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A statistical 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 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 of X and its probability p(x), and then adding all these products together, giving ${\displaystyle \mu =\sum xp(x)....}$. 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.

Sub population

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A subset of a population that shares one or more additional properties is called a sub population. For example, if the population is all Egyptian people, a sub population is all Egyptian males; if the population is all pharmacies in the world, a sub population is all pharmacies in Egypt. By contrast, a sample is a subset of a population that is not chosen to share any additional property.

Descriptive statistics may yield different results for different sub populations. For instance, a particular medicine may have different effects on different sub populations, and these effects may be obscured or dismissed if such special sub populations are not identified and examined in isolation.

Similarly, one can often estimate parameters more accurately if one separates out sub populations: the distribution of heights among people is better modeled by considering men and women as separate sub populations, for instance.

Populations consisting of sub populations can be modeled by mixture models, which combine the distributions within sub populations into an overall population distribution. Even if sub populations are well-modeled by given simple models, the overall population may be poorly fit by a given simple model – poor fit may be evidence for the existence of sub populations. For example, given two equal sub populations, both normally distributed, if they have the same standard deviation but different means, the overall distribution will exhibit low kurtosis relative to a single normal distribution – the means of the sub populations fall on the shoulders of the overall distribution. If sufficiently separated, these form a bimodal distribution; otherwise, it simply has a wide peak. Further, it will exhibit overdispersion relative to a single normal distribution with the given variation. Alternatively, given two sub populations with the same mean but different standard deviations, the overall population will exhibit high kurtosis, with a sharper peak and heavier tails (and correspondingly shallower shoulders) than a single distribution.

Analyzing sub populations can help in understanding how certain factors affect different segments of the population, which might not be apparent when looking at the population as a whole. For instance, the mean of a sub population, which can be viewed as a moment of the distribution, may differ significantly from that of the entire population, highlighting unique characteristics of that subgroup.

The identification and analysis of sub populations can be crucial for making accurate statistical inferences. As noted by Wang et al. (2008), sub population analysis is essential for increasing the precision of estimates and controlling for potential confounding variables that might otherwise distort the results of a study. This approach is particularly important when sub populations exhibit distinct characteristics that need to be accounted for in statistical models. Furthermore, Murari et al. (2009) discuss the application of these concepts in biological modeling, emphasizing the importance of recognizing sub populations to better understand complex biological systems and improve the accuracy of computational models.

Universe of data

A statistical population is one of the fundamental concepts in statistics and data science, referring to the complete set of items or events that share a common characteristic, from which data can be gathered and analyzed. It is often considered the "universe" of data, encompassing all possible subjects of interest—every person, object, or event that fits the criteria being studied. For instance, the population in a study on global healthcare could include every person on Earth.

Populations can be finite or infinite. Finite populations are countable, such as the number of students in a school. Infinite populations, on the other hand, are uncountable or so large that they're treated as infinite, like the number of grains of sand on a beach or the number of stars in the universe. When studying a population, researchers can either collect data from every member (a census) or a subset (a sample). Censuses are more accurate but expensive and time-consuming. Sampling is more practical but introduces the need for statistical methods to ensure the sample represents the population well.

The target population is the entire group the researcher is interested in, while the accessible population is the portion that can actually be studied due to practical constraints. For example, a study might target all high school students in a country but only access students from a few schools.

The concept of a population is crucial for statistical inference. Inferences made from samples are generalized to the population. The goal is to make accurate conclusions about the population based on sample data, often involving probabilities and confidence intervals.

Populations can change over time. A dynamic population is one where the membership can change, such as the population of a city, which fluctuates due to births, deaths, and migration. This complicates studies, requiring adjustments in statistical models.

Populations are described by parameters like the mean (average), variance, and proportion. These parameters are often unknown and are estimated through sample statistics. For example, the mean income of a country’s population might be estimated from a sample of taxpayers.

In ecology, a population refers to a group of organisms of the same species living in a particular geographic area. Ecologists study population dynamics, including birth rates, death rates, and migration patterns, to understand species survival and ecosystem health.

In the era of big data, the concept of population is evolving. With access to massive datasets, researchers can sometimes work with entire populations of data (e.g., all Twitter users). This reduces reliance on sampling, but it also introduces challenges in data management and interpretation. When defining and studying populations, especially human populations, ethical considerations are paramount. Issues like informed consent, privacy, and the potential for bias in selecting samples must be carefully managed to ensure that research is both valid and respectful of participants' rights.

Importance of Population

The Australian Government Bureau of Statistics notes:

"It is important to understand the target population being studied, so you can understand who or what the data are referring to. If you have not clearly defined who or what you want in your population, you may end up with data that are not useful to you."

See also

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

References

1. "Glossary of statistical terms: Population". Statistics.com. Retrieved 22 February 2016.
2. Weisstein, Eric W. "Statistical population". MathWorld.
3. 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.
4. "Glossary of statistical terms: Sample". Statistics.com. Retrieved 22 February 2016.
5. Feller, William (1950). Introduction to Probability Theory and its Applications, Vol I. Wiley. p. 221. ISBN 0471257087.
6. Elementary Statistics by Robert R. Johnson and Patricia J. Kuby, p. 279
7. Weisstein, Eric W. "Population Mean". mathworld.wolfram.com. Retrieved 2020-08-21.
8. Schaum's Outline of Theory and Problems of Probability by Seymour Lipschutz and Marc Lipson, p. 141
9. West, Brady T.; Berglund, Patricia; Heeringa, Steven G. (2008-12). "A Closer Examination of Subpopulation Analysis of Complex-Sample Survey Data". The Stata Journal: Promoting communications on statistics and Stata. 8 (4): 520–531. doi:10.1177/1536867X0800800404. ISSN 1536-867X. {{cite journal}}: Check date values in: |date= (help)
10. Huang, Shuguang (2010-07). "Statistical Issues in Subpopulation Analysis of High Content Imaging Data". Journal of Computational Biology. 17 (7): 879–894. doi:10.1089/cmb.2009.0071. ISSN 1066-5277. {{cite journal}}: Check date values in: |date= (help)
11. "What Is a Population in Statistics?". ThoughtCo. Retrieved 2024-08-28.

External links

• Statistical Terms Made Simple

• Articles with too few wikilinks from August 2024
• Statistical theory