The Goodness of fit of a statistical model describes how well it fits a set of observations. Measures of goodness of fit typically summarize the discrepancy between observed values and the values expected under the model in question. Such measures can be used in statistical hypothesis testing, e.g. to test for normality of residuals, to test whether two samples are drawn from identical distributions (see Kolmogorov-Smirnov test), or whether outcome frequencies follow a specified distribution (see Pearson's chi-square test). A statistical model is used in applied statistics. ... One may be faced with the problem of making a definite decision with respect to an uncertain hypothesis which is known only through its observable consequences. ... The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. ... In statistics and optimization, the concepts of error and residual are easily confused with each other. ... In statistics, the Kolmogorov-Smirnov test is used to determine whether two underlying probability distributions differ from each other or whether an underlying probability distribution differs from a hypothesized distribution, in either case based in finite samples. ... Pearsons chi-square test (χ2) is one of a variety of chi-square tests – statistical procedures whose results are evaluated by reference to the chi-square distribution. ...

Example

The chi-square statistic is a sum of differences between observed and expected outcome frequencies, each squared and divided by the expectation: Pearsons chi-square test (χ2) is one of a variety of chi-square tests – statistical procedures whose results are evaluated by reference to the chi-square distribution. ... In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are...

where:

O = an observed frequency

E = an expected (theoretical) frequency, asserted by the null hypothesis

The resulting value can be compared to the chi-square distribution to determine the goodness of fit. In statistics, a null hypothesis is a hypothesis set up to be nullified or refuted in order to support an alternative hypothesis. ... This article is about the mathematics of the chi-square distribution. ...

In order to determine the degrees of Freedom of the Chi-Squared distribution, one takes the total number of observed frequencies and subtracts one. For example, if there are eight different frequencies, one would compare to a chi-squared with seven degrees of freedom.

There is also a reduced chi-squared statistic, which is weighted based on measurement error.

where σ² is the variance of the observation. ^[1] This article is about mathematics. ...

Binomial Case

A binomial experiment is a sequence of independent trials in which the trials can result in one of two outcomes, success or failure. There are n trials each with probability of success, denoted by p.
Provided that np_i 5 for every i (where i=1,2,...,k), then

.

This has approximately a chi-squared distribution with k-1 df. The fact that df= k - 1 is a consequence of the restriction . We know there are k observed cell counts, however, once any k - 1 are known, the remaining one is uniquely determined. Basically, one can say, there are only k - 1 freely determined cell counts, thus df= k - 1.

References

1. ^ http://www.sns.gov/workshops/sns_hfir_users/posters/Laub_Chi-Square_Data_Fitting.pdf