NationMaster - Encyclopedia: Inferential statistics

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Inferential statistics or statistical induction is a branch of statistics that consists of generalizing from samples to populations performing hypothesis testing and making predictions. Wikipedia does not have an article with this exact name. ... The topics below are usually included in the area of interpreting statistical data. ... Statistics is a type of data analysis which practice includes the planning, summarizing, and interpreting of observations of a system possibly followed by predicting or forecasting of future events based on a mathematical model of the system being observed. ... Sampling is that part of statistical practice concerned with the selection of individual observations intended to yield some knowledge about a population of concern, especially for the purposes of statistical inference. ... A hypothesis (assumption in ancient Greek) is a proposed explanation for a phenomenon. ... The words test and testing have many meanings: Test and experiment forms part of the scientific method, to verify or falsify an already-formed expectation with an observation. ... A prediction is a statement or claim that a particular event will occur in the future. ...

Two schools of inferential statistics are frequency probability using maximum likelihood estimation, and Bayesian inference. The following is an example of the latter. Statistical regularity has motivated the development of the relative frequency concept of probability. ... Maximum likelihood estimation (MLE) is a popular statistical method used to make inferences about parameters of the underlying probability distribution of a given data set. ... Bayesian inference is a statistical inference in which probabilities are interpreted not as frequencies or proportions or the like, but rather as degrees of belief. ...

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

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

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

ways (see binomial coefficient). In mathematics, particularly in combinatorics, the binomial coefficient of the natural number n and the integer k is defined to be the natural number and (Here, for a natural number m, m! denotes the factorial of m. ...

Fixing $(N,n,I)$ , this expression is the unnormalized deduction distribution function of $i$ . In probability theory and statistics, the hypergeometric distribution is a discrete probability distribution that describes the number of successes in a sequence of n draws from a finite population without replacement. ...

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

Mean ± standard deviation

The mean value ± the standard deviation of the deduction distribution is used for estimating $i$ knowing $(N,n,I)$ In mathematics, there are numerous methods for calculating the average or central tendency of a list of n numbers. ... In probability and statistics, the standard deviation is the most commonly used measure of statistical dispersion. ... In probability theory and statistics, the hypergeometric distribution is a discrete probability distribution that describes the number of successes in a sequence of n draws from a finite population without replacement. ...

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

where

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

The mean value ± the standard deviation of the induction distribution is used for estimating $I$ knowing $(N,n,i)$ In mathematics, there are numerous methods for calculating the average or central tendency of a list of n numbers. ... In probability and statistics, the standard deviation is the most commonly used measure of statistical dispersion. ...

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

Thus deduction is translated into induction by means of the involution In mathematics, an involution is a function that is its own inverse, so that f(f(x)) = x for all x in the domain of f. ...

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

Example

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

$Iapprox -1-f(-2,-3,-1)=frac{1}{2}pmfrac{1}{2}$

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

(The frequency probability solution to this problem is $Iapprox frac{Ni}{n}=frac{0}{0}$ giving no meaning.) Statistical regularity has motivated the development of the relative frequency concept of probability. ...

Limiting cases

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 $P=frac{I}{N}$ as a parameter, See binomial (disambiguation) for a list of other topics using that name. ...

$iapprox nPleft (1pmsqrt{frac{frac{1}{P}-1}{n}}right )$

and the induction distribution of $P$ tends towards the beta distribution In probability theory and statistics, the beta distribution is a continuous probability distribution with the probability density function (pdf) defined on the interval [0, 1]: where and are parameters that must be greater than zero and is the beta function. ...

$Papproxfrac{i+1pmsqrt{frac{(i+1)(n-i+1)}{n+3}}}{n+2}.$

(The frequency probability solution to this problem is $P approx frac{i}{n}$ : the probability is estimated by the relative frequency.) Statistical regularity has motivated the development of the relative frequency concept of probability. ... The word probability derives from the Latin probare (to prove, or to test). ... Estimation is generally the calculation of an approximate or uncertain result, often based on approximate, uncertain, incomplete, or noisy data. ... In a series of observations, or trials, the relative frequency of occurrence of an event is calculated as: The of an event over a long series of trials is the conceptual foundation of the frequency interpretation of probability. ...

Example

The population is big and the sample is empty. $n=i=0$ . The beta distribution formula gives $P approx(50 pm 29)%$ . In probability theory and statistics, the beta distribution is a continuous probability distribution with the probability density function (pdf) defined on the interval [0, 1]: where and are parameters that must be greater than zero and is the beta function. ...

(The frequency probability solution to this problem is $P approx frac{i}{n}=frac{0}{0}$ giving no meaning.) Statistical regularity has motivated the development of the relative frequency concept of probability. ...

Poisson and Gamma

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, In probability theory and statistics, the hypergeometric distribution is a discrete probability distribution that describes the number of successes in a sequence of n draws from a finite population without replacement. ... In probability theory and statistics, the Poisson distribution is a discrete probability distribution. ...

$i approx M pm sqrt{M}$

and the induction distribution of $M$ tends towards the gamma distribution In probability theory and statistics, the gamma distribution is a continuous probability distribution. ...

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

Example

The population is big and the sample is big but contains no special items. $i=0$ . The gamma distribution formula gives $Mapprox 1 pm 1$ . In probability theory and statistics, the gamma distribution is a continuous probability distribution. ...

(The frequency probability solution to this problem is $Mapprox 0$ which is misleading. Even if you have not been wounded you may still be vulnerable). Statistical regularity has motivated the development of the relative frequency concept of probability. ...

See also

Induction (philosophy)

Categories: Articles to be merged | Statistics Induction or inductive reasoning, sometimes called inductive logic, is the process of reasoning in which the premises of an argument support the conclusion, but do not ensure it. ...

Results from FactBites:

Defining and Conceptualizing Descriptive and Inferential Statistics (387 words)

Inferential Statistics: A method that takes chance factors into account when samples are used to reach conclusions (or make inferences about) populations.

The goal of inferential statistics is to help researchers decide between the two interpretations.

Inferential statistics begins with actual data (sample data) from the experiment above and ends with a probability statement (i.e., the probability of obtaining data like those above if there is no effect of vitamin C on cognitive ability in the population)

World Bank Group | Social Analysis | Inferential Statistics (2005 words)

Inferential statistics are so named because they permit us to infer the characteristics of the population from a representative sample.

Because the logic is the same regardless of the inferential statistics used, once we understand the logic, it is easier for us to understand how a given statistic works and when and how it is appropriate to use its.

In statistical tables, the significance level is often indicated with asterisks, a single asterisk for the.05 level and two asterisks for the.1 levels.

More results at FactBites »

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