In sampling surveys, it is sometimes a good idea to break up the population into subdivisions before any sampling were to take place. For example, from a population $U$ of automobile insurance policies, the claim frequencies (loosely speaking, this is the ratio of the number of claims to the number of policies in the $U$ ) are found in the following table:

1. Decide what subdivisions are to be analyzed from within $U$ and what information (or statistics) within the subdivisions should be ``preserved''. For example, if we want to analyze our data by gender, then we would have two subdivisions to study. If there is more than one categorical variable, then we would look at all the possible combinations of the these variables.
2. Make sure all the possible combinations are mutually exclusive events;
3. Divide $U$ into $k$ subdivisions, or strata, $U_i$ , where $k$ is the total number of possible combinations described above. From the first two steps, we have $$U=U_1\cup U_2\cup\ldots U_k \mbox{ such that }U_i\cap U_j=\varnothing,$$ for all $i\neq j$ and $1\leq i,j\leq k$ . In addition, if we let $N_i=\lvert U_i \rvert$ , then $$N=\sum_{i=1}^{k}N_i.$$
4. Draw a sample $S_i$ from each stratum $U_i$ .

• When each $S_i$ is a simple random sample within each $U_i$ , then we call this procedure a stratified random sampling.
• Each stratum corresponds to a number $$W_i:=\frac{N}{N_i},$$ called a stratum weight.
• Suppose each sample $S_i$ contains $n_i$ units ($\lvert S_i \rvert = n_i$ ) and that $n=\sum_{i=1}^{k}n_i$ . We call the stratified sampling proportional if, for each $i$ , $$\frac{n}{n_i}=W_i.$$