You are here
Homestratified sampling
Primary tabs
stratified sampling
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:
male drivers  female drivers  all drivers 
$10%$  $7%$  $9%$ 
Suppose that we would like to take a sample from $U$ 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 $U$ of $N$ units:
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 $U_{i}$.
Remarks.

When each $S_{i}$ is a simple random sample within each $U_{i}$, then we call this procedure a stratified random sampling.

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 $i$,
$\frac{n}{n_{i}}=W_{i}.$
Mathematics Subject Classification
62D05 no label found Forums
 Planetary Bugs
 HS/Secondary
 University/Tertiary
 Graduate/Advanced
 Industry/Practice
 Research Topics
 LaTeX help
 Math Comptetitions
 Math History
 Math Humor
 PlanetMath Comments
 PlanetMath System Updates and News
 PlanetMath help
 PlanetMath.ORG
 Strategic Communications Development
 The Math Pub
 Testing messages (ignore)
 Other useful stuff
 Corrections