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stratified sampling
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{{Short description|Sampling from a population which can be partitioned into subpopulations}}{{more citations needed|date=December 2020}}In statistics, stratified sampling is a method of sampling from a population which can be partitioned into subpopulations.(File:Stratified_sampling.PNG|thumb|Stratified sampling example)In statistical surveys, 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 and mutually exclusive: every element in the population must be assigned to one and only one stratum. Then simple random sampling is applied within each stratum. The objective is to improve the precision 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 computational statistics, stratified sampling is a method of variance reduction when Monte Carlo methods are used to estimate population statistics from a known population.JOURNAL, Botev, Z., Ridder, A., Variance Reduction, Wiley StatsRef: Statistics Reference Online, 2017, 1â6, 10.1002/9781118445112.stat07975, 9781118445112, - the content below is remote from Wikipedia
- it has been imported raw for GetWiki
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 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) â The sampling fraction of each stratum is proportionate to both the proportion (as above) and 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 overall sampling variance.
Advantages
The reasons to use stratified sampling rather than simple random sampling includeWEB, 6.1 How to Use Stratified Sampling {{!, STAT 506|url =weblink|website = onlinecourses.science.psu.edu|access-date = 2015-07-23}}- 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 for groups within the population â stratified sampling verifies we have enough samples from the strata of interest.
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 significantlyâe.g. using an F test). 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{x} = frac{1}{N} sum_{h=1}^L N_h bar{x}_h
s_bar{x}^2 = sum_{h=1}^L left(frac{N_h}{N}right)^2 left(frac{N_h - n_h}{N_h - 1}right)frac{s_h^2}{n_h}
where
L ={}number of strata
N ={}the sum of all stratum sizes
N_h ={}size of stratum h
bar{x}_h ={}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_h-n_h) / (N_h-1), which equals 1-frac{n_h - 1}{N_h - 1}, is a finite population correction and N_h must be expressed in "sample units". Foregoing the finite population correction gives:
s_bar{x}^2 = sum_{h=1}^L left(frac{N_h}{N}right)^2 frac{s_h^2}{n_h}
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:WEB, Hunt, Neville, Tyrrell, Sidney, 2001,weblinkweblink" title="archive.today/20131013132818weblink">weblink dead, 13 October 2013, Stratified Sampling, Webpage at Coventry University, 12 July 2012,- male, full-time: 90
- male, part-time: 18
- female, full-time: 9
- female, part-time: 63
- total: 180
- % male, full-time = 90 ÷ 180 = 50%
- % male, part-time = 18 ÷ 180 = 10%
- % female, full-time = 9 ÷ 180 = 5%
- % female, part-time = 63 ÷ 180 = 35%
- 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.
- male, full-time = 90 à (40 ÷ 180) = 20
- male, part-time = 18 à (40 ÷ 180) = 4
- female, full-time = 9 à (40 ÷ 180) = 2
- female, part-time = 63 à (40 ÷ 180) = 14
See also
- Opinion poll
- Multistage sampling
- Statistical benchmarking
- Stratified sample size
- Stratification (clinical trials)
References
Further reading
- BOOK, Särndal, Carl-Erik, Stratified Sampling, Model Assisted Survey Sampling, New York, Springer, 2003, 100â109, 0-387-40620-4, etal,
- content above as imported from Wikipedia
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- "stratified sampling" does not exist on GetWiki (yet)
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