Stratified Sampling Variance

When sampling from a population with distinct subgroups (strata), we can often achieve a more precise estimate of the population mean by using stratified sampling. By independently sampling a fixed number of units from each stratum, we ensure every group is represented. The overall variance of the estimator then depends on the variation within the strata, completely removing the variance between the strata from our uncertainty.

Interactive Experiment

Use the Stratified Sampling Variance interactive module to observe how the choice of strata affects the variance of our estimate.

1. Set the Population preset to "Ordered" and the Partition to "Contiguous". Note that the Stratified Variance (0.333333333333) is lower than the SRS Without Replacement Variance (1.16666666667).
2. Change the Partition to "Interleaved". Notice how Stratum 1 now contains (1, 3, 5) and Stratum 2 contains (2, 4, 6). The within-stratum variance increases drastically. The Stratified Variance is now 1.33333333333, which is worse (higher) than SRS!
3. Switch the Population preset to "Separated" and return Partition to "Contiguous". Here, the population is (0, 0, 0, 10, 10, 10). With a "Contiguous" partition, the variance within each stratum is 0. Consequently, the Stratified Variance drops exactly to 0. Every sample pair yields exactly the true mean.
4. Adjust the First stratum index and Second stratum index to see a specific realized estimator and observe its location in the histogram compared to the true mean. The dashed line marks the selected estimate. The bars show exact probabilities, with each of the 9 stratified pairs weighted 1/9 and each of the 15 unordered SRS pairs weighted 1/15; the expandable table gives their complete counts.