Statistical population

In statistics, a population is a set of similar items or events which is of interest for some question or experiment. A statistical population can be a group of existing objects (e.g. the set of all stars within the Milky Way galaxy) or a hypothetical and potentially infinite group of objects conceived as a generalization from experience (e.g. the set of all possible hands in a game of poker). A common aim of statistical analysis is to produce information about some chosen population.

In statistical inference, a subset of the population (a statistical sample) is chosen to represent the population in a statistical analysis. Moreover, the statistical sample must be unbiased and accurately model the population (every unit of the population has an equal chance of selection). The ratio of the size of this statistical sample to the size of the population is called a sampling fraction. It is then possible to estimate the population parameters using the appropriate sample statistics.

Mean

The population mean, or population expected value, is a measure of the central tendency either of a probability distribution or of a random variable characterized by that distribution. In a discrete probability distribution of a random variable ${\displaystyle X}$, 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 ${\displaystyle x}$ of ${\displaystyle X}$ and its probability ${\displaystyle p(x)}$, and then adding all these products together, giving ${\displaystyle \mu =\sum x\cdot p(x)....}$. An analogous formula applies to the case of a continuous probability distribution. Not every probability distribution has a defined mean (see the Cauchy 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. The sample mean may differ from the population mean, especially for small samples. The law 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.

See also

• Data collection system
• Horvitz–Thompson estimator
• Sample (statistics)
• Sampling (statistics)
• Stratum (statistics)

References

1. Haberman, Shelby J. (1996). "Advanced Statistics". Springer Series in Statistics. doi:10.1007/978-1-4757-4417-0. ISSN 0172-7397.
2. "Glossary of statistical terms: Population". . Retrieved 22 February 2016.
3. Weisstein, Eric W. "Statistical population". MathWorld.
4. 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 from the original on 2005-02-09.
5. "Glossary of statistical terms: Sample". . Retrieved 22 February 2016.
6. Levy, Paul S.; Lemeshow, Stanley (2013-06-07). Sampling of Populations: Methods and Applications. John Wiley & Sons. ISBN 978-1-118-62731-0.
7. Feller, William (1950). Introduction to Probability Theory and its Applications, Vol I. Wiley. p. 221. ISBN 0471257087.
8. Elementary Statistics by Robert R. Johnson and Patricia J. Kuby, p. 279
9. Weisstein, Eric W. "Population Mean". mathworld.wolfram.com. Retrieved 2020-08-21.
10. Schaum's Outline of Theory and Problems of Probability by Seymour Lipschutz and Marc Lipson, p. 141

External links

• Statistical Terms Made Simple