Population Parameter

A Population Parameter is a numeric value that characterizes a population or statistical model.

• AKA: Statistical Measure.
• Context:
  • It can usually be a fixed (theoretical or unknown) value which can be specific under the Null Hypothesis.
  • It can be associated to a Parameter Estimate (produced by parameter estimation).
  • …
• Example(s):
  • If [math]\displaystyle{ f(X|\mu,\sigma^2)=\frac{1}{\sqrt{2\pi\sigma^2}} exp(-\frac{X-\mu}{2\sigma^2}) }[/math] is the population distribution of the random variable [math]\displaystyle{ X }[/math], then [math]\displaystyle{ \sigma^2 }[/math] and [math]\displaystyle{ \mu }[/math] are population parameters.
  • Population Mean.
  • Population Median.
  • Population Variance.
  • …
• Counter-Example(s):
  • Sample Statistic.
  • Point Estimate.
  • Sample Mean.
  • Sample Variance.
• See: Multiset Scoring Function, Point Estimate, Confidence Interval, Sample size.

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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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