Both the Chi-Square distribution and F-distribution investigate population variance. Population variance (a parameter) and sample variance (a statistic) describe the heterogeneity of a population and sample, respectively. Both these distributions investigate hypotheses about population variances using sample variance.
While the complex mathematical forms of the Chi-Square distribution and F-distribution look strikingly different, their mathematics have some commonalities. Of main interest to statisticians is the relationship between Chi-Square distribution and the F-distribution. The F-distribution takes two variables (v1 and v2), whereas the Chi-Square distribution takes one variable (v). In fact, the F-distribution multiplied by the variable "v1" asymptotically approaches the Chi-Square distribution as "v2" approaches infinity. In the opposite direction, the ratio two Chi-Square distributions being divided by the ratio of their associated variables is an F-distribution. That is, if "X1" and "X2" are two Chi-Square distributions associated with variables "v1" and "v2" respectively, then "X1/X2" multiplied by "v2/v1" is an F-distribution.
Unlike other statistical distributions like the Bernoulli distribution or the Binomial distribution, the Chi-Square distribution and F-distribution do not occur in nature. That is to say, these two distributions are entirely theoretical. Yet these two distributions are essential in applied statistics, especially in studies of variance. In short, both the Chi-Square distribution and the F-distribution make the impossible assumption of a distribution according to a theoretical spreading of individual data points (on some variable), yet this does not impact their use in the real world.
When researchers apply the Chi-Square distribution and F-distribution, they are making an implicit assumption. This assumption is that the data from which researchers are sampling is normally distributed. In other words, the applications of both distributions rely on assumptions of normality for the population. In fact, both of these distributions can be linked back to the normal distribution (F is Chi-Square, Chi-Square is Exponential, Exponential is Gamma and Gamma is Normal). One consequence of this fact is that if a researcher were to find that a population is not normally distributed, then his basis for using either of these two distributions would be lost.