# F-Statistic

Jump to navigation
Jump to search

A F-Statistic is a test statistic used in analysis of variance or regression analysis.

**AKA:**f value, F-statistic score, F-ratio.**Context:**- It can be defined the ratio between the variability between groups divided by variability within groups.
- It can also be defined as ratio:

- [math]\displaystyle{ f= \frac{s_1^2/\sigma_1^2}{s_2^2/\sigma_2^2} }[/math]

- where [math]\displaystyle{ s_1 }[/math] is the standard deviation of random sample drawn from a population with standard deviation [math]\displaystyle{ \sigma_1 }[/math], [math]\displaystyle{ s_2 }[/math] is the standard deviation of an independent random sample drawn from a population with standard deviation [math]\displaystyle{ \sigma_2 }[/math]. Both populations are assumed to follow as normal distribution.

- The probability distribution of all possible values of a F-Statistic is a F-distribution.
- It can (often) be used in an F-Test.

- …

**Example(s):**- AnANOVA test can produce a f value.

**Counter-Example(s):**- a t-Statistic.

**See:**ANOVA, Regression Analysis, Normal Distribution.

## References

### 2016

- (Wikipedia, 2016) ⇒ http://www.wikiwand.com/en/F-test
*Retrieved 2016-10-09*- QUOTE: The formula for the one-way
**ANOVA***F*-test statistic is

- QUOTE: The formula for the one-way

- [math]\displaystyle{ F = \frac{\text{explained variance}}{\text{unexplained variance}} , }[/math]
- or

- [math]\displaystyle{ F = \frac{\text{between-group variability}}{\text{within-group variability}}. }[/math]
- The "explained variance", or "between-group variability" is

- [math]\displaystyle{
\sum_i n_i(\bar{Y}_{i\cdot} - \bar{Y})^2/(K-1)
}[/math]
- where [math]\displaystyle{ \bar{Y}_{i\cdot} }[/math] denotes the sample mean in the
*i*^{th}group,*n*_{i}is the number of observations in the*i*^{th}group,[math]\displaystyle{ \bar{Y} }[/math] denotes the overall mean of the data, and*K*denotes the number of groups. - The "unexplained variance", or "within-group variability" is

- where [math]\displaystyle{ \bar{Y}_{i\cdot} }[/math] denotes the sample mean in the
- [math]\displaystyle{
\sum_{ij} (Y_{ij}-\bar{Y}_{i\cdot})^2/(N-K),
}[/math]
- where
*Y*_{ij}is the*j*^{th}observation in the*i*^{th}out of*K*groups and*N*is the overall sample size. This*F*-statistic follows the*F*-distribution with*K*−1,*N*−*K*degrees of freedom under the null hypothesis. The statistic will be large if the between-group variability is large relative to the within-group variability, which is unlikely to happen if the population means of the groups all have the same value. - Note that when there are only two groups for the one-way ANOVA
*F*-test,*F*=*t*^{2}where*t*is the Student's*t*statistic.

- where

- http://www.statisticshowto.com/f-statistic/
- An F statistic is a value you get when you run an ANOVA test or a regression analysis to find out if the means between two populations are significantly different. It’s similar to a T statistic from a T-Test; A-T test will tell you if a single variable is statistically significant and an F test will tell you if a group of variables are jointly significant.

- http://www.mathworks.com/help/stats/f-statistic-and-t-statistic.html
- QUOTE: In linear regression, the F-statistic is the test statistic for the analysis of variance (ANOVA) approach to test the significance of the model or the components in the model.
Definition: The F-statistic in the linear model output display is the test statistic for testing the statistical significance of the model. The F-statistic values in the anova display are for assessing the significance of the terms or components in the model.

- QUOTE: In linear regression, the F-statistic is the test statistic for the analysis of variance (ANOVA) approach to test the significance of the model or the components in the model.