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upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia sampling from a population which can be partitioned into subpopulations this article needs more citations please help improve this article by adding citations to reliable sources unsourced material may be challenged and removed find sources stratified sampling news newspapers books scholar jstor december 2020 learn how and when to remove this message in statistics stratified sampling is a method of sampling from a population which can be partitioned into subpopulations 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 sampling is done in each stratum for example by simple random sampling 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 strategies edit 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 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 neyman allocation is a strategy of this type 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 both mean and variance can be corrected for disproportionate sampling costs using stratified sample sizes example edit 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 advantages edit the reasons to use stratified sampling rather than simple random sampling include 2 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 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 disadvantages edit 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 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 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 exist in different groups of data disappear or even reverse when the groups are combined mean and standard error edit the mean and variance of stratified random sampling are given by 2 x 1 n h 1 l n h x h displaystyle bar x frac 1 n sum _ h 1 l n_ h bar x _ h s x 2 h 1 l n h n 2 n h n h n h 1 s h 2 n h displaystyle 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 displaystyle l number of strata n displaystyle n the sum of all stratum sizes n h displaystyle n_ h size of stratum h displaystyle h x h displaystyle bar x _ h sample mean of stratum h displaystyle h n h displaystyle n_ h number of observations in stratum h displaystyle h s h displaystyle s_ h sample standard deviation of stratum h displaystyle h note that the term n h n h n h 1 displaystyle n_ h n_ h n_ h 1 which equals 1 n h 1 n h 1 displaystyle 1 frac n_ h 1 n_ h 1 is a finite population correction and n h displaystyle n_ h must be expressed in sample units forgoing the finite population correction gives s x 2 h 1 l n h n 2 s h 2 n h displaystyle 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 displaystyle w_ h n_ h n is the population weight of stratum h displaystyle h sample size allocation edit 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 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 see also edit mathematics portal opinion poll multistage sampling statistical benchmarking stratified sample size stratification clinical trials references edit botev z ridder a 2017 variance reduction wiley statsref statistics reference online pp 1 6 doi 10 1002 9781118445112 stat07975 hdl 1959 4 unsworks_50616 isbn 9781118445112 1 2 6 1 how to use stratified sampling stat 506 pennstate statistics online courses 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 retrieved 12 july 2012 cite web cs1 maint deprecated archival service link further reading edit särndal carl erik et al 2003 stratified sampling model assisted survey sampling new york springer pp 100 109 isbn 0 387 40620 4 v t e seven basic tools of quality cause and effect diagram check sheet control chart histogram pareto chart scatter diagram stratification quality business v t e statistics outline index descriptive statistics continuous data center mean arithmetic arithmetic geometric contraharmonic cubic generalized power geometric harmonic heronian heinz lehmer median mode dispersion average absolute deviation coefficient of variation interquartile range percentile range standard deviation variance shape central limit theorem moments kurtosis l moments skewness count data index of dispersion summary tables contingency table frequency distribution grouped data dependence partial correlation pearson product moment correlation rank correlation kendall s τ spearman s ρ scatter plot graphics bar chart biplot box plot control chart correlogram fan chart forest plot histogram pie chart q q plot radar chart run chart scatter plot stem and leaf display violin plot heatmap scatter plot matrix ecdf plot line chart statistical data processing transformations data transformation log transformation power transform box cox transformation yeo johnson transformation variance stabilizing transformation anscombe transform fisher transformation scaling and normalization feature scaling normalization standardization z score min max normalization unit vector normalization data cleaning data cleaning outlier winsorizing truncation missing data data reduction dimensionality reduction principal component analysis factor analysis time series preprocessing differencing detrending seasonal adjustment stationarity transformation data collection study design effect size missing data optimal design population replication sample size determination statistic statistical power survey methodology sampling cluster stratified opinion poll questionnaire standard error controlled experiments blocking factorial experiment interaction random assignment randomized controlled trial randomized experiment scientific control adaptive designs adaptive clinical trial stochastic approximation up and down designs observational studies cohort study cross sectional study natural experiment quasi experiment statistical inference statistical theory population statistic probability distribution sampling distribution order statistic empirical distribution density estimation statistical model model specification l p space parameter location scale shape parametric family likelihood monotone location scale family exponential family completeness sufficiency statistical functional bootstrap u v optimal decision loss function efficiency statistical distance divergence asymptotics robustness sensitivity analysis frequentist inference point estimation estimating equations maximum likelihood method of moments m estimator minimum distance unbiased estimators mean unbiased minimum variance rao blackwellization lehmann scheffé theorem median unbiased plug in interval estimation confidence interval pivot likelihood interval prediction interval tolerance interval resampling bootstrap jackknife testing hypotheses 1 2 tails power uniformly most powerful test permutation test randomization test multiple comparisons parametric tests likelihood ratio g test score lagrange multiplier wald z test normal specific tests parametric student s t test f test goodness of fit chi squared kolmogorov smirnov anderson darling lilliefors jarque bera normality shapiro wilk model selection cross validation aic bic rank statistics sign sample median signed rank wilcoxon hodges lehmann estimator rank sum mann whitney nonparametric anova 1 way kruskal wallis 2 way friedman ordered alternative jonckheere terpstra van der waerden test bayesian inference bayesian probability prior posterior credible interval bayes factor bayesian estimator maximum posterior estimator correlation regression analysis correlation pearson product moment partial correlation confounding variable coefficient of determination regression analysis errors and residuals regression validation mixed effects models simultaneous equations models multivariate adaptive regression splines mars template least squares and regression analysis linear regression simple linear regression ordinary least squares general linear model bayesian regression non standard predictors nonlinear regression nonparametric semiparametric isotonic robust homoscedasticity and heteroscedasticity generalized linear model exponential families logistic bernoulli binomial poisson regressions partition of variance analysis of variance anova analysis of variance ancova manova degrees of freedom categorical multivariate time series survival analysis categorical cohen s kappa contingency table graphical model log linear model mcnemar s test cochran mantel haenszel statistics multivariate regression manova principal components canonical correlation discriminant analysis cluster analysis classification structural equation model factor analysis multivariate distributions elliptical distributions normal time series general decomposition trend stationarity seasonal adjustment exponential smoothing cointegration structural break granger causality specific tests dickey fuller johansen q statistic ljung box durbin watson breusch godfrey time domain autocorrelation acf partial pacf cross correlation xcf arma model arima model box jenkins autoregressive conditional heteroskedasticity arch vector autoregression var autoregressive model ar frequency domain spectral density estimation fourier analysis least squares spe...