Just because proportions and probabilities are measured with the same units (percentages) does not mean that they're equivalent or convertible. A proportion is a measurement representing the size of a subset relative to its whole. A probability is a measurement of the likelihood that a given event will occur. If you flip a coin ten times and get four heads (proportion = 40%) that doesn't mean that the probability that the next flip will be a head is also 40%, or 60%, or any other percent.

I say all that to point out the shoddy analysis of a recent Associated Press poll about President Obama by Ken Thomas and Jennifer Agiesta. The key offense is here:

Entering 2012, President Barack Obama's re-election prospects are essentially a 50-50 proposition, according to a new Associated Press-GfK poll. It found that most Americans say the president deserves to be voted out of office even though they have concerns about the Republican alternatives. ...

The poll found Americans were evenly divided over whether they expect Obama to be re-elected next year.

For the first time, the poll found that a majority of adults, 52 percent, said Obama should be voted out of office while 43 percent said he deserves another term.

1. There's a huge difference between (a) "Americans say the president deserves to be voted out of office" and (b) "Americans were evenly divided over whether they expect Obama to be re-elected". (A) is a question about what should happen, in the mind of the individual being polled, while (b) is a question about what the person believes will happen based on the votes of millions of people. For example, it's very possible to think that Obama should win re-election but will not.

2. 52 percent to 43 percent is not "evenly divided" by any stretch of the imagination. If there is any 50/50 data in the poll that the "evenly divided" is referring to the authors never mention it. A 52%-43% election would be an historic landslide for the victor. (In 2008, Obama beat John McCain 52.9% to 45.7% in the "popular vote", a seven-point spread.)

3. If 52% of people really think that Obama should not be re-elected and 43% think he should be, that doesn't mean that Obama has a 52% probability of losing the election and a 43% probability of winning. If the poll response accurately represents the view of the people who show up to vote, then Obama has a 100% probability of losing the election. If Obama gets 50% of the votes minus one, and his opponent gets 50% plus one, Obama will 100% lose.

4. Journalists really need to understand statistics before they "analyze" them.

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