Determine the distribution of the population from which data is to be sampled. This distribution depends on the type of data you intend to sample. For example, sampling height measurements from a group of people is a normal (or Gaussian) distribution. Sampling the number of heads in a series of coin flips is a binomial distribution.
Write the probability density function of the distribution. If you are not familiar with the probability density function for the data that you plan to measure, you can find the mathematical form in most statistical textbooks. For example, if your data is in the form of success/fail, as for acceptances to colleges or other such procedures, this is a binomial distribution, which has the probability density function of nCk*p^k*(1-p)^(n-k), where nCk is the combinatoric function, n!/[k!(n-k)!]; p is the probability of a success; n is the number of trials and k is the number of successes.
Put the basic information (not actual data) and estimator into the probability density function. For example, if you are running 10 trials (applying to 10 universities) and have an estimator of a probability of 70% for getting into a university, you will have the resulting probability density function: f(k | n, p) = 10Ck*(.7)^k*(.3)^(10-k).
Write the likelihood function for the probability density function. Reorder the function used previously so the parameter of the probability density function is the main variable (i.e. f(k | n, p) becomes f(p | k, n)). The equation’s right-hand side will remain the same with the exception of the values for “p” returning to the variable “p.” Thus for the example, you would write f(p | k, n) = 10Ck*(p)^k*(1-p)^(10-k).
Put the data into the likelihood function. For the example, the data is the number of successes. For example, if you received seven acceptance letters from 10 colleges, your data is k = 7. Thus, your equation will be f(p | k, n) = 10C7*(p)^7*(1-p)^3.
Plot the finished likelihood function. The plot will be a curve.
Find the maximum value of the likelihood function on the plot. This is the maximum likelihood value. For example, for the curve f(p | k, n) = 10C7*(p)^7*(1-p)^3, the maximum value appears at p = .7. Thus, the maximum likelihood value is p=.7 for this example.