Probability & Statistics

Stratified Sampling Variance

Choose two strata in a six-item population and compare every stratified estimate with equal-size simple random sampling.

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Stratified Sampling Variance

Concept Overview

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.

Worked Example: Contiguous Strata

Suppose we have a population of 6 items with values (1, 2, 3, 4, 5, 6). We partition this population into two equal contiguous strata of size 3:

  • Stratum 1: values (1, 2, 3)
  • Stratum 2: values (4, 5, 6)

We want to estimate the true mean by sampling one item uniformly from each stratum and averaging the two values.

  1. Calculate True Mean: The population sum is 1+2+3+4+5+6 = 21. The true mean is 21 / 6 = 3.5.
  2. Calculate Stratum Variances: The variance of a finite population stratum of size Nh is (1/Nh) Σ (yi - μh)2.
    For Stratum 1: Mean is 2. Variance is ((1-2)2 + (2-2)2 + (3-2)2) / 3 = (1 + 0 + 1) / 3 = 2/3.
    For Stratum 2: Mean is 5. Variance is ((4-5)2 + (5-5)2 + (6-5)2) / 3 = (1 + 0 + 1) / 3 = 2/3.
  3. Calculate Stratified Estimator Variance: Because the strata are equal in size (each representing 1/2 of the total population) and we select exactly 1 item from each (sample size 1 per stratum of size 3), the variance of the stratified estimator (mean of the two samples) is given by:
    V(yst) = (1/4) * (V1 + V2) = (1/4) * (2/3 + 2/3) = 1/3 ≈ 0.3333.
  4. Compare with Simple Random Sampling (SRS): The population variance is ((1-3.5)2 + ... + (6-3.5)2) / 6 = 17.5 / 6 = 35/12. When selecting a simple random sample of size 2 without replacement, the variance of the sample mean is ((6-2) / (6-1)) * (35/12) / 2 = (4/5) * (35/24) = 7/6 ≈ 1.1667.
    The variance under stratified sampling (1/3) is much smaller than under SRS (7/6) because the contiguous strata group similar values together, minimizing within-stratum variance.

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.

Limitations and Assumptions

There are practical considerations when applying stratified sampling.

  • Finite Population Frame: This module enumerates the exact design variance for a small finite population (N=6). The same finite-design formulas remain valid for larger populations, but practical within-stratum variances usually need to be estimated from the observed sample.
  • Strata Design: The benefit depends on the partition and allocation; grouping similar values often helps in this equal-allocation design. As shown in the interleaved partition example, a poor choice of strata can yield a variance larger than standard simple random sampling.
  • Equal Allocation: The provided model strictly assumes two strata of exactly equal size (3 items each) and a sample of exactly 1 item from each. More complex designs often use proportional or optimal (Neyman) allocation for unequally sized strata.

References

For further reading, see the following textbook material:

Experience it interactively

Adjust parameters, observe in real time, and build deep intuition with Riano’s interactive Stratified Sampling Variance module.

Try Stratified Sampling Variance on Riano →

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