## The Stratification Principle

If your only objective of stratification is to produce estimators with small variances, then we want to stratify such that within each stratum, the units are as similar as possible. In a survey of the human population, stratification may be based on socioeconomic factors or geographic regions.

For example, to estimate the average starting income for recent Penn State graduates, it would make sense to stratify by the department since the starting income for graduates of the same department would be similar.

## Allocation in Stratified Random Sampling

The question is, given a total sample size of *n*, how do we allocate these among *L* strata?

### Try it!

The best allocation scheme is affected by the following three factors:

- the total number of elements in each stratum,
- the variability of the measurements within each stratum, and
- the cost associated with obtaining an observation from each stratum.

If we don't have all this information, but we know the total number, we can use a simplistic allocation. This is a proportional allocation that will maintain a steady sampling fraction throughout the population.

\(n_h=\dfrac{n\cdot N_h}{N}\)

This does not take into consideration the variability within each stratum and is not the optimal choice.

If the cost of sampling from each stratum is the same, then the optimal allocation (the allocation with the lowest variances) is:

\(n_h=\dfrac{n \cdot N_h \sigma_h}{\sum\limits_{k=1}^L N_k \sigma_k}\)

read text section 11.8 for proof

However, if the cost of sampling differs from stratum to stratum and the total cost is:

\(c=c_0+c_1n_1+c_2n_2+...+c_Ln_L\)

where \(c_0\) is the overhead cost, \(c_h\) is the cost per unit for stratum *h*. The optimal allocation is:

\(n_h=\dfrac{(c-c_0)N_h \sigma_h/\sqrt{c_h}}{\sum\limits_{k=1}^L N_k \sigma_k \sqrt{c_k}}\)

**Note!**

- the sample size is directly proportional to \(N_h\) and \(\sigma_h\), i.e., allocate a larger sample size to the larger and more variable stratum.
- the sample size is inversely proportional to \(\sqrt{c_h}\), i.e., this allocates smaller sample sizes to the more expensive stratum.

In order to use the optimal allocation, one must be able to estimate σ_{h}

Let's take a look at this in the context of the TV Example...

### Try it!

Optimal allocation:

\(n_h=\dfrac{n \cdot N_h \sigma_h}{\sum\limits_{k=1}^L N_k \sigma_k}\)

where,

\(N_1=155, \sigma_1=5\)

\(N_2=62, \sigma_2=15\)

\(N_3=93, \sigma_3=10\)

Then,

\(n_1=\dfrac{40 \times 155 \times 5}{155 \times 5+62 \times 15+93 \times 10}=11.7647\)

\(n_2=\dfrac{40 \times 62 \times 15}{155 \times 5+62 \times 15+93 \times 10}=14.1176\)

\(n_3=\dfrac{40 \times 93 \times 10}{155 \times 5+62 \times 15+93 \times 10}=14.1177\)

Thus we will choose \(n_1=12, n_2=14\) and \(n_3=14\).

Remember, it is important that \(n_1+n_2+n_3=40\) in this case.