Sampling distribution - wikipedia Sampling distribution In statistics, a sampling distribution or finite - sample distribution is the probability distribution of a given statistic based on a random sample. Sampling distributions are important in statistics because they provide a major simplification en route to statistical inference. More specifically, they allow analytical considerations to be based on the sampling distribution of a statistic, rather than on the joint probability distribution of all the individual sample values. Contents 1 Introduction 2 Standard error 3 Examples 4 Statistical inference 5 References 6 External links Introduction The sampling distribution of a statistic is the distribution of that statistic, considered as a random variable, when derived from a random sample of size n. It may be considered as the distribution of the statistic for all possible samples from the same population of a given size. The sampling distribution depends on the underlying distribution of the population, the statistic being considered, the sampling procedure employed, and the sample size used. There is often considerable interest in whether the sampling distribution can be approximated by an asymptotic distribution, which corresponds to the limiting case either as the number of random samples of finite size, taken from an infinite population and used to produce the distribution, tends to infinity, or when just one equally - infinite - size `` sample '' is taken of that same population. For example, consider a normal population with mean μ and variance σ2. Assume we repeatedly take samples of a given size from this population and calculate the arithmetic mean x  ̄ ( displaystyle scriptstyle ( \bar ( x ) ) ) for each sample -- this statistic is called the sample mean. Each sample has its own average value, and the distribution of these averages is called the `` sampling distribution of the sample mean ''. This distribution is normal N ( μ, σ 2 / n ) ( displaystyle scriptstyle ( mathcal ( N ) ) ( mu, , sigma ^ ( 2 ) / n ) ) ( n is the sample size ) since the underlying population is normal, although sampling distributions may also often be close to normal even when the population distribution is not ( see central limit theorem ). An alternative to the sample mean is the sample median. When calculated from the same population, it has a different sampling distribution to that of the mean and is generally not normal ( but it may be close for large sample sizes ). The mean of a sample from a population having a normal distribution is an example of a simple statistic taken from one of the simplest statistical populations. For other statistics and other populations the formulas are more complicated, and often they do n't exist in closed - form. In such cases the sampling distributions may be approximated through Monte - Carlo simulations, bootstrap methods, or asymptotic distribution theory. Standard error The standard deviation of the sampling distribution of a statistic is referred to as the standard error of that quantity. For the case where the statistic is the sample mean, and samples are uncorrelated, the standard error is : σ x  ̄ = σ n ( displaystyle sigma _ ( \bar ( x ) ) = ( \frac ( sigma ) ( sqrt ( n ) ) ) ) where σ ( displaystyle sigma ) is the standard deviation of the population distribution of that quantity and n is the sample size ( number of items in the sample ). An important implication of this formula is that the sample size must be quadrupled ( multiplied by 4 ) to achieve half ( 1 / 2 ) the measurement error. When designing statistical studies where cost is a factor, this may have a role in understanding cost -- benefit tradeoffs. Examples Population Statistic Sampling distribution Normal : N ( μ, σ 2 ) ( displaystyle ( mathcal ( N ) ) ( mu, sigma ^ ( 2 ) ) ) Sample mean X  ̄ ( displaystyle ( \bar ( X ) ) ) from samples of size n X  ̄ ∼ N ( μ, σ 2 n ) ( displaystyle ( \bar ( X ) ) sim ( mathcal ( N ) ) ( Big ( ) mu, , ( \frac ( sigma ^ ( 2 ) ) ( n ) ) ( Big ) ) ) Bernoulli : Bernoulli ⁡ ( p ) ( displaystyle operatorname ( Bernoulli ) ( p ) ) Sample proportion of `` successful trials '' X  ̄ ( displaystyle ( \bar ( X ) ) ) n X  ̄ ∼ Binomial ⁡ ( n, p ) ( displaystyle n ( \bar ( X ) ) sim operatorname ( Binomial ) ( n, p ) ) Two independent normal populations : N ( μ 1, σ 1 2 ) ( displaystyle ( mathcal ( N ) ) ( mu _ ( 1 ), sigma _ ( 1 ) ^ ( 2 ) ) ) and N ( μ 2, σ 2 2 ) ( displaystyle ( mathcal ( N ) ) ( mu _ ( 2 ), sigma _ ( 2 ) ^ ( 2 ) ) ) Difference between sample means, X  ̄ 1 − X  ̄ 2 ( displaystyle ( \bar ( X ) ) _ ( 1 ) - ( \bar ( X ) ) _ ( 2 ) ) X  ̄ 1 − X  ̄ 2 ∼ N ( μ 1 − μ 2, σ 1 2 n 1 + σ 2 2 n 2 ) ( displaystyle ( \bar ( X ) ) _ ( 1 ) - ( \bar ( X ) ) _ ( 2 ) sim ( mathcal ( N ) ) ! left ( mu _ ( 1 ) - mu _ ( 2 ), , ( \frac ( sigma _ ( 1 ) ^ ( 2 ) ) ( n_ ( 1 ) ) ) + ( \frac ( sigma _ ( 2 ) ^ ( 2 ) ) ( n_ ( 2 ) ) ) \right ) ) Any absolutely continuous distribution F with density ƒ Median X ( k ) ( displaystyle X_ ( ( k ) ) ) from a sample of size n = 2k − 1, where sample is ordered X ( 1 ) ( displaystyle X_ ( ( 1 ) ) ) to X ( n ) ( displaystyle X_ ( ( n ) ) ) f X ( k ) ( x ) = ( 2 k − 1 ) ! ( k − 1 ) ! 2 f ( x ) ( F ( x ) ( 1 − F ( x ) ) ) k − 1 ( displaystyle f_ ( X_ ( ( k ) ) ) ( x ) = ( \frac ( ( 2k - 1 ) ! ) ( ( k - 1 ) ! ^ ( 2 ) ) ) f ( x ) ( Big ( ) F ( x ) ( 1 - F ( x ) ) ( Big ) ) ^ ( k - 1 ) ) Any distribution with distribution function F Maximum M = max X k ( displaystyle M = max X_ ( k ) ) from a random sample of size n F M ( x ) = P ( M ≤ x ) = ∏ P ( X k ≤ x ) = ( F ( x ) ) n ( displaystyle F_ ( M ) ( x ) = P ( M leq x ) = prod P ( X_ ( k ) leq x ) = left ( F ( x ) \right ) ^ ( n ) ) Statistical inference In the theory of statistical inference, the idea of a sufficient statistic provides the basis of choosing a statistic ( as a function of the sample data points ) in such a way that no information is lost by replacing the full probabilistic description of the sample with the sampling distribution of the selected statistic. In frequentist inference, for example in the development of a statistical hypothesis test or a confidence interval, the availability of the sampling distribution of a statistic ( or an approximation to this in the form of an asymptotic distribution ) can allow the ready formulation of such procedures, whereas the development of procedures starting from the joint distribution of the sample would be less straightforward. In Bayesian inference, when the sampling distribution of a statistic is available, one can consider replacing the final outcome of such procedures, specifically the conditional distributions of any unknown quantities given the sample data, by the conditional distributions of any unknown quantities given selected sample statistics. Such a procedure would involve the sampling distribution of the statistics. The results would be identical provided the statistics chosen are jointly sufficient statistics. This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. ( May 2014 ) References Jump up ^ Mooney, Christopher Z. ( 1999 ). Monte Carlo simulation. Thousand Oaks, Calif. : Sage. ISBN 9780803959439. Merberg, A. and S.J. Miller ( 2008 ). `` The Sample Distribution of the Median ''. Course Notes for Math 162 : Mathematical Statistics, on the web at http://web.williams.edu/Mathematics/sjmiller/public_html/BrownClasses/162/Handouts/MedianThm04.pdf, pgs 1 -- 9. External links Generate sampling distributions in Excel Mathematica demonstration showing the sampling distribution of various statistics ( e.g. Σx2 ) for a normal population Statistics Outline Index Descriptive statistics Continuous data Center Mean arithmetic geometric harmonic Median Mode Dispersion Variance Standard deviation Coefficient of variation Percentile Range Interquartile range Shape Central limit theorem Moments Skewness Kurtosis L - moments Count data Index of dispersion Summary tables Grouped data Frequency distribution Contingency table Dependence Pearson product - moment correlation Rank correlation Spearman 's rho Kendall 's tau Partial correlation Scatter plot Graphics Bar chart Biplot Box plot Control chart Correlogram Fan chart Forest plot Histogram Pie chart Q -- Q plot Run chart Scatter plot Stem - and - leaf display Radar chart Data collection Study design Population Statistic Effect size Statistical power Sample size determination Missing data Survey methodology Sampling stratified cluster Standard error Opinion poll Questionnaire Controlled experiments Design control optimal Controlled trial Randomized Random assignment Replication Blocking Interaction Factorial experiment Uncontrolled studies Observational study Natural experiment Quasi-experiment Statistical inference Statistical theory Population Statistic Probability distribution Sampling distribution Order statistic Empirical distribution Density estimation Statistical model L 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 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 Wald Score Specific tests Z ( normal ) Student 's t - test Goodness of fit Chi - squared Kolmogorov -- Smirnov Anderson -- Darling Lilliefors Jarque -- Bera Normality ( Shapiro -- Wilk ) Likelihood - ratio test 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 ) 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 model validation Mixed effects models Simultaneous equations models Multivariate adaptive regression splines ( MARS ) Linear regression Simple linear regression Ordinary least squares General linear model Bayesian regression Non-standard predictors Nonlinear regression Nonparametric Semiparametric Isotonic Robust Heteroscedasticity Homoscedasticity Generalized linear model Exponential families Logistic ( Bernoulli ) / Binomial / Poisson regressions Partition of variance Analysis of variance ( ANOVA, anova ) Analysis of covariance Multivariate ANOVA Degrees of freedom Categorical / Multivariate / Time - series / Survival analysis Categorical Cohen 's kappa Contingency table Graphical model Log - linear model McNemar 's test 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 ) Frequency domain Spectral density estimation Fourier analysis Wavelet Survival Survival function Kaplan -- Meier estimator ( product limit ) Proportional hazards models Accelerated failure time ( AFT ) model First hitting time Hazard function Nelson -- Aalen estimator Test Log - rank test Applications Biostatistics Bioinformatics Clinical trials / studies Epidemiology Medical statistics Engineering statistics Chemometrics Methods engineering Probabilistic design Process / quality control Reliability System identification Social statistics Actuarial science Census Crime statistics Demography Econometrics National accounts Official statistics Population statistics Psychometrics Spatial statistics Cartography Environmental statistics Geographic information system Geostatistics Kriging Portal Commons WikiProject Retrieved from `` https://en.wikipedia.org/w/index.php?title=Sampling_distribution&oldid=801887238 '' Categories : Statistical inference Sampling ( statistics ) Hidden categories : Use dmy dates from September 2015 Articles needing additional references from May 2014 All articles needing additional references Talk Contents Interaction About Wikipedia Català Deutsch Español Euskara فارسی Basa Sunda Українська Edit links This page was last edited on 22 September 2017, at 15 : 08. 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Question: the sampling distribution of a proportion is called
Answer: probability distribution