Assessment |
Biopsychology |
Comparative |
Cognitive |
Developmental |
Language |
Individual differences |
Personality |
Philosophy |
Social |

Methods |
Statistics |
Clinical |
Educational |
Industrial |
Professional items |
World psychology |

**Statistics:**
Scientific method ·
Research methods ·
Experimental design ·
Undergraduate statistics courses ·
Statistical tests ·
Game theory ·
Decision theory

**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.

## Deduction and inductionEdit

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

- $ {I \choose i}{{N-I} \choose {n-i}} $

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*.

## Mean ± standard deviationEdit

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

- $ i \approx f(N,n,I) $

where

- $ f(N,n,I)=\frac{nI\pm\sqrt{\frac{nI(N-n)(N-I)}{N-1}}}{N}. $

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

- $ I \approx -1-f(-2-n,-2-N,-1-i). $

Thus deduction is translated into induction by means of the involution

- $ (N,n,I,i) \leftrightarrow (-2-n,-2-N,-1-i,-1-I). $

#### ExampleEdit

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

- $ I\approx -1-f(-2,-3,-1)=\frac{1}{2}\pm\frac{1}{2} $

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

(The frequency probability solution to this problem is $ I\approx \frac{Ni}{n}=\frac{0}{0} $ giving no meaning.)

## Limiting casesEdit

### Binomial and BetaEdit

In the limiting case where *N* is a large number, the deduction distribution of *i* tends towards the binomial distribution with the probability $ P=\frac{I}{N} $ as a parameter,

- $ i\approx nP\left (1\pm\sqrt{\frac{\frac{1}{P}-1}{n}}\right ) $

and the induction distribution of $ \ P $ tends towards the beta distribution

- $ P\approx\frac{i+1\pm\sqrt{\frac{(i+1)(n-i+1)}{n+3}}}{n+2}. $

(The frequency probability solution to this problem is $ P \approx \frac{i}{n} $: the probability is estimated by the relative frequency.)

#### ExampleEdit

The population is big and the sample is empty. *n=i=*0. The beta distribution formula gives $ P \approx(50 \pm 29)\% $.

(The frequency probability solution to this problem is $ P \approx \frac{i}{n}=\frac{0}{0} $ giving no meaning.)

### Poisson and GammaEdit

In the limiting case where $ \frac{N}{n} $ and $ \ n $ are large numbers, the deduction distribution of *i* tends towards the poisson distribution with the intensity $ M=\frac{nI}{N} $ as a parameter,

- $ i \approx M \pm \sqrt{M} $

and the induction distribution of *M* tends towards the gamma distribution

- $ M \approx i+1 \pm \sqrt{i+1}. $

#### ExampleEdit

The population is big and the sample is big but contains no special items. *i* = 0. The gamma distribution formula gives $ M\approx 1 \pm 1 $.

(The frequency probability solution to this problem is $ M\approx 0 $ which is misleading. Even if you have not been wounded you may still be vulnerable).

## See also Edit

This page uses Creative Commons Licensed content from Wikipedia (view authors). |