American presidential elections are decided by electoral votes, not by popular vote. Each electoral district is treated as an independent race and the winner is the candidate who wins the most electoral votes.

The website FiveThirtyEight tracks polls and estimates outcomes by electoral district. It then appears to run Monte Carlo simulations to estimate the distribution of outcomes in Electoral College votes, from which it projects a likelihood that each candidate will win. Those probabilities should be the percentage of times each candidate won in the simulated elections.

This project repeats the Monte Carlo simulation, given some assumptions about the data in the FiveThirtyEight HTML file. FiveThirtyEight shows bars for the range that 80% of election outcomes are expected to come under. I use this to back out the mean and standard deviation of each district’s outcome distribution, and using this description of the distribution, I perform a Monte Carlo simulation.

At least, that is the plan. The simulation produces an improbably high expectation of victory for Hillary Clinton that does not match FiveThirtyEight’s simulation results.

1. Getting the standard deviation and the mean

   Given the left side of the bar and the width, and knowing that the bar represents 80% of the outcomes, I set the mean to the midpoint of the bar and set the standard deviation to the width of the bar divided by the symmetric Z-score that captures 80% of outcomes (i.e. 2 * 1.2816).

2. Getting the probability of victory

   Given the mean, standard deviation, and the estiomated position of the 50% mark, I back out the Z-score that the candidate’s opponent’s popular vote must be below for the candidate to win. Given this Z-score, I can back out the probability of victory for the candidate.

3. Simulating the districts

   Given the probability of winning for the candidate in each district, I sample from a linear distribution from 0 to 1. In the cases that a random linear variable exceeds the candidate’s probability of winning, the candidate loses that district. If the random linear variable is below, then the candidate wins that district.

4. The distribution of outcomes

   Having run the simulation millions of times, we obtain a distribution of electoral college results that we can plot. The count of outcomes are normalized against the number of simulations to produce projected Electoral College votes on the x-axis and the probability of that outcome on the y-axis.

The method used by Nate Silver at FiveThirtyEight.com is markedly different from the method used here. In particular, Nate Silver samples from a Student t-distribution with 10 degrees of freedom (instead of a Gaussian distribution) and performs more adjustments to the poll data. A complete description can be found here.