Statistical parameter

A statistical parameter is a parameter that indexes a family of probability distributions. It can be regarded as a numerical characteristic of a population or a model.

Among parameterized families of distributions are the normal distributions, the Poisson distributions, the binomial distributions, and the exponential 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 he 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).

Even if a family of distributions is not specified, quantities such as the mean and variance can still be regarded as parameters of the distribution of the population from which a sample is drawn. Statistical procedures can still attempt to make inferences about such population parameters. Parameters of this type are given names appropriate to their roles, including:

• location parameter
• dispersion parameter or scale parameter
• shape parameter

Where a probability distribution has a domain over a set of objects that are themselves probability distributions, the term concentration parameter is used for quantities that index how variable the outcomes would be.

Quantities such as regression coefficients, are statistical parameters in the above sense, since they index the family of conditional probability distributions that describe how the dependent variables are related to the independent variables.

Analogies

A parameter is to a population as a statistic is to a sample.

For people unfamiliar with statistics as a whole, a parameter is to a population as a human is to the Earth. A statistic is to a sample as a grape is to a bowl. However, notice that the bowl does not contain all the grapes that one might care about, such as the ones still growing in the vineyard. A sample contains only some of the items, while a population (such as all humans on Earth, or all humans located in a certain country) contains all of the items of interest.

Misuse of statistical terminology

Many people who do not have a statistical background or never received any kind of education in statistics often use the word statistic to mean both statistical parameter and statistic. Not only is this confusing to somebody who is aware of the difference in terminology, it also makes layman language unclear and very genercized. The meaning of certain sentences becomes blunted when statistics and parameters are the same word.

Examples

• Class statistics - Size is exactly 30, only 1 teacher, and the topic is always going to be Environmental Science and Policy Management.
• US statistics - About 300,000,000 people...
• Website statistics - Exactly 7293 views and 244 downloads of files.

The problem with "exact" measurements is that exact measurements measure something about populations (these measurements are parameters), while statistics only measure a small part of the entire population (that portion of a population refers to a sample).

See also

• Precision (statistics), another parameter not specific to any one distribution
• Parametrization (i.e., coordinate system)
• Parsimony (with regards to the trade-off of many or few parameters in data fitting)