There are several methods to calculate the F statistic in a one-way Analysis of Variance (ANOVA) test - I learned the method in the video below from the good folks in the Mathematics Support Services at NCI. All the methods are based on calculating F by taking into account the differences within each sample and between each sample when you want to see if there are differences between three or more samples.
ANOVA involves a lot of calculations, especially in larger sample sizes - but the good news is that it is mostly just adding, multiplying, dividing, and squaring numbers. The video below took several takes to get right, and at 16 mins 27 secs is bar far my longest video on YouTube. I hope my viewers will not get bored before the end and drop off!