Population Parameter

A Population Parameter is a numeric value that characterizes a complete statistical population or defines a probability distribution within a statistical model.

• AKA: Statistical Parameter, True Parameter, Parameter Value, Population Characteristic, Distribution Parameter, Model Parameter, True Value, Theoretical Parameter.
• Context:
  • It can typically represent True Characteristics of entire statistical populations that are often unknown and must be estimated.
  • It can typically serve as the Target Value that sample statistics attempt to estimate through statistical inference.
  • It can typically define Probability Distributions by specifying their mathematical form and statistical properties.
  • It can typically remain fixed for a given population under statistical assumptions.
  • It can typically be specified under Null Hypotheses in hypothesis testing tasks.
  • It can often index Parametric Families of probability distributions.
  • It can often be Unobservable in practice due to population size or measurement constraints.
  • It can often require Census Data or complete enumeration for exact determination.
  • It can often be approximated through Parameter Estimation Tasks using sample data.
  • It can often have associated Parameter Spaces defining allowable values.
  • It can range from being a Location Population Parameter to being a Scale Population Parameter, depending on its distributional role.
  • It can range from being a Shape Population Parameter to being a Threshold Population Parameter, depending on its distribution characteristic.
  • It can range from being a Univariate Population Parameter to being a Multivariate Population Parameter, depending on its dimensionality.
  • It can range from being a Discrete Population Parameter to being a Continuous Population Parameter, depending on its value space.
  • It can range from being a Natural Population Parameter to being a Canonical Population Parameter, depending on its parameterization form.
  • It can be associated with a Point Estimate produced by parameter estimation tasks.
  • It can be estimated within Confidence Intervals through interval estimation tasks.
  • It can serve as the Estimand in estimation tasks and statistical inference tasks.
  • It can be contrasted with Sample Statistic Values which estimate it from sample data.
  • It can be tested through Statistical Hypothesis Testing Tasks using test statistics.
  • ...
• Example(s):
  • Central Tendency Population Parameters, such as:
    • Population Mean Value (μ) representing average of all population values.
    • Population Median Value representing middle value of entire population.
    • Population Mode Value representing most frequent value in population.
  • Dispersion Population Parameters, such as:
    • Population Variance Value (σ²) measuring spread of all population values.
    • Population Standard Deviation Value (σ) as square root of population variance.
    • Population Range Value as difference between maximum and minimum.
  • Distribution-Specific Population Parameters, such as:
    • Normal Distribution Parameters: mean (μ) and variance (σ²).
    • Poisson Distribution Parameter: rate parameter (λ).
    • Binomial Distribution Parameters: trial count (n) and success probability (p).
    • Exponential Distribution Parameter: rate parameter (λ).
    • Beta Distribution Parameters: shape parameters (α, β).
    • Gamma Distribution Parameters: shape (k) and scale (θ).
  • Shape Population Parameters, such as:
    • Population Skewness Parameter measuring asymmetry.
    • Population Kurtosis Parameter measuring tail heaviness.
    • Location Shape Parameter defining distribution position.
  • Association Population Parameters, such as:
    • Population Correlation Coefficient (ρ) measuring linear relationship.
    • Population Covariance measuring joint variability.
    • Population Regression Coefficient (β) in linear models.
  • Proportion Population Parameters, such as:
    • Population Proportion (p) of specific characteristic.
    • Population Prevalence of condition or trait.
    • Population Risk Ratio comparing groups.
  • Economic Population Parameters, such as:
    • Population Gini Coefficient measuring inequality.
    • Population Elasticity measuring responsiveness.
    • Population Growth Rate for demographic analysis.
  • Machine Learning Model Parameters, such as:
    • Neural Network Weight Parameters in deep learning models.
    • Support Vector Machine Parameters defining decision boundaries.
    • Decision Tree Split Parameters determining branches.
  • Bayesian Population Parameters, such as:
    • Prior Distribution Parameters encoding beliefs.
    • Posterior Distribution Parameters after data observation.
    • Hyperparameters in hierarchical models.
  • Mathematical notation examples:
    • If f(X|μ,σ²) = (1/√(2πσ²))exp(-(X-μ)²/(2σ²)) is the population distribution, then μ and σ² are population parameters.
    • In Y = β₀ + β₁X + ε, the coefficients β₀ and β₁ are population parameters.
  • ...
• Counter-Example(s):
  • Sample Statistic Value, which is computed from sample data rather than entire population.
  • Point Estimate, which approximates rather than equals the true value.
  • Sample Mean, which estimates rather than equals population mean.
  • Sample Variance, which estimates rather than equals population variance.
  • Test Statistic, which is calculated from samples for hypothesis testing.
  • Descriptive Statistic, which summarizes sample data rather than population.
  • Random Variable, which represents stochastic outcomes rather than fixed values.
  • Sampling Distribution, which describes sample statistic behavior rather than population characteristics.
  • Machine Learning Hyperparameter, which is set before training rather than characterizing data.
• See: Statistical Model, Parameter, Sample Statistic Value, Point Estimate, Confidence Interval, Parameter Estimation Task, Statistical Inference Task, Statistical Hypothesis, Statistical Hypothesis Testing Task, Test Statistic, Estimand, Population Aggregation Function, Statistical Population, Likelihood Function, Method of Moments Algorithm, Maximum Likelihood Estimate, Degrees of Freedom (Statistics), Standard Error (SE) Measure, Statistical Population Contrast, Random Sample-based Statistical Measure.

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References

2022

• (Wikipedia, 2022) ⇒ https://en.wikipedia.org/wiki/Statistical_parameter Retrieved:2022-8-18.
  • In statistics, as opposed to its general use in mathematics, a parameter is any measured quantity of a statistical population that summarises or describes an aspect of the population, such as a mean or a standard deviation. If a population exactly follows a known and defined distribution, for example the normal distribution, then a small set of parameters can be measured which completely describes the population, and can be considered to define a probability distribution for the purposes of extracting samples from this population.

    A parameter is to a population as a statistic is to a sample; that is to say, a parameter describes the true value calculated from the full population, whereas a statistic is an estimated measurement of the parameter based on a subsample. Thus a "statistical parameter" can be more specifically referred to as a population parameter.

2016

• (Wikipedia) ⇒ http://en.wikipedia.org/wiki/Statistical_parameter
  • A statistical parameter or population parameter is a quantity that indexes a family of probability distributions. It can be regarded as a numerical characteristic of a population or a statistical model.

(...) Among parameterized families of distributions are the normal distributions, the Poisson distributions, the binomial distributions, and the exponential family of distributions. The family of normal distributions has two parameters, the mean and the variance: if these are specified, the distribution is known exactly. The family of chi-squared distributions, on the other hand, has only one parameter, the number of degrees of freedom.

In statistical inference, parameters are sometimes taken to be unobservable, and in this case the statistician's task is to infer what they can about the parameter based on observations of random variables distributed according to the probability distribution in question, or, more concretely stated, based on a random sample taken from the population of interest. In other situations, parameters may be fixed by the nature of the sampling procedure used or the kind of statistical procedure being carried out (for example, the number of degrees of freedom in a Pearson's chi-squared test).

2012

• http://www.stats.ox.ac.uk/~tomas/html_links/0809/Lecture23.pdf
  • QUOTE: Both estimation and testing are concerned with a parameter [math]\displaystyle{ \theta }[/math], which should (if possible) be a meaningful quantity. … A statistic [math]\displaystyle{ t = t(\mathbf{x}) }[/math] is any number calculated from the sample. Since the sample is a random observation of [math]\displaystyle{ X_1, X_2, ...,X_n }[/math], we can regard [math]\displaystyle{ t }[/math] as a sample of the random variable [math]\displaystyle{ T = t(X) }[/math]. The distribution of T is called the sampling distribution. … A statistic [math]\displaystyle{ T }[/math] is an estimator of (population parameter) [math]\displaystyle{ \theta }[/math] if its intention is to be close to the (unknown) value of [math]\displaystyle{ \theta }[/math]. To perform statistical inference for an estimator [math]\displaystyle{ T }[/math] of [math]\displaystyle{ \theta }[/math] we will often need to derive its distribution.

    Suppose the population has mean [math]\displaystyle{ \mu }[/math] and variance [math]\displaystyle{ \sigma^2 }[/math]. Then we can often use : [math]\displaystyle{ \bar{X}_n \sim N(\mu,\frac{\sigma^2}{n}) }[/math].

2011

• http://www.cliffsnotes.com/math/statistics/sampling/populations-samples-parameters-and-statistics
  • QUOTE: A parameter is a characteristic of a population. A statistic is a characteristic of a sample. Inferential statistics enables you to make an educated guess about a population parameter based on a statistic computed from a sample randomly drawn from that population (see Figure 1).

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