# F-Statistic

• 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:
$\displaystyle{ f= \frac{s_1^2/\sigma_1^2}{s_2^2/\sigma_2^2} }$
where $\displaystyle{ s_1 }$ is the standard deviation of random sample drawn from a population with standard deviation $\displaystyle{ \sigma_1 }$, $\displaystyle{ s_2 }$ is the standard deviation of an independent random sample drawn from a population with standard deviation $\displaystyle{ \sigma_2 }$. Both populations are assumed to follow as normal distribution.

## References

### 2016

$\displaystyle{ F = \frac{\text{explained variance}}{\text{unexplained variance}} , }$
or
$\displaystyle{ F = \frac{\text{between-group variability}}{\text{within-group variability}}. }$
The "explained variance", or "between-group variability" is
$\displaystyle{ \sum_i n_i(\bar{Y}_{i\cdot} - \bar{Y})^2/(K-1) }$
where $\displaystyle{ \bar{Y}_{i\cdot} }$ denotes the sample mean in the ith group, ni is the number of observations in the ith group,$\displaystyle{ \bar{Y} }$ denotes the overall mean of the data, and K denotes the number of groups.
The "unexplained variance", or "within-group variability" is
$\displaystyle{ \sum_{ij} (Y_{ij}-\bar{Y}_{i\cdot})^2/(N-K), }$
where Yij is the jth observation in the ith 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=t2 where t is the Student's t statistic.

• 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.