Stratified sampling

# Overview
In statistics, stratified sampling is a method of sampling from a population.
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 or systematic 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.

# Stratified sampling strategies
- Proportionate allocation uses a sampling fraction in each of the strata that is proportional to that of the total population. If the population consists of 60% in the male stratum and 40% in the female stratum, then the relative size of the two samples (three males, two females) should reflect this proportion.
- 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.
Similarly, if population density varies greatly within a region, stratified sampling will ensure that estimates can be made with equal accuracy in different parts of the region, and that comparisons of sub-regions can be made with equal statistical power. For example, in Ontario a survey taken throughout the province might use a larger sampling fraction in the less populated north, since the disparity in population between north and south is so great that a sampling fraction based on the provincial sample as a whole might result in the collection of only a handful of data from the north.
Randomized stratification can also be used to improve population representativeness in a study.

# Advantages over other sampling methods
- focuses on important subpopulations and 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, or introduces bias.
- looks randomly within specific sub headings.

# Practical example
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 = 50
- % male, part time = ( 18 / 180 ) x100 = 10
- % female, full time = (9 / 180 ) x 100 = 5
- % female, part time = (63 / 180) 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.