Stratified Random Sampling - Study Notes

SRS is a random sampling procedure without regard to subgroups.
Two common implementations described in the transcript:
- Method A: assign each student a unique number from $1$ to $N = 420$ ; use a random number generator to select $n = 42$ distinct numbers (ignore repeats). The students with those numbers are surveyed.
- Method B: place the 420 student names in a hat and draw out $42$ names without replacement.
Key point: Each student has an equal probability of being chosen in SRS, provided the sampling is properly random.
What about representation by grade? This is where stratified sampling provides a benefit (see next section).

Definition: The population is subdivided into groups called strata that are homogeneous with respect to a characteristic that might affect the sample. Then a simple random sample is drawn within each stratum.
Why stratify? To ensure representation across key subgroups and to reduce potential bias from over- or under-sampling any single subgroup.
Key terms:
- Strata (plural)
- Stratum (singular)
General rule: You can have as many strata as you please, but they should be roughly homogeneous within and distinct across.
How the school could implement stratified sampling in the given scenario:
- Since 42 is 10% of the total population, survey 10% from each grade (each stratum).
- Allocated sample sizes by grade: fresh: $n1 = 10$ , sophs: $n2 = 11$ , juniors: $n3 = 12$ , seniors: $n4 = 9$ .
- Within each grade, perform SRS (e.g., draw names from a hat or use a random-number method) to obtain those counts.
Resulting idea: This procedure guarantees a cross-section that mirrors the grade distribution and mitigates bias from any single grade’s opinion.
The transcript explicitly states: "Since 42 is 10% of the school's population, survey 10% of each grade. With the freshman, sophomore, junior, and senior classes, randomly select ten, eleven, twelve, and nine."

Strata are homogeneous groups with respect to a characteristic thought to influence the overall sample.
The main idea is to prevent the entire sample from being dominated by a single characteristic.
After splitting into strata, you still perform a simple random sample within each stratum and then combine the results.
Summary wording from the transcript:
- "What you've just described to me … is a sampling method called a stratified random sample. It's a sampling method where the population is subdivided into groups called strata."
- "The strata are homogenous with some respect, characteristic that we think might affect the overall sample."
- "Basically, we don't want too many of the sample to be having this characteristic. But then a simple random sample is carried out within each stratum."
Practical takeaway: Stratification helps achieve a representative cross-section and can reduce sampling error when strata are chosen to be internally homogenous.

Concept: create strata based on value category (low, middle, high).
Procedure: for 15 balls, put all low-valued balls in one hat, all middle-valued in another, and all high-valued in a third; randomly sample from each hat.
Example allocation: draw 2 from each stratum.
Result: a stratified random sample of size 6 with exactly 2 from each category (2 low, 2 middle, 2 high).
Purpose: demonstrates that stratification can enforce representation across predefined categories, yielding a balanced sample across strata.

Population and strata sizes:
- $N = \big(N1 + N2 + N3 + N4\big) = 420$
- $N1 = 100,\, N2 = 110,\, N3 = 120,\, N4 = 90$
Overall sample size:
- $n = 42$
Stratified sample sizes (proportional to stratum size, as demonstrated):
- $n1 = 10,\, n2 = 11,\, n3 = 12,\, n4 = 9$
- These satisfy $nh \,\approx\, n \cdot \frac{Nh}{N} = 42 \cdot \frac{Nh}{420} = 0.10 \cdot Nh$ and $\sum{h=1}^4 nh = 42$ .
General estimator for a population quantity using stratified sampling (conceptual, connects to foundational principles):
- If each stratum provides an estimate $\hat{Y}_h$ based on its sample, then the overall stratified estimator is
- $\hat{Y}{\text{str}} = \sum{h=1}^H Wh \hat{Y}h, \quad Wh = \frac{Nh}{N},$
- where $N = \sum{h=1}^H Nh$ and $H$ is the number of strata.
Allocation note (optional): If you allocate by proportional allocation, then the sum of the stratum samples equals the total sample size by design. Alternative allocations (e.g., equal allocation) are possible but have different variance properties.

Real-world relevance: Stratified sampling is useful when you know that opinions may differ by subgroup (e.g., grade level) and you want to ensure representation from each subgroup.
Ethical and practical considerations:
- Ensures fair representation across grades, reducing potential bias from over- or under-representation.
- Requires a reliable sampling frame for each stratum (e.g., lists by grade).
- More complex to implement than a simple random sample; requires careful planning and documentation.
Connections to foundational principles:
- Probability and random sampling theory: equal-probability sampling at the final stage within strata.
- Variance reduction principle: stratification can decrease estimator variance when within-stratum variance is smaller than overall variance.
- Weighting and pooling: combining stratum estimates with weights proportional to stratum size to form a population-level estimate.

Stratified random sampling: divide the population into homogeneous strata, then perform SRS within each stratum and combine results.
Key terminology: strata (plural), stratum (singular).
In the school example: strata = grades; sample sizes per stratum reflect 10% of each grade, yielding a cross-section across freshmen, sophomores, juniors, and seniors.
The billiards metaphor illustrates how stratification guarantees representation across predefined categories.