The GRI is built on Total Variation Distance (TVD), one of the most fundamental metrics in probability theory. For discrete distributions \(P\) and \(Q\) over a shared set of categories:
\[ \text{TVD}(P, Q) = \frac{1}{2} \sum_{i=1}^{k} |p_i - q_i| \]
TVD has several properties that make it ideal for representativeness measurement:
The GRI is defined as the complement of TVD:
\[ \text{GRI}(p, q) = 1 - \text{TVD}(p, q) = 1 - \frac{1}{2} \sum_{i=1}^{k} |p_i - q_i| \]
This maps to a 0–1 scale where 1.0 = perfect match and 0.0 = complete mismatch.
Inputs:
Process:
The GRI captures how closely the sample matches the population, but not how many population segments are represented at all. The Diversity Score fills this gap:
\[ \text{Diversity} = \frac{|\{i : p_i > 0 \text{ and } q_i \geq X\}|}{|\{i : q_i \geq X\}|} \]
where \(X = \frac{1}{2N}\) is a dynamic relevance threshold that scales with sample size. This measures the fraction of meaningfully-sized population strata that appear at least once in the sample.
The GRI is computed across 13 dimensions organized in three tiers:
These combine geography with demographics, producing the finest-grained assessment:
| Dimension | Strata Count | Benchmark Source |
|---|---|---|
| Country x Gender x Age | 2,699 | UN WPP 2023 |
| Country x Religion | 1,607 | Pew GLS 2010 |
| Country x Environment | 449 | UN WUP 2018 |
Replace countries with 20 world regions, reducing cardinality:
Assess one demographic factor at a time:
Each dimension tells a different story. A survey might achieve strong continental representation (GRI = 0.88) while poorly representing the Country x Gender x Age intersection (GRI = 0.37)—the scorecard makes these trade-offs transparent.
With thousands of strata and finite samples, perfect GRI scores are mathematically impossible. We use Monte Carlo simulation to estimate the maximum achievable GRI at any sample size by generating optimally-allocated samples.
At \(N = 1{,}000\):
| Dimension | Max GRI | Max Diversity |
|---|---|---|
| Country x Gender x Age | 0.792 | 0.925 |
| Country x Religion | 0.938 | 0.970 |
| Country x Environment | 0.950 | 0.977 |
These ceilings enable efficiency ratios—the percentage of the theoretical maximum that a real survey achieves—providing a fairer basis for comparison than raw scores alone.
| Dataset | Source | Year | Use |
|---|---|---|---|
| World Population Prospects | UN DESA Population Division | 2023 | Country x Gender x Age benchmarks |
| Global Religious Landscape | Pew Research Center | 2010 | Country x Religion benchmarks |
| World Urbanization Prospects | UN DESA Population Division | 2018 | Country x Environment benchmarks |
All benchmark data is included in the repository under data/raw/benchmark_data/ with full attribution in Sources.csv.
The religious composition data from Pew dates to 2010 and may not reflect current demographics in rapidly changing regions. Urban/rural classifications use national-level data that can mask within-country variation. These limitations are inherent to the best available global demographic data and should be considered when interpreting GRI scores.