Not all statistics are created equal: Proof from Mixed Martial Arts

A few years ago, taking testimony in a court case, an attorney asked me,

“Tell me, doctor, have you heard the saying, ‘Lies, damned lies and statistics’? Isn’t it true what they say, that you can lie with statistics?”

I answered,

“Not to me, you can’t.”

My point that day was that if the person evaluating your statistical argument knows their stuff, you are not going to be able to use statistics to prove a false argument. This past week provided a prime example of that.

On March 3rd, darling daughter number three fought for the world title in mixed martial arts in the 135 pound division. Prior to the fight, several websites had picked her opponent to win, “based on the numbers”. They argued that the sports book odds (which favored Ronda three to one) were influenced by hype, trash-talking, looks, you name it and that if you looked at actual numbers her opponent would win.

What were these numbers? They used statistics like this:

• Average number of punches landed in a fight: The opponent had more
• Average percentage of successful punches in a fight: The opponent had more
• Average number of kicks landed in a fight: The opponent had more
• Average number of the opponents’ punches avoided: The opponent had more
• Average number of successful take down attempts: The opponent had more
• Number of technical knock-outs: The opponent had more
• Percent of submissions: Ronda had more. In fact, she had 100%.

According to these “statistical analyses”, on every dimension but one, Ronda was the weaker fighter and thus, they predicted, she would lose. They pointed out that she had won all of her matches the same way and was therefore clearly a limited fighter. They advised the readers of their blogs and websites to take advantage of these ridiculous odds and place some serious money on the opponent.

There is only one statistic that matters

There were a couple of problems with this analysis. Foremost is that not all statistics are created equal. A submission ends the match and gives you a win. So, even if Ronda’s opponent manages to land 6 punches to her 4 before the submission occurs, once Ronda dislocates the other woman’s elbow and wins by submission, the number of punches is irrelevant. The percentage of times Ronda’s matches have come to a decision – 0%.

One reason Ronda has not landed a bunch of kicks and punches is that she had ended all of her matches up to this point in under a minute. That doesn’t give a lot of time to punch or kick. What about the percentage of successful punches? Surely that is relevant, no? The number of punches, kicks and take downs only is relevant when it comes to a decision. Ronda has been criticized for the fact that she is willing to “eat a couple of punches” on her way in to get into the clinch and throw her opponent, transitioning into a submission. She does this deliberately figuring that hey, she may get hit in the face once but after she does she is going to be close enough to grab you, throw you and break your arm, so it was a calculated risk. She gets a lot of press for her looks and athletic accomplishments, but when the writer from Sports Illustrated asked me to tell her something most people don’t know about Ronda, I told her,

“She’s really good at math.”

One way to understand the error of these armchair statisticians, and why they were so far wrong, is to realize they had failed to realize theirs was an implied conditional probability.  We all know that, as this lovely site from Yale University points out:

“If events A and B are not independent, then the probability of the intersection of A and B (the probability that both events occur) is defined by P(A and B) = P(A)P(B|A).”

On the condition that Ronda had not already won by arm bar, these other variables could predict a decision or technical knock out. A and B are definitely not independent.

So maybe, the probability of her opponent winning by a decision, was 60% , if it went to a decision, making a the 3 to 1 odds the bookies were giving on Ronda winning look outrageous, as some sites called it. However, if her odds of losing to Ronda by submission were 80%, then the odds of her opponent actually winning a decision were 12% – 60% of the 20% of the time it went to a decision. Now, I just made up those numbers of 80%, 60% and so on. The point is that you need to consider the probability of Ronda winning by submission is considerably higher than 0% and calculate your probability of her opponent winning given the inverse of that probability.

In case you were wondering, Ronda won the fight in the first round by arm barring her opponent into submission.

Congratulations to Ronda Rousey, 135 lb champion of the world.