Editing F-test (section)

=== One-way analysis of variance ===
The formula for the one-way '''ANOVA''' ''F''-test [[test statistic|statistic]] is
:<math>F = \frac{\text{explained variance}}{\text{unexplained variance}} ,</math>

or

:<math>F = \frac{\text{between-group variability}}{\text{within-group variability}}.</math>

The "explained variance", or "between-group variability" is

:<math>
\sum_{i=1}^{K} n_i(\bar{Y}_{i\cdot} - \bar{Y})^2/(K-1)
</math>

where <math>\bar{Y}_{i\cdot}</math> denotes the [[average|sample mean]] in the ''i''-th group, <math>n_i</math> is the number of observations in the ''i''-th group, <math>\bar{Y}</math> denotes the overall mean of the data, and <math>K</math> denotes the number of groups.

The "unexplained variance", or "within-group variability" is

:<math>
\sum_{i=1}^{K}\sum_{j=1}^{n_{i}} \left( Y_{ij}-\bar{Y}_{i\cdot} \right)^2/(N-K),
</math>

where <math>Y_{ij}</math> is the ''j''<sup>th</sup> observation in the ''i''<sup>th</sup> out of <math>K</math> groups and <math>N</math> is the overall sample size.  This ''F''-statistic follows the [[F-distribution|''F''-distribution]] with degrees of freedom <math>d_1=K-1</math> and <math>d_2=N-K</math> 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 [[expected value|population means]] of the groups all have the same value.
[[File:5% F table.jpg|thumb|F Table: Level 5% Critical values, containing degrees of freedoms for both denominator and numerator ranging from 1-20]]
The result of the F test can be determined by comparing calculated F value and critical F value with specific significance level (e.g. 5%). The F table serves as a reference guide containing critical F values for the distribution of the F-statistic under the assumption of a true null hypothesis. It is designed to help determine the threshold beyond which the F statistic is expected to exceed a controlled percentage of the time (e.g., 5%) when the null hypothesis is accurate. To locate the critical F value in the F table, one needs to utilize the respective degrees of freedom. This involves identifying the appropriate row and column in the F table that corresponds to the significance level being tested (e.g., 5%).<ref>{{Citation |last=Siegel |first=Andrew F. |title=Chapter 15 - ANOVA: Testing for Differences Among Many Samples and Much More |date=2016-01-01 |url=https://www.sciencedirect.com/science/article/pii/B9780128042502000158 |work=Practical Business Statistics (Seventh Edition) |pages=469–492 |editor-last=Siegel |editor-first=Andrew F. |access-date=2023-12-10 |publisher=Academic Press |doi=10.1016/b978-0-12-804250-2.00015-8 |isbn=978-0-12-804250-2|url-access=subscription }}</ref>

How to use critical F values: 

If the F statistic < the critical F value

* Fail to reject null hypothesis
* Reject alternative hypothesis
* There is no significant differences among sample averages 
* The observed differences among sample averages could be reasonably caused by random chance itself
* The result is not statistically significant 

If the F statistic > the critical F value

* Accept alternative hypothesis
* Reject null hypothesis
* There is significant differences among sample averages 
* The observed differences among sample averages could not be reasonably caused by random chance itself
* The result is statistically significant 

Note that when there are only two groups for the one-way ANOVA ''F''-test, <math>F = t^{2}</math>where ''t'' is the [[Student's t-test|Student's <math>t</math> statistic]].

==== Advantages ====

* Multi-group comparison efficiency: facilitating simultaneous comparison of multiple groups, enhancing efficiency particularly in situations involving more than two groups.
* Clarity in variance comparison: offering a straightforward interpretation of variance differences among groups, contributing to a clear understanding of the observed data patterns.
* Versatility across disciplines: demonstrating broad applicability across diverse fields, including social sciences, natural sciences, and engineering.

==== Disadvantages ====

* Sensitivity to assumptions: the F-test is highly sensitive to certain assumptions, such as homogeneity of variance and normality which can affect the accuracy of test results.
* Limited scope to group comparisons: the F-test is tailored for comparing variances between groups, making it less suitable for analyses beyond this specific scope.
* Interpretation challenges: the F-test does not pinpoint specific group pairs with distinct variances. Careful interpretation is necessary, and additional post hoc tests are often essential for a more detailed understanding of group-wise differences.