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In statistics, **stratified sampling** is a method of sampling from a population which can be partitioned into subpopulations.

- Example
- Stratified sampling strategies
- Advantages
- Disadvantages
- Mean and standard error
- Sample size allocation
- See also
- References
- Further reading

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.^{ [1] }

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.

*Proportionate allocation*uses a sampling fraction in each of the strata that is 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 (*x*=_{1}*m/n*males,*x*=_{2}*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 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. 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 or systematic sampling.

The reasons to use stratified sampling rather than simple random sampling include^{ [2] }

- If measurements within strata have lower standard deviation (as compared to the overall standard deviation in the population), stratification gives 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 population parameters 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. 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.

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. by means of 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 deleterious effect on the performance of any analysis on the dataset, e.g. classification.^{ [3] } In that regard, minimax sampling ratio can be used to make the dataset robust with respect to uncertainty in the underlying data generating process.^{ [3] }

Combining sub-strata to ensure adequate numbers can lead to Simpson's paradox, where trends that actually exist in different groups of data disappear or even reverse when the groups are combined.

The mean and variance of stratified random sampling are given by:^{ [2] }

where,

- number of strata

- the sum of all stratum sizes

- size of stratum

- sample mean of stratum

- number of observations in stratum

- sample standard deviation of stratum

Note that the term ( − ) / (), which equals (1 − / ), is a finite population correction and must be expressed in "sample units". Foregoing the finite population correction gives:

where the = / is the population weight of stratum .

For 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:^{ [4] }

- 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

In statistics, **cluster sampling** is a sampling plan used when mutually homogeneous yet internally heterogeneous groupings are evident in a statistical population. It is often used in marketing research. In this sampling plan, the total population is divided into these groups and a simple random sample of the groups is selected. The elements in each cluster are then sampled. If all elements in each sampled cluster are sampled, then this is referred to as a "one-stage" cluster sampling plan. If a simple random subsample of elements is selected within each of these groups, this is referred to as a "two-stage" cluster sampling plan. A common motivation for cluster sampling is to reduce the total number of interviews and costs given the desired accuracy. For a fixed sample size, the expected random error is smaller when most of the variation in the population is present internally within the groups, and not between the groups.

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 of the set, while a high standard deviation indicates that the values are spread out over a wider range.

In probability theory and statistics, **variance** is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers is spread out from their average value. Variance has a central role in statistics, where some ideas that use it include descriptive statistics, statistical inference, hypothesis testing, goodness of fit, and Monte Carlo sampling. Variance is an important tool in the sciences, where statistical analysis of data is common. The variance is the square of the standard deviation, the second central moment of a distribution, and the covariance of the random variable with itself, and it is often represented by , , , , or .

The **weighted arithmetic mean** is similar to an ordinary arithmetic mean, except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The notion of weighted mean plays a role in descriptive statistics and also occurs in a more general form in several other areas of mathematics.

In statistics, quality assurance, and survey methodology, **sampling** is the selection of a subset of individuals from within a statistical population to estimate characteristics of the whole population. Statisticians attempt to collect samples that are representative of the population in question. Sampling has lower costs and faster data collection than measuring the entire population and can provide insights in cases where it is infeasible to sample an entire population.

A ** Z-test** is any statistical test for which the distribution of the test statistic under the null hypothesis can be approximated by a normal distribution. Z-tests test the mean of a distribution. For each significance level in the confidence interval, the

In statistics, an **effect size** is a number measuring the strength of the relationship between two variables in a population, or a sample-based estimate of that quantity. It can refer to the value of a statistic calculated from a sample of data, the value of a parameter for a hypothetical population, or to the equation that operationalizes how statistics or parameters lead to the effect size value. Examples of effect sizes include the correlation between two variables, the regression coefficient in a regression, the mean difference, or the risk of a particular event happening. Effect sizes complement statistical hypothesis testing, and play an important role in power analyses, sample size planning, and in meta-analyses. The cluster of data-analysis methods concerning effect sizes is referred to as estimation statistics.

In statistics and optimization, **errors** and **residuals** are two closely related and easily confused measures of the deviation of an observed value of an element of a statistical sample from its "theoretical value". The **error** of an observed value is the deviation of the observed value from the (unobservable) *true* value of a quantity of interest, and the **residual** of an observed value is the difference between the observed value and the *estimated* value of the quantity of interest. The distinction is most important in regression analysis, where the concepts are sometimes called the **regression errors** and **regression residuals** and where they lead to the concept of studentized residuals.

In statistics, a **sampling distribution** or **finite-sample distribution** is the probability distribution of a given random-sample-based statistic. If an arbitrarily large number of samples, each involving multiple observations, were separately used in order to compute one value of a statistic for each sample, then the sampling distribution is the probability distribution of the values that the statistic takes on. In many contexts, only one sample is observed, but the sampling distribution can be found theoretically.

The ** t-test** is any statistical hypothesis test in which the test statistic follows a Student's

The **standard error** (**SE**) of a statistic 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 of the mean** (**SEM**).

In mathematics, **Monte Carlo integration** is a technique for numerical integration using random numbers. It is a particular Monte Carlo method that numerically computes a definite integral. While other algorithms usually evaluate the integrand at a regular grid, Monte Carlo randomly chooses points at which the integrand is evaluated. This method is particularly useful for higher-dimensional integrals.

**Sample size determination** is the act of choosing the number of observations or replicates to include in a statistical sample. The sample size is an important feature of any empirical study in which the goal is to make inferences about a population from a sample. In practice, the sample size used in a study is usually determined based on the cost, time, or convenience of collecting the data, and the need for it to offer sufficient statistical power. In complicated studies there may be several different sample sizes: for example, in a stratified survey there would be different sizes for each stratum. In a census, data is sought for an entire population, hence the intended sample size is equal to the population. In experimental design, where a study may be divided into different treatment groups, there may be different sample sizes for each group.

In mathematics, more specifically in the theory of Monte Carlo methods, **variance reduction** is a procedure used to increase the precision of the estimates that can be obtained for a given simulation or computational effort. Every output random variable from the simulation is associated with a variance which limits the precision of the simulation results. In order to make a simulation statistically efficient, i.e., to obtain a greater precision and smaller confidence intervals for the output random variable of interest, variance reduction techniques can be used. The main ones are common random numbers, antithetic variates, control variates, importance sampling, stratified sampling, moment matching, conditional Monte Carlo and quasi random variables. For simulation with black-box models subset simulation and line sampling can also be used. Under these headings are a variety of specialized techniques; for example, particle transport simulations make extensive use of "weight windows" and "splitting/Russian roulette" techniques, which are a form of importance sampling.

In statistics, the **jackknife** is a resampling technique especially useful for variance and bias estimation. The jackknife pre-dates other common resampling methods such as the bootstrap. The jackknife estimator of a parameter is found by systematically leaving out each observation from a dataset and calculating the estimate and then finding the average of these calculations. Given a sample of size , the jackknife estimate is found by aggregating the estimates of each -sized sub-sample.

**Balanced repeated replication** is a statistical technique for estimating the sampling variability of a statistic obtained by stratified sampling.

**Stratification** of clinical trials is the partitioning of subjects and results by a factor other than the treatment given.

In survey methodology, the **design effect** is the ratio between the variances of two estimators to some parameter of interest. Specifically the ratio of an actual variance of an estimator that is based on a sample from some sampling design, to the variance of an alternative estimator that would be calculated (hypothetically) using a sample from a simple random sample (SRS) of the same number of elements. It measures the expected effect of the design structure on the variance of some estimator of interest. The design effect is a positive real number that can indicate an inflation, or deflation in the variance of an estimator for some parameter, that is due to the study not using SRS.

In statistics, the **Horvitz–Thompson estimator**, named after Daniel G. Horvitz and Donovan J. Thompson, is a method for estimating the total and mean of a pseudo-population in a stratified sample. Inverse probability weighting is applied to account for different proportions of observations within strata in a target population. The Horvitz–Thompson estimator is frequently applied in survey analyses and can be used to account for missing data, as well as many sources of unequal selection probabilities.

In statistics, **stratified randomization** is a method of sampling which first stratifies the whole study population into subgroups with same attributes or characteristics, known as strata, then followed by simple random sampling from the stratified groups, where each element within the same subgroup are selected unbiasedly during any stage of the sampling process, randomly and entirely by chance. Stratified randomization is considered a subdivision of stratified sampling, and should be adopted when shared attributes exist partially and vary widely between subgroups of the investigated population, so that they require special considerations or clear distinctions during sampling. This sampling method should be distinguished from cluster sampling, where a simple random sample of several entire clusters is selected to represent the whole population, or stratified systematic sampling, where a systematic sampling is carried out after the stratification process. Stratified random sampling is sometimes also known as "**quota random sampling**".

- ↑ Botev, Z.; Ridder, A. (2017). "Variance Reduction".
*Wiley StatsRef: Statistics Reference Online*: 1–6. doi:10.1002/9781118445112.stat07975. ISBN 9781118445112. - 1 2 "6.1 How to Use Stratified Sampling | STAT 506".
*onlinecourses.science.psu.edu*. Retrieved 2015-07-23. - 1 2 Shahrokh Esfahani, Mohammad; Dougherty, Edward R. (2014). "Effect of separate sampling on classification accuracy".
*Bioinformatics*.**30**(2): 242–250. doi: 10.1093/bioinformatics/btt662 . PMID 24257187. - ↑ Hunt, Neville; Tyrrell, Sidney (2001). "Stratified Sampling".
*Webpage at Coventry University*. Archived from the original on 13 October 2013. Retrieved 12 July 2012.

- Särndal, Carl-Erik; et al. (2003). "Stratified Sampling".
*Model Assisted Survey Sampling*. New York: Springer. pp. 100–109. ISBN 0-387-40620-4.

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