What is the empirical frequency? I have generated $1.000.000$ repetitions of the experiment of drawing $20$ iid. Bernoulli random variables $X_1, ..., X_{20}$ (20 coins) with bias $1/2$ in R. 
I then wish to plot the empirical frequency of observing $1/20 \sum_{i=1}^{20} X_i \geq \alpha$ for $\alpha \in (0.5, 0.55, 0.6, ..., 0.95, 1)$ 
Can anyone explain what empirical frequency is and how to calculate it? 
 A: Empirical probability is the number of times some event was observed in the data you have divided by the total sample size. Empirical frequency is simply the counts.
A: Empirical Frequency is merely the observed frequency.  If you have a known fair coin and you flip it once and it turns up heads then the emprical frequency of the coin being heads is 1.  It is a different concept than the Probability the coin will come up heads.  To calculate Empirical Frequency, you just take the number of times the the result you are measuring comes up and divide that by the number of times you run the experiment.  
In the case of a coin flip; if you flip the coin 8 times and heads comes up 5 of those times then the Empirical Frequency is 5/8.  That's it.  In the case of a die roll; if you roll a six sided die 9 times and get 6 twice, and you want to know what the Empirical Frequency of getting a 6 on this die is; then the Empirical Frequency would be 6/9 which equals 2/3.  Even if the coin in the first example was a fair coin and the die in the second example was fair; the Empirical Frequency only addresses what was observed.  
This is closely related to the Maximum Likelihood estimate.  This is the formula:
$$
\frac{1}{N} \Sigma X_i
$$
$ \frac{1}{N} $ just means "one result of how ever many outcomes." In the case of a die this would be $\frac {1}{2} $ or a six sided die it would be $\frac {1}{6} $.  Basically, it is the probability for a fair die or coin.
$ \Sigma X_i$ Means the sum $\Sigma$ of the results you are measuring $ X_i $. So, in the example with the six sided die, if you are measuring the empirical frequency of getting six on this die and you get six 5 times then $ X_i $ would be 5. 
