Inferential statistics

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Inferential statistics or statistical induction comprises the use of statistics to make inferences concerning some unknown aspect (usually a parameter) of a population.

Two schools of inferential statistics are frequency probability using maximum likelihood estimation, and Bayesian inference. The following is an example of the latter.

Contents 1 Deduction and induction 2 Mean ± standard deviation 2.1 Example 3 Limiting cases 3.1 Binomial and Beta 3.1.1 Example 3.2 Poisson and Gamma 3.2.1 Example 4 See also

[edit] Deduction and induction

From a population containing N items of which I are special, a sample containing n items of which i are special can be chosen in

ways (see multiset and binomial coefficient).

Fixing (N,n,I), this expression is the unnormalized deduction distribution function of i.

Fixing (N,n,i) , this expression is the unnormalized induction distribution function of I.

[edit] Mean ± standard deviation

The mean value ± the standard deviation of the deduction distribution is used for estimating i knowing (N,n,I)

where

The mean value ± the standard deviation of the induction distribution is used for estimating I knowing (N,n,i)

Thus deduction is translated into induction by means of the involution

[edit] Example

The population contains a single item and the sample is empty. (N,n,i)=(1,0,0). The induction formula gives

confirming that the number of special items in the population is either 0 or 1.

(The frequency probability solution to this problem is giving no meaning.)

[edit] Limiting cases

[edit] Binomial and Beta

In the limiting case where N is a large number, the deduction distribution of i tends towards the binomial distribution with the probability as a parameter,

and the induction distribution of tends towards the beta distribution

(The frequency probability solution to this problem is : the probability is estimated by the relative frequency.)

[edit] Example

The population is big and the sample is empty. n=i=0. The beta distribution formula gives .

(The frequency probability solution to this problem is giving no meaning.)

[edit] Poisson and Gamma

In the limiting case where and are large numbers, the deduction distribution of i tends towards the poisson distribution with the intensity as a parameter,

and the induction distribution of M tends towards the gamma distribution

[edit] Example

The population is big and the sample is big but contains no special items. i = 0. The gamma distribution formula gives .

(The frequency probability solution to this problem is which is misleading. Even if you have not been wounded you may still be vulnerable).

[edit] See also

• Descriptive statistics
• Induction (philosophy)

This page uses content from the English-language version of Wikipedia. The original article was at Inferential statistics. The list of authors can be seen in the page history. As with Psychology Wiki, the text of Wikipedia is available under the GNU Free Documentation License.