What, me worry about margins of error?
The Wall Street Journal's poll of the seven swing states finds that either former President Donald Trump or Vice President Kamala Harris leads by 2% or less, except in Nevada, where Trump is up by 6%. The Journal says the margin of error in each state is +/- 4%. Although Trump leads by 6% in Nevada, the Journal says this lead too is "within the margin of error."
Say what? In the technical notes, we read:
"A candidate’s lead—the difference between two candidates’ percentages in a poll—has its own margin of error. This is because the margin of error for a lead is calculated to account for the margins of error around both candidates’ percentages.
"In most cases, a candidate’s apparent lead must be at least two times the poll’s basic margin of error to say a candidate is actually in the lead. In this case, the poll’s basic margin of error of 4 percentage points would require a lead of 8 percentage points to clearly show a lead."
This is a misunderstanding. Trump really is winning in Nevada.
A simple example may clear up the confusion. Suppose that I toss a fair coin. The chances of a head are one-half. And the chances of a tail are one half.
Suppose that we observe a head. What was the probability of a head?
The Journal would reason something like this: "Well, the chance of a head was one-half, and the chance of a tail was one-half. Either outcome could occur, and their probabilities are independent. That is, the chance of a head does not affect the chance of a tail. The probability of two independent events is the product of their probabilities. Therefore the chance of a head is one-half times one-half, or one-fourth."
Uh, no. The probability of a head is one-half. There are two outcomes, heads and tails. If a head occurs, a tail cannot. Given the head, the probability of a tail is not one-half. It is zero.
The same thing in political polls. We ask the interviewee if she would vote for Trump or Harris. If she chooses Trump, she cannot simultaneously choose Harris. So the only margin of error -- which measures the dispersion in responses for Harris in the sample -- that matters is for Harris. Once Harris is chosen, the choice of Trump is no longer a random variable. Its standard error, which determines the margin of error, is zero.
The margin of error for a Trump victory is 4%, not 8%. The +/- 4%, which The Journal incorrectly calls a margin of error, describes the probability that either Trump leads by up to 4% or Harris leads by up to 4% in the poll sample, given that the race is actually a tie.
What The Journal probably has in mind is something like the difference in votes for Harris in two periods. For example, we may observe that Harris took 50% of the sample this month and only 45% last month. We want to know whether the difference, 5%, is more than a fluke. In this case, we can reasonably treat the two events -- a Harris win last month and a Harris win this month -- as independent. That Harris won last month need not affect her chances of winning this month. So, in determining whether the two Harris shares differ significantly, we should consider the probability of each share independently.
But that is not the case for the event in which an interviewer says this month that she favors Harris. There is not an independent probability that she favors Trump. The Trump probability is zero.
In practical terms, The Journal's error matters little, this time. But in a race this close, one must take care to interpret future poll results correctly.
The major newspapers -- The New York Times, The Washington Post, and The Wall Street Journal -- are abominable at reporting political poll statistics. They have misreported the Presidential race at every stage. And their mistakes have probably changed the race, by misleading donors and campaign coordinators. -- Leon Taylor, Seymour, Indiana tayloralmaty@gmail.com
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