Stratified Sampling
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In
statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, stratified sampling is a method of sampling from a
population Population typically refers to the number of people in a single area, whether it be a city or town, region, country, continent, or the world. Governments typically quantify the size of the resident population within their jurisdiction using a ...
which can be partitioned into
subpopulation In statistics, a population is a Set (mathematics), set of similar items or events which is of interest for some question or experiment. A statistical population can be a group of existing objects (e.g. the set of all stars within the Milky Way g ...
s. In
statistical survey Survey methodology is "the study of survey methods". As a field of applied statistics concentrating on human-research surveys, survey methodology studies the sampling of individual units from a population and associated techniques of survey da ...
s, when subpopulations within an overall population vary, it could be advantageous to sample each subpopulation (stratum) independently. Stratification is the process of dividing members of the population into homogeneous subgroups before sampling. The strata should define a partition of the population. That is, it should be ''
collectively exhaustive In probability theory and logic, a set of events is jointly or collectively exhaustive if at least one of the events must occur. For example, when rolling a six-sided die, the events 1, 2, 3, 4, 5, and 6 balls of a single outcome are collect ...
'' and ''
mutually exclusive In logic and probability theory, two events (or propositions) are mutually exclusive or disjoint if they cannot both occur at the same time. A clear example is the set of outcomes of a single coin toss, which can result in either heads or tails ...
'': every element in the population must be assigned to one and only one stratum. Then
simple random sampling In statistics, a simple random sample (or SRS) is a subset of individuals (a sample (statistics), sample) chosen from a larger Set (mathematics), set (a statistical population, population) in which a subset of individuals are chosen randomization, ...
is applied within each stratum. The objective is to improve the precision of the sample by reducing
sampling error In statistics, sampling errors are incurred when the statistical characteristics of a population are estimated from a subset, or sample, of that population. Since the sample does not include all members of the population, statistics of the sample ( ...
. It can produce a
weighted mean The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The ...
that has less variability than the
arithmetic mean In mathematics and statistics, the arithmetic mean ( ) or arithmetic average, or just the ''mean'' or the ''average'' (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The colle ...
of a
simple random sample In statistics, a simple random sample (or SRS) is a subset of individuals (a sample) chosen from a larger set (a population) in which a subset of individuals are chosen randomly, all with the same probability. It is a process of selecting a sample ...
of the population. In
computational statistics Computational statistics, or statistical computing, is the bond between statistics and computer science. It means statistical methods that are enabled by using computational methods. It is the area of computational science (or scientific computi ...
, stratified sampling is a method of
variance reduction In mathematics, more specifically in the theory of Monte Carlo methods, variance reduction is a procedure used to increase the precision of the estimates obtained for a given simulation or computational effort. Every output random variable fro ...
when
Monte Carlo method Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be determi ...
s are used to estimate population statistics from a known population.


Example

Assume that we need to estimate the average number of votes for each candidate in an election. Assume that a country has 3 towns: Town A has 1 million factory workers, Town B has 2 million office workers and Town C has 3 million retirees. We can choose to get a random sample of size 60 over the entire population but there is some chance that the resulting random sample is poorly balanced across these towns and hence is biased, causing a significant error in estimation (when the outcome of interest has a different distribution, in terms of the parameter of interest, between the towns). Instead, if we choose to take a random sample of 10, 20 and 30 from Town A, B and C respectively, then we can produce a smaller error in estimation for the same total sample size. This method is generally used when a population is not a homogeneous group.


Stratified sampling strategies

#''Proportionate allocation'' uses a
sampling fraction In sampling theory, the sampling fraction is the ratio of sample size to population size or, in the context of stratified sampling, the ratio of the sample size to the size of the stratum. The formula for the sampling fraction is :f=\frac, where ' ...
in each of the strata that are proportional to that of the total population. For instance, if the population consists of ''n'' total individuals, ''m'' of which are male and ''f'' female (and where ''m'' + ''f'' = ''n''), then the relative size of the two samples (''x1'' = ''m/n'' males, ''x2'' = ''f/n'' females) should reflect this proportion. #''Optimum allocation'' (or ''disproportionate allocation'') - The sampling fraction of each stratum is proportionate to both the proportion (as above) and the
standard deviation In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while ...
of the distribution of the variable. Larger samples are taken in the strata with the greatest variability to generate the least possible overall sampling variance. A real-world example of using stratified sampling would be for a political
survey Survey may refer to: Statistics and human research * Statistical survey, a method for collecting quantitative information about items in a population * Survey (human research), including opinion polls Spatial measurement * Surveying, the techniq ...
. If the respondents needed to reflect the diversity of the population, 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 population than a survey of
simple random sampling In statistics, a simple random sample (or SRS) is a subset of individuals (a sample (statistics), sample) chosen from a larger Set (mathematics), set (a statistical population, population) in which a subset of individuals are chosen randomization, ...
or systematic sampling.


Advantages

The reasons to use stratified sampling rather than
simple random sampling In statistics, a simple random sample (or SRS) is a subset of individuals (a sample (statistics), sample) chosen from a larger Set (mathematics), set (a statistical population, population) in which a subset of individuals are chosen randomization, ...
include # If measurements within strata have a lower standard deviation (as compared to the overall standard deviation in the population), stratification gives a smaller error in estimation. # For many applications, measurements become more manageable and/or cheaper when the population is grouped into strata. # When it is desirable to have estimates of the population
parameters A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
for groups within the population - stratified sampling verifies we have enough samples from the strata of interest. If the 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 In statistics, the power of a binary hypothesis test is the probability that the test correctly rejects the null hypothesis (H_0) when a specific alternative hypothesis (H_1) is true. It is commonly denoted by 1-\beta, and represents the chances ...
. For example, in
Ontario Ontario ( ; ) is one of the thirteen provinces and territories of Canada.Ontario is located in the geographic eastern half of Canada, but it has historically and politically been considered to be part of Central Canada. Located in Central Ca ...
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.


Disadvantages

Stratified sampling is not useful when the population cannot be exhaustively partitioned into disjoint subgroups. It would be a misapplication of the technique to make subgroups' sample sizes proportional to the amount of data available from the subgroups, rather than scaling sample sizes to subgroup sizes (or to their variances, if known to vary significantly—e.g. using an
F Test An ''F''-test is any statistical test in which the test statistic has an ''F''-distribution under the null hypothesis. It is most often used when comparing statistical models that have been fitted to a data set, in order to identify the model th ...
). Data representing each subgroup are taken to be of equal importance if suspected variation among them warrants stratified sampling. If subgroup variances differ significantly and the data needs to be stratified by variance, it is not possible to simultaneously make each subgroup sample size proportional to subgroup size within the total population. For an efficient way to partition sampling resources among groups that vary in their means, variance and costs, see "optimum allocation". The problem of stratified sampling in the case of unknown class priors (ratio of subpopulations in the entire population) can have a deleterious effect on the performance of any analysis on the dataset, e.g. classification. In that regard, minimax sampling ratio can be used to make the dataset robust with respect to uncertainty in the underlying data generating process. Combining sub-strata to ensure adequate numbers can lead to Simpson's paradox, where trends that exist in different groups of data disappear or even reverse when the groups are combined.


Mean and standard error

The mean and variance of stratified random sampling are given by: :\bar = \frac \sum_^L N_h \bar :s_\bar^2 = \sum_^L \left(\frac\right)^2 \left(\frac\right)\frac where, :L = number of strata :N = the sum of all stratum sizes :N_h = size of stratum h :\bar = sample mean of stratum h :n_h = number of observations in stratum h :s_h = sample standard deviation of stratum h Note that the term (N_hn_h) / (N_h), which equals (1 − n_h / N_h), is a
finite population correction The standard error (SE) of a statistic (usually an estimate of a parameter) is the standard deviation of its sampling distribution or an estimate of that standard deviation. If the statistic is the sample mean, it is called the standard error o ...
and N_h must be expressed in "sample units". Foregoing the finite population correction gives: :s_\bar^2 = \sum_^L \left(\frac\right)^2 \frac where the w_h = N_h/N is the population weight of stratum h.


Sample size allocation

For the proportional allocation strategy, the size of the sample in each stratum is taken in proportion to the size of the stratum. 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 calculate the percentage of each group of the total. *% male, full-time = 90 ÷ 180 = 50% *% male, part-time = 18 ÷ 180 = 10% *% female, full-time = 9 ÷ 180 = 5% *% female, part-time = 63 ÷ 180 = 35% This tells us that of our sample of 40, *50% (20 individuals) should be male, full-time. *10% (4 individuals) should be male, part-time. *5% (2 individuals) should be female, full-time. *35% (14 individuals) should be female, part-time. Another easy way without having to calculate the percentage is to multiply each group size by the sample size and divide by the total population size (size of entire staff): * male, full-time = 90 × (40 ÷ 180) = 20 * male, part-time = 18 × (40 ÷ 180) = 4 * female, full-time = 9 × (40 ÷ 180) = 2 * female, part-time = 63 × (40 ÷ 180) = 14


See also

*
Opinion poll An opinion poll, often simply referred to as a survey or a poll (although strictly a poll is an actual election) is a human research survey of public opinion from a particular sample. Opinion polls are usually designed to represent the opinions ...
*
Multistage sampling In statistics, multistage sampling is the taking of samples in stages using smaller and smaller sampling units at each stage. Multistage sampling can be a complex form of cluster sampling because it is a type of sampling which involves dividing ...
*
Statistical benchmarking {{Unreferenced, date=October 2007 In statistics, benchmarking is a method of using auxiliary information to adjust the sampling weights used in an estimation process, in order to yield more accurate estimates of totals. Suppose we have a popula ...
* Stratified sample size *
Stratification (clinical trials) Stratification of clinical trials is the partitioning of subjects and results by a factor other than the treatment given. Stratification can be used to ensure equal allocation of subgroups of participants to each experimental condition. This may ...


References


Further reading

* {{DEFAULTSORT:Stratified Sampling Sampling (statistics) Sampling techniques Variance reduction