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Define the population weights
N_h is the number of finite population units in stratum h, N is their total and x-bar_h is the stratum sample mean.
| Symbol | Definition |
|---|---|
| W_h | Population-unit share; all shares sum to one |
| x-bar_h | Sample mean from a positive within-stratum sample |
| H | Number of defined nonoverlapping strata |
Start from a population identity
A total across the population is the sum of the stratum totals. Dividing by N expresses the population mean as the sum of each stratum mean multiplied by its share of population units. Replacing each unknown stratum mean with its sample estimate gives the stated estimator.
Inspect a disproportionate allocation
If population shares are 0.75 and 0.25 and the stratum estimates are 4 and 12, the estimated population mean is 0.75 × 4 + 0.25 × 12 = 6. Equal numbers of sampled units do not change those population shares.
Do not silently change the estimand
These weights target an average per population unit. A market-value-weighted bond characteristic or a sales-weighted company measure needs the corresponding target weights. A count-based formula should not be relabelled as an investment-index replication guarantee.