Neuroscience Animations

John H. Krantz, Hanover College, krantzj@hanover.edu

The F Distribution

Brief description and instructions (DRAFT):

Background:

For comparing more than two means simultaneously, the Analysis of Variance (ANOVA) test is done. The ANOVA test uses the F statistic:

F = (Var₁)/(Var₂)

While ANOVA tables often hide this fact, the F statistics is simply the ratio of two variance measures. Mean Squares are a different name for variance. The two degrees of freedom are those associated with each variance: df₁ is the degrees of freedom of the variance from the numerator (Var₁) while df₂ is the degrees of freedom associated with the denominator. In the ANOVA, the numerator variance is obtained from variation that results from the different conditions while the denominator variance is the variation that results from the variation in participants and measurement error.

Well, since we are taking the ration of two variances, we can hardly assume that an F statistic will have the same distribution of possible and likely values as t or the normal distribution (z)

Using the illustration:

This applet allows you to see how the F values that can be expected when the null hypothesis is true varies. The method used will be through collecting samples and plotting the results to develop a frequency histogram. The x-axis will show plot the value, and the y-axis will plot how often that value is obtained.

There will be 4 histograms on the screen. The upper left histograms will plot the frequency of individual sample values. The bottom three histograms will plot F values obtained from taking two samples from the same population and calculating the F from those two samples. The difference will be the degrees of freedoms used for the samples. An F distribution will be fit to the data.

To start the sampling, press the Start button in the upper left corner of the screen. The number of samples taken (actual number of values for the sample distributions in the top row and the number of samples of size n in the sampling distributions in the lower row) is shown below the Reset button on the upper left corner. As each sample is taken, the value or the mean will be plotted on its proper histogram. You can see the histogram develop as the number of samples increase. To stop the sampling press the Stop button and to start over, press the Reset button. If you change the sample size the next time you start the sampling, the sample will be reset.

There are also advanced option available in the menus. You can change the bin size of the histograms, the speed the samples are taken and the distribution of the population from which the samples are taken. It might be important to change the population distribution and see if the same F distribution is still found. In other words, does the Central Limit Theorem still hold?

Click here to open the applet. It will open a new window that will fill your screen.

References: