Instatistics, apopulation is aset of similar items orevents 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 population with finitely many values in thesupport[4] of the population distribution is afinite population with population size. A population with infinitely many values in the support is calledinfinite population.
A common aim of statistical analysis is to produceinformation about some chosen population.[5]Instatistical inference, a subset of the population (a statisticalsample) is chosen to represent the population in a statistical analysis.[6] 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.[7]
For finite populations, sampling from the population typically removes the sampled value from the populationdue to drawing samples without replacement. This introduces a violation of the typicalindependent and identically distribution assumption so that sampling from finite populations requires "finite population corrections" (which can be derived from thehypergeometric distribution). As a rough rule of thumb,[8] if the sampling fraction is below 10% of the population size, then finite population corrections can approximately be neglected.
Thepopulation mean, or populationexpected value, is a measure of thecentral tendency either of aprobability distribution or of arandom variable characterized by that distribution.[9] In adiscrete probability distribution of a random variable, 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 of and its probability, and then adding all these products together, giving.[10][11] 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.[12]
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