PlanetMath: stratified sampling

(more info)

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stratified sampling

(Definition)

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 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 ) are found in the following table:

male drivers	female drivers	all drivers
$10\%$	$7\%$	$9\%$

Suppose that we would like to take a sample from

so that when the sample units are divided up into male drivers and female drivers, the respective sample claim frequencies are more or less $10\%$ and $7\%$ . How would we do this? If a simple random sample is taken directly from the population, we may get the total claim frequency (for all drivers) to be more or less $9\%$ , but when the sample is broken down into two groups by gender, we may no longer guarantee that the claim frequencies by gender match (more or less) those calculated from the population. To insure that the sample taken preserves claim frequencies by gender, we would take a stratified sampling.

Formally, in stratified sampling, the following steps are taken, in order, from a population

units:

Decide what subdivisions are to be analyzed from within 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.
Make sure all the possible combinations are mutually exclusive events;
Divide into subdivisions, or strata, , where is the total number of possible combinations described above. From the first two steps, we have
$\displaystyle U=U_1\cup U_2\cup\ldots U_k$ such that $\displaystyle U_i\cap U_j=\varnothing,$
for all $i\neq j$ and $1\leq i,j\leq k$ . In addition, if we let $N_i=\lvert U_i \rvert$ , then
$\displaystyle N=\sum_{i=1}^{k}N_i.$
Draw a sample from each stratum .

Remarks.

When each is a simple random sample within each , then we call this procedure a stratified random sampling.
Each stratum corresponds to a number
$\displaystyle W_i:=\frac{N}{N_i},$
called a stratum weight.
Suppose each sample contains 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 ,
$\displaystyle \frac{n}{n_i}=W_i.$