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Statistical population

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(Redirected fromPopulation mean)

Complete set of items that share at least one property in common

For the number of people, seePopulation.

Instatistics, apopulation is aset of similar items or events which is of interest for some question orexperiment.^[1]^[2] A statistical population can be a group of existing objects (e.g. the set of all stars within theMilky Way galaxy) or ahypothetical and potentiallyinfinite group of objects conceived as a generalization from experience (e.g. the set of all possible hands in a game of poker).^[3] A common aim of statistical analysis is to produceinformation about some chosen population.^[4]

Instatistical inference, a subset of the population (a statisticalsample) is chosen to represent the population in a statistical analysis.^[5] Moreover, the statistical sample must beunbiased andaccurately model the population. The ratio of the size of this statistical sample to the size of the population is called asampling fraction. It is then possible toestimate thepopulation parameters using the appropriatesample statistics.^[6]

Mean

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Thepopulation mean, or populationexpected value, is a measure of thecentral tendency either of aprobability distribution or of arandom variable characterized by that distribution.^[7] In adiscrete probability distribution of a random variable $X {\displaystyle X}$ , the mean is equal to the sum over every possible value weighted by the probability of that value; that is, it is computed by taking the product of each possible value $x {\displaystyle x}$ of $X {\displaystyle X}$ and its probability $p(x)$ , and then adding all these products together, giving $\mu =\sum x\cdot p(x)....$ .^[8]^[9] An analogous formula applies to the case of acontinuous probability distribution. Not every probability distribution has a defined mean (see theCauchy distribution for an example). Moreover, the mean can be infinite for some distributions.

For a finite population, the population mean of a property is equal to the arithmetic mean of the given property, while considering every member of the population. For example, the population mean height is equal to the sum of the heights of every individual—divided by the total number of individuals. Thesample mean may differ from the population mean, especially for small samples. Thelaw of large numbers states that the larger the size of the sample, the more likely it is that the sample mean will be close to the population mean.^[10]

References

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^Haberman, Shelby J. (1996)."Advanced Statistics".Springer Series in Statistics.doi:10.1007/978-1-4757-4417-0.ISBN 978-1-4419-2850-4.ISSN 0172-7397.
^"Glossary of statistical terms: Population".Statistics.com. Retrieved22 February 2016.
^Weisstein, Eric W."Statistical population".MathWorld.
^Yates, Daniel S.; Moore, David S; Starnes, Daren S. (2003).The Practice of Statistics (2nd ed.). New York:Freeman.ISBN 978-0-7167-4773-4. Archived fromthe original on 2005-02-09.
^"Glossary of statistical terms: Sample".Statistics.com. Retrieved22 February 2016.
^Levy, Paul S.; Lemeshow, Stanley (2013-06-07).Sampling of Populations: Methods and Applications. John Wiley & Sons.ISBN 978-1-118-62731-0.
^Feller, William (1950).Introduction to Probability Theory and its Applications, Vol I. Wiley. p. 221.ISBN 0471257087.
^Elementary Statistics by Robert R. Johnson and Patricia J. Kuby,p. 279
^Weisstein, Eric W."Population Mean".mathworld.wolfram.com. Retrieved2020-08-21.
^Schaum's Outline of Theory and Problems of Probability by Seymour Lipschutz and Marc Lipson,p. 141

External links

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Statistical Terms Made Simple

Statistics

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

Dependence

Graphics

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

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 Score/Lagrange multiplier Wald

Specific tests

Z-test(normal) Student'st-test F-test
Goodness of fit	Chi-squared G-test 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) Van der Waerden test

Bayesian inference

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)
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, anova) Analysis of covariance Multivariate ANOVA Degrees of freedom

Categorical / Multivariate / Time-series / Survival analysis

Categorical

Multivariate

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 Least-squares spectral analysis Wavelet Whittle likelihood

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 Jurimetrics National accounts Official statistics Population statistics Psychometrics
Spatial statistics	Cartography Environmental statistics Geographic information system Geostatistics Kriging