A specific probability figure is quoted: for the last five years of data, assuming offensive plays to fumbles follows the Normal distribution, the probability of the Patriots amassed such a record by chance would occur "once in 16,233.77 instances." In other words, it's very, very unlikely. So, does this constitute proof the Patriots cheated?

There are two assumptions that are important to understand: that plays per fumble follows a Normal distribution (a bell curve), and that the Patriots were an average team.

The first assumption is a fairly standard one, but there's not really enough evidence presented in the article to know whether or not it's true. The graphs presented do look like bell curve, but I would like to see a lot more data. In fact, many distributions look quite similar to the Normal distribution but are different enough to make a big difference in calculating the probability of unusual events. In other words, one in 16,233.77 might be significantly higher or lower than it should be if the distribution is not Normal.

The second assumption is of a different sort. It is part of the hypothesis being tested. "If the Patriots are an average team, how likely would they have this unusual statistic just by chance?" An average team could get a whole lot of lucky breaks, but as the analysis showed, it would be very unusual. If, on the other hand, the Patriots are an above-average team, then they would be much likelier to have an unusual fumble statistic. Perhaps they are so good at avoiding fumbles that there's only a 1 in 20 chance of them accumulating such an unusual statistic -- the statistic is so unusual that even a good team still needs to get several lucky breaks to accumulate it. There still might be an element of luck, but at a more believable level.

As Mr. Sharp clearly acknowledges:

Could the Patriots be so good that they just defy the numbers? As my friend theorized: Perhaps they’ve invented a revolutionary in-house way to protect the ball, or perhaps they’ve intentionally stocked their skill positions with players who don’t have a propensity to fumble. Or perhaps, still, they call plays that intentionally result in a lower percentage of fumbles. Or maybe it’s just that they play with deflated footballs on offense. It could be any combination of the above.

This is exactly right. One in 16,233.77 should be understood to be the probability an average team would have this fumble statistic by chance. It indicates the Patriots were either an above-average team in ball protection or deflating balls or both. The statistic really just helps us rule out chance. We know that it really isn't just a matter of good luck. But it tells us nothing about the mixture of causes.

The best analogy I can think of is a grand jury versus a petit jury. The statistic alerts us that something unusual has gone on. We have evidence that says, "Proceed with an investigation." But the statistic doesn't enable us to tell what flavor of unusual it is, whether it is simply great playing or it is cheating. Only a more detailed analysis and investigation will tell.

On the other hand, I can easily imagine other teams deflate balls and will never be caught by this sort of statistical analysis. How? If a team has below-average ball protection and the team starts to cheat by deflating balls, its efforts will bring it back to the average. Its results will not look unusual at all. This kind of statistical analysis will only pick up on cheating that pushes a team far above the norm.

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