)$
• Rate ratio: $r = q_a/q_b$
• Updated odds: Initial odds × Rate ratio

Differential registration multiplies the initial odds by the rate ratio. If $r > 1$, group $a

Measuring Absolute Share Changes

While odds ratios provide relative measures, absolute share changes matter for practical policy discussions:

$$P'_a - p = \frac{p(1-p)(q_a - q_b)}{p \cdot q_a + (1-p) \cdot q_b}$$

This formula yields several insights:

1. Direction: The sign of the share change equals the sign of $(q_a - q_b)$
2. Magnitude: The change is largest when $p \approx 0.5$ (groups are equally sized)
3. Bounds: The term $p(1-p)$ ensures changes are naturally bounded

When registration rates are similar ($q_a \approx q_b \approx q$), we can approximate:

$$P'_a - p \approx \frac{p(1-p)}{q}(q_a - q_b)$$

This shows that share changes are proportional to rate differences, scaled by the variance-like term $p(1-p)/q$.

Sensitivity Analysis

How responsive are registered shares to changes in registration rates? Taking derivatives:

$\frac{\partial P'_a}{\partial q_a} = \frac{p(1-p) \cdot q_b}{D^2}$

$\frac{\partial P'_a}{\partial q_b} = -\frac{p(1-p) \cdot q_a}{D^2}$

where $D = p \cdot q_a + (1-p) \cdot q_b$ is the overall registration rate.

These sensitivities reveal that:

• Effects are symmetric in magnitude but opposite in sign
• Sensitivity is highest when groups are balanced ($p \approx 0.5$)
• Higher overall registration rates ($D$) dampen the effects of rate changes

Multi-Group Generalization

Real populations contain multiple groups. For groups $g = 1, \ldots, G$ with shares $p_g$ and rates $q_g$:

$P'g = \frac{p_g \cdot q_g}{\sum{j=1}^{G} p_j \cdot q_j}$

Pairwise odds comparisons reduce to the two-group rule:

$$\frac{P'_g}{P'_h} = \frac{p_g}{p_h} \cdot \frac{q_g}{q_h}$$

This shows that the two-group intuition extends naturally to complex demographic landscapes.

A Real-World Example: California's Voter Registration

Let's apply this framework to California's voting-eligible population (VEP):

The aggregate "registered mass" is:

$D = 0.628 \times 0.729 + 0.230 \times 0.628 + 0.142 \times 0.620 \approx 0.690$

This yields registered voter shares of:

• Whites: $(0.628 \times 0.729) / 0.690 \approx 66.3%$ (up 3.5 percentage points)
• Latinos: $(0.230 \times 0.628) / 0.690 \approx 20.9%$ (down 2.1 percentage points)
• Others: $(0.142 \times 0.620) / 0.690 \approx 12.8%$ (down 1.4 percentage points)

Despite a 10-point registration rate gap between Whites and Latinos, the compositional shift is more modest—Whites increase their share by 3.5 percentage points rather than 10. This illustrates a crucial point: rate gaps don't translate one-to-one into compositional distortions.

Why Distortions Are Bounded

The mathematical framework reveals why even substantial rate gaps produce bounded compositional effects:

1. Initial shares matter: Small groups remain relatively small even with higher registration rates. A group that's 5% of the population might double its registration rate and still be under 10% of registrants.
2. The denominator effect: The overall registration rate ($D$) appears in the denominator of all share calculations, dampening extreme shifts.
3. Competition for shares: In a multi-group setting, one group's gain is necessarily others' losses, creating natural bounds on any single group's change.