I have multiple student graduation rates from the same school. Each year, there are variations in the student population and each year there are different graduation rates. Is there a way to compare the graduation rate from year to year to see the following:

1. Graduation rate is not changing due to random chance?
2. The population levels are not effecting the scores?

To expand on #2, I mean, lets say in year 1, I have 50 children, then in year 2, I have 55 children. This means the graduation rate in year 2, each child weighs a bit less than year 1.

Thank you,

Take $p$ as the probability of a single student graduating (Bernoulli distribution), then the number of students graduating would be a binomial distribution, $Binom(N, p)$; so basically you are testing if $p$ is constant from one year to another. One way to test this is likelihood ratio test.
The reduced-model (or the null model) is the simpler model where $p$ is constant. You can easily find the estimate $\hat{p}$ under this estimation by MLE or method of moments, knowing that the mean of distribution is $p \times N$, and calculate the likelihood value using this estimate of $p$.
There would be multiple ways to define the full-model (or the alternative model); one would be to divide the sample points into halves, and take $p_0$ as the probability of graduation for the first half, and $p_1$ as the probability of graduation for the second half. Estimate each $p_i$ separately, and calculate the likelihood as the product of the likelihood of the first half with $\hat{p}_0$ and the second half with $\hat{p}_1$. (or, alternatively one can say log likelihood is the summation of log likelihood of the first half and the second half).