## Wednesday, November 9, 2016

### Powerlifters vs Gym Rats, take 2

(This is a redo of the numbers in my previous powerlifters vs gym rats post, with assumptions that are less favorable to powerlifters.)

First, since we need some sort of metric to compare athletes, I'll unbiasedly 😀 choose the average of three lifts, bench press, deadlift, and squat, as a percentage of the bodyweight of the athlete. Call that metric $S$.

We'll use a standard Normal for the distribution of this metric, by subtracting the mean (100 percent of bodyweight for non-powerlifters, assuming that the average gym rat can bench, deadlift, and squat their own bodyweight) and dividing by the standard deviation (say 15 percent of bodyweight, using the scientific approach of judging 10 to be too little and 20 to be too much). In other words, for non-powerlifters, $z \doteq (S-100)/15.$

As in the previous post, we'll assume that powerlifters are 1 percent of the gym rats; but instead of the powerlifters having a mean at 2 (in $z$ space, 130 in $S$ space), they only have a one-SD advantage, that is their mean is at 1 (in $z$ space, 115 in $S$ space). In other words

$\qquad z \sim \mathcal{N}(0,1)\qquad$ for non-powerlifters
$\qquad z \sim \mathcal{N}(1,1)\qquad$ for powerlifters

Using these assumptions we can now compute the percentage of powerlifters that exist in a gym population above a given threshold; we can also compute the median score of all athletes who score above that threshold (click for larger):

Note that the conditional median that we're using here is lower  than the conditional mean, as the conditional distribution is skewed to the right, i.e. has a long right tail. The choice of the median is more informative for skewed distributions as a "sense of what we'll see in the gym."*

It's interesting to note that this is the median of the combined distribution of powerlifters and other gym rats, weighted by their proportion in the population above the threshold, so the difference between this median and the threshold is a non-monotonic function of the threshold as the curvature and the weight of the distribution of each type of athlete change significantly in the $1-8$ range of the table.

Under these weaker assumptions (pun intended), only when the threshold for inclusion passes 5 standard deviations from the other gym goers' mean do powerlifters become the majority of the qualifying athletes. Unless the gym is full of football players (that's american football), weightlifters, and strongman competitors, I think these assumptions are too unfavorable to powerlifters.

Here are some strong athletes moving metal, for variety (NSFW language):

"While they squat I eat cookies" has to be the most powerlifter-y sentence ever.

Update Nov 11, 2016: Here's the percentage of powerlifters in the population of qualifying athletes for different assumptions about the advantage of powerlifters (i.e. the mean of the powerlifters' distribution in standard deviation units); click for larger:

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* Unless there are CrossFit-ers in the gym, in which case what we typically see in the gym is dangerous, counter-productive nonsense.