How do investment managers/prop traders/hedge fund analysts respond to the abysmal statistics that compare active money management to indices?

First of all, the statistics are a bit misleading because in many instances the asset manager doesn't want to beat the index.  For example, most pension funds would like a volatility that is lower than the SP 500 which means a lower yield. Second, when you are dealing with large enough amounts of money, then passive index investing is not possible.  If your holdings are large enough then anytime you buy and sell something, you will move the market and in that situation your goal is not to beat the market, but not to get killed by the market.  For a large holding, getting market returns is non trivial, because any time you do something, the market will react to reduce your return.

Most investment managers are rain dancers. They dance, some money rains on them, they say hey I can manage a fund, raise a bunch of cash and the rain stops. When they dance again and all they get is a blizzard. This happens during every market cycle, and was self-evident during the dotcom bubble when upper-class college grads started their own venture funds. Some investment managers are scientists. They model things, and then make predictions. For example, if I know what you are going to buy next, I can trade ahead of you. That's HFT. If I know that decreased access to capital is going to jack up corporate interest rates, I can short junk bonds. That's fixed income. If I know that a company is going to create a product that can totally monopolize an entirely new market, I can invest in it. That's VC. Saying that it's all random is both useful and useless at the very same time. It is all random most of the time. But given the current situation you are in, and the current situation others are in, you can use past experience to model future events and arbitrage the difference between the model and reality in doing so. That's real edge, it can be created, and there's no taking it away.

The fact that active managers do not usually beat the market isn't surprising when you consider that, collectively, active managers are the market. To be more specific: the market is made up of active managers plus passive (index-following) managers.  The return of the market as a whole is the average of the return of these two styles.  Since the return of passive strategies is (more or less) the return of the whole market, the return of active strategies must also (more or less) be that of the whole market (if C = (A+B)/2, and C=A, then C=B). But active managers charge higher fees.  Therefore they will collectively tend to underperform passive strategies and the market as a whole.  Note that I have not said anything at all about the skills of active managers.  It may be that every active manager is a Warren Buffet-like genius -- it is simply impossible, as a matter of arithmetic, for active managers to collectively beat the market.  If all active managers are geniuses this will be reflected in the average market return, not in the outperformance of the average active manager.

1) It's incorrect to compare some vague "average investment managers" to an index. Like in any other profession, there is a distribution of abilities in investing - there are excellent managers, good, mediocre, and weak ones. Instead of averaging them out, take top 20% of the managers and see how they performed against the passive indexes. 2) "Statistical anomaly" argument is just ridiculous. From this point of view, all accomplishments can be attributed to plain dumb luck. All billionaires are just lucky. All sport champions and Oscar winning actors are just lucky. Warren Buffet is just lucky. If many people try, somebody has to be a winner, right? Statistically speaking... Of course, talent, skills, perseverance, hard work have nothing to do with this. Anyone who uses the "statistical anomaly" argument to explain results that contradict their theories is guilty of the attribution bias and abuse of statistical science.

Your statement "almost all active money managers fail to beat their respective indices" is categorically untrue, plenty of money managers beat their benchmark indices.  Some don't and some do, certainly not ALMOST ALL on either side. Regarding Medallion it would be a pretty odd question for a fund of funds or a somewhat knowledgeable investor to ask a question about a single fund and it's relative performance that year.  They typically understand there are outliers in terms of fund performance every year and there always will be.