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'''Statistics''' can be described as "the practice or science of collecting and analysing numerical data in large quantities, especially for the purpose of inferring proportions in a whole from those in a representative sample."<ref>Soanes, C. and Stevenson, A. (eds.) (2005) 'Oxford Dictionary of English (2nd edition revised)' Oxford University Press, Oxford, U.K.</ref> It involves all stages of data collection and processing from the initial collection, to the analysis and ultimately to the conclusions and interpretations of the data. It is used in all research oriented disciplines from physics, chemistry and biology to economics, anthropology and psychology as well as many thousands of other fields. It is also used in businesses and governments.   

'''Statistics''' can be described as "the practice or science of collecting and analysing numerical data in large quantities, especially for the purpose of inferring proportions in a whole from those in a representative sample."<ref>Soanes, C. and Stevenson, A. (eds.) (2005) 'Oxford Dictionary of English (2nd edition revised)' Oxford University Press, Oxford, U.K.</ref> It involves all stages of data collection and processing from the initial collection, to the analysis and ultimately to the conclusions and interpretations of the data. It is used in all research oriented disciplines from physics, chemistry and biology to economics, anthropology and psychology as well as many thousands of other fields. It is also used in businesses and governments.   

Statistics analyzes data in two primary ways, the first is called [[descriptive statistics]] which describes and summarizes the data. Often this will include things like: [[mean | the mean]], [[standard error]], or [[standard deviation]] each of these is an example of a [[statistic]]. Also statistics can attempt to infer relationships between the data collected and various hypothesis or populations, this is called [[inferential statistics]]. Both descriptive and inferential statistics comprise applied statistics. There is also a discipline called mathematical statistics, which is concerned with the theoretical basis of the subject.

Statistics analyzes data in two primary ways, the first is called [[descriptive statistics]] which describes and summarizes the data. Often this will include things like: [[mean|the mean]], [[standard error]], or [[standard deviation]] each of these is an example of a [[statistic]]. Also statistics can attempt to infer relationships between the data collected and various hypothesis or populations, this is called [[inferential statistics]]. Both descriptive and inferential statistics comprise applied statistics. There is also a discipline called mathematical statistics, which is concerned with the theoretical basis of the subject.

Statistics takes its name from the fact that it was traditionally taught to monarchs to enable them to manage affairs of state. <ref>http://www.etymonline.com/index.php?term=statistics</ref>

Statistics takes its name from the fact that it was traditionally taught to monarchs to enable them to manage affairs of state.<ref>http://www.etymonline.com/index.php?term=statistics</ref>

   

   

==Frequentist Approaches==

==Frequentist Approaches==

[[Frequency probability | Frequentist]] approaches are often referred to as classical approaches because it is the oldest and most used method of statistical analysis. The heart of this approach is to try and understand data as a relative frequency or ratio of a particular occurrence out of a total possible number of occurrences. For example, a frequentist would describe the number of times a coin turns up heads as a ratio of total number of heads out of total number of flips.  

[[Frequency probability|Frequentist]] approaches are often referred to as classical approaches because it is the oldest and most used method of statistical analysis. The heart of this approach is to try and understand data as a relative frequency or ratio of a particular occurrence out of a total possible number of occurrences. For example, a frequentist would describe the number of times a coin turns up heads as a ratio of total number of heads out of total number of flips.  

===Descriptive statistics===

===Descriptive statistics===

===Inferential statistics===

===Inferential statistics===

Frequentist approaches to inferential statistics primarily involve trying to compare descriptive statistics of two data sets to determine if they are [[statistically significant | significantly]] different. One of the most common approaches is to test a given data set against a [[null hypothesis]] or the data set that would be created if the values were the result of random chance alone. For example, if a given head came up 9 times as heads and 1 time as tails you would compare the number of heads, 9, to the number of heads that would be expected if chance alone was operating, or 5.  

Frequentist approaches to inferential statistics primarily involve trying to compare descriptive statistics of two data sets to determine if they are [[statistically significant|significantly]] different. One of the most common approaches is to test a given data set against a [[null hypothesis]] or the data set that would be created if the values were the result of random chance alone. For example, if a given head came up 9 times as heads and 1 time as tails you would compare the number of heads, 9, to the number of heads that would be expected if chance alone was operating, or 5.  

Testing against the null hypothesis is sometimes referred to as an [[omnibus]] test since it is testing the idea that a given data set is the result of anything other than chance. Often it is much more desirable to test specific data sets against each other.  

Testing against the null hypothesis is sometimes referred to as an [[omnibus]] test since it is testing the idea that a given data set is the result of anything other than chance. Often it is much more desirable to test specific data sets against each other.  

==Bayesian Approaches==

==Bayesian Approaches==

[[Bayesian inference | Bayesian statistics]] is a method of applying [[Bayes theorem]] to data analysis. One of the biggest difference between Bayesian approaches and frequentist approaches is that Bayesians attempt to determine the probability that a given hypothesis is true given the data, while frequentist attempt to define the probability of getting the data given that a particular hypothesis is true.  

[[Bayesian inference|Bayesian statistics]] is a method of applying [[Bayes theorem]] to data analysis. One of the biggest difference between Bayesian approaches and frequentist approaches is that Bayesians attempt to determine the probability that a given hypothesis is true given the data, while frequentist attempt to define the probability of getting the data given that a particular hypothesis is true.  

Bayesian approaches are becoming more and more popular in science because what most people are interested in is the probability of the proposed hypothesis, not the probability of the data. It also does not need to make prior assumptions about the data such as [[Normal distribution | normality]] and [[homogeneity of variance]]. However, Bayesian methods have come under fire from many frequentist proponents. This has led to very heated debate in statistical circles, though this has largely died now, about the respective validity of both methods. The primary complaint leveled at Bayesian statistics is that it must use a [[prior probability]] of a hypothesis in its analysis. This prior is intended to build contextual information into the analysis, but it may be seen by its critics as subjective or arbitrary.  Commonly used prior distributions include the [[uniform distribution]] and [[beta distribution]].   

Bayesian approaches are becoming more and more popular in science because what most people are interested in is the probability of the proposed hypothesis, not the probability of the data. It also does not need to make prior assumptions about the data such as [[Normal distribution|normality]] and [[homogeneity of variance]]. However, Bayesian methods have come under fire from many frequentist proponents. This has led to very heated debate in statistical circles, though this has largely died now, about the respective validity of both methods. The primary complaint leveled at Bayesian statistics is that it must use a [[prior probability]] of a hypothesis in its analysis. This prior is intended to build contextual information into the analysis, but it may be seen by its critics as subjective or arbitrary.  Commonly used prior distributions include the [[uniform distribution]] and [[beta distribution]].   

===Descriptive statistics===

===Descriptive statistics===

In order to compare hypothesis [[Bayesian model selection]] is often used. This is when each hypothesis you want to test is assigned a prior probability, and then the likelihood of the data given each hypothesis being test is calculated. You can then us Bayes equation to determine the relative probabilities that each hypothesis is correct. This method is almost always testing relative probabilites since to calculate an absolute probability would require knowing every possible hypothesis. Usually this is not possible, but sometimes the subset is finite enough it can be tested.  

In order to compare hypothesis [[Bayesian model selection]] is often used. This is when each hypothesis you want to test is assigned a prior probability, and then the likelihood of the data given each hypothesis being test is calculated. You can then us Bayes equation to determine the relative probabilities that each hypothesis is correct. This method is almost always testing relative probabilites since to calculate an absolute probability would require knowing every possible hypothesis. Usually this is not possible, but sometimes the subset is finite enough it can be tested.  

Because of the large number of calculations needed for model selection Bayesian approaches have only became practical and popular with the advent of computers. But even with the most modern computers available many Bayesian models remain computational intractable. Recent developments in applying [[Markov chain]] [[Monte Carlo method | Monte Carlo methods]] to these problems have led to promising results.

Because of the large number of calculations needed for model selection Bayesian approaches have only became practical and popular with the advent of computers. But even with the most modern computers available many Bayesian models remain computational intractable. Recent developments in applying [[Markov chain]] [[Monte Carlo method]]s to these problems have led to promising results.

==Non-parametric and Bootstrapping methods==

==Non-parametric and Bootstrapping methods==