In statistics, stratified sampling is a method of sampling from a population. A graph of a bell curve in a normal distribution showing statistics used in educational assessment, comparing various grading methods. ... Sampling is that part of statistical practice concerned with the selection of individual observations intended to yield some knowledge about a population of concern, especially for the purposes of statistical inference. ...

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. In statistics, given a set of data, X = { x1, x2, ..., xn} and corresponding weights, W = { w1, w2, ..., wn} the weighted mean is calculated as Note that if all the weights are equal, the weighted mean is the same as the arithmetic mean. ... In mathematics and statistics, the arithmetic mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided by the number of items in the list. ...

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. In sampling theory, sampling fraction is the ratio of sample size to population size. ... In probability and statistics, the standard deviation is the most common measure of statistical dispersion. ... Statistical surveys are used to collect quantitative information about items in a population. ... In statistics, a simple random sample from a population is a sample chosen randomly, in which each member of the population has the same probability of being chosen. ... Systematic sampling is the selection of every kth element from a sampling frame, where k, the sampling interval, is calculated as: k = Number in population / Number in sample Using this procedure each element in the population has a known and equal probability of selection. ...

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.

The power of a statistical test is the probability that the test will reject a false null hypothesis, or in other words that it will not make a Type II error. ... One may be faced with the problem of making a definite decision with respect to an uncertain hypothesis which is known only through its observable consequences. ...

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.

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.