Stratified sampling

Stratified sampling is a method of sampling from a population in statistics.

When sub-populations vary considerably, it is advantageous to sample each subpopulation (stratum) independently. Stratification is the process of grouping members of the population into relatively homogeneous subgroups before sampling. The strata should be mutually exclusive : every element in the population must be assigned to only one stratum. The strata should also be collectively exhaustive : no population element can be excluded. Then random sampling is applied within each stratum. This often improves the representativeness of the sample by reducing sampling error. It can produce a weighted mean that has less variability than the arithmetic mean of a simple random sample of the population.

There are several possible strategies:
 * 1) Proportionate allocation uses a sampling fraction in each of the strata that is proportional to that of the total population. If the population consist of 60% in the male stratum and 40% in the female stratum, then the relative size of the two samples (one males, one females) should reflect this proportion.
 * 2) Optimum allocation (or Disproportionate allocation) - Each stratum is proportionate to the standard deviation of the distribution of the variable. Larger samples are taken in the strata with the greatest variability to generate the least possible sampling variance.

A real-world example of using stratified sampling would be for a US political survey. If we wanted the respondents to reflect the diversity of the population of the United States, the researcher would specifically seek to include participants of various minority groups such as race or religion, based on their proportionality to the total population as mentioned above. A stratified survey could thus claim to be more representative of the US population than a survey of simple random sampling or systematic sampling.

Advantages:
 * focuses on important subpopulations but ignores irrelevant ones
 * improves the accuracy of estimation
 * efficient
 * sampling equal numbers from strata varying widely in size may be used to equate the statistical power of tests of differences between strata.

Disadvantages:
 * can be difficult to select relevant stratification variables
 * not useful when there are no homogeneous subgroups
 * can be expensive
 * requires accurate information about the population.

Choice of Sample Size for each Stratum


 * In general the size of the sample in each stratum is taken in proportion to the size of the stratum. This is called proportional allocation.  Suppose that in a company there are the following staff:
 * male, full time 	90
 * male, part time 	18
 * female, full time 	9
 * female, part time 	63
 * Total                  180

and we are asked to take a sample of 40 staff, stratified according to the above categories.

The first step is to find the total number of staff (180) and calculate the percentage in each group.


 * % male, full time = ( 90 / 180 ) x 100 = 0.5 x 100 = 50
 * % male, part time = ( 18 / 180 ) x100 = 0.1 x 100 = 10
 * % female, full time = (9 / 180 ) x 100 = 0.05 x 100 = 5
 * % female, part time = (63/180)x100 = 0.35 x 100 = 35

This tells us that of our sample of 40,
 * 50% should be male, full time.
 * 10% should be male, part time.
 * 5% should be female, full time.
 * 35% should be female, part time.


 * 50% of 40 is 20.
 * 10% of 40 is 4.
 * 5% of 40 is 2.
 * 35% of 40 is 14.

Sometimes there is greater variability in some strata compared with others. In this case, a larger sample should be drawn from those strata with greater variability.