Commentary on Correlation

If it is a real correlation, while a direct causative relationship is not guaranteed, it is guaranteed that they are causally related. 

Being born and growing hair is highly correlated. Hair doesn't grow on anything that isn't (and won't be) born. However, it's not caused by being born - rather, both hair growth and birth are caused by being conceived. The correlation does imply attachment to the same causal chain.

If it's not causally related in some way, it is not a real correlation, and it means the data is wrong. A better set will show a proper 0 correlation. Put another way, we can see the above silly correlation break if we consider the dataset of haired creatures which aren't born yet.

 

Wording again: using my previous example, flipping the free-money coin a bunch of times will be all but 1.0 correlated with your money going up. Nobody is going to quit before they get a tails. It's only a 0.5 on a per-flip basis.

Wording the third: if a lifetime of wealth is 0.5 correlated with coin-flipping, then this means of events that caused your net worth, half of them were coin-flips, and the other half were something else. 

Scientists often say 'half of wealth is explained by' but this isn't actually correct, due to the magnitude issue. You get the same 0.5 correlation whether every other dollar was earned by coin-flips by many individuals, or whether half of individuals earned all their wealth via gambling, and the other half didn't touch the stuff. Even if all the coin-flippers are basically poor and the wagies are basically rich, as long as half the time the wealth goes up from coin-flipping, the correlation is going to be 0.5. 

Imagine 99 folk earned all their money by working hard flipping as many coins as possible (just to be perverse, really), and another guy won the lottery instead. The 99 have $100 and the other guy has $10,100.
I'm probably simplifying, but this dataset approximately says that working is correlated with the average net wealth of $200 at 0.5 and winning the lottery "explains" the other 0.5. The problem is the scientists. They have mixed apples and oranges. If you don't win the lottery your wealth is 1.0 correlated with coin-flipping, and if you do win the lottery, it will be highly correlated with winning the lottery. 

Maybe we can hope in real datasets humans kinda normalize themselves, but, uh, they don't. Real life, unlike my nice clean coin-flipping example, is not well-normalized. Nutrition ""scientists"" are particularly horrible for this. 

Again: imagine a group all with an IQ of 100. In this case, 80 of those points were generated due to genes, and the other 20 were generated due to other stuff, such as not being cracked on the dome with a sledgehammer. 0.8 correlation between genes/parents and IQ.

Reimagine. This time, for 80% all 100 points were due to genes, and for 20%, it was entirely due to something else, maybe schooling lol. Schololing. Again, 0.8 correlation with genes, even though that's completely untrue for every single individual.
We can kinda hope humans aren't this diverse, but except on crude biological measures, they, uh, are. Haha, oops.

Correlation not causation? Not in individual cases, no. E.g in this latter model, genes do not necessarily explain any part of IQ for individual samples. Depending on the actual causation involved; again, if there was no attachment to a similar causal chain, there would be no correlation. 

(Technically if everyone has exactly the same 100 IQ, you get an undefined correlation, so imagine the relevant perturbations, I guess. One guy did get domed, and another guy had 120 and would have lost 24 from being domed, but didn't.)
(In a less-toy dataset not everyone would have exactly $100, some would have more, and some less, depending entirely on how much they worked.)

It's a real problem when the scientist doesn't know which are apples and which are oranges; if you don't already know, no statistical tomfooleries you can learn you. You have to understand the data before you analyze it. Trying to analyze the data in an attempt to understand it simply will not work. 

P-hacking, in short: recategorize the data until you find a properly fraudulent grouping of oranges and apples that show a statistically significant correlation. If your standard is p < 0.05, that means you will on average have to try 20 Pearson calculations before you find a spurious correlation. Gathering the data takes a long time, but generating a paper from the data is a footnote, guaranteed to work on nearly any set of data. Takes a few hours, maybe. 

Journals are State-funded and thus see no harm in failing to produce actual results. Mandarinism. They do see an incentive to get articles written about them in magazines and newspapers, though. 

What statistics give you is magnitudes of stuff you already know, which then lets you do accounting. If you know that more meat = more better, it's hard to tell if the extra money you're spending on food is worth it, unless the magnitude is so huge the correlation doesn't matter. 

What you need to know is whether the benefit of going to university, relative to IQ/conscientiousness matched controls, is worth the price. It seems they do make more money, but is it still more net of the forgone earnings, foregone interest, and spending on student loan interest payments? Unlike the meat thing you can't just take three weeks and try it. It's a long-term decision and requires long-term data.

That said the marginal gain might not be worth the analysis costs...the net financial benefits are clearly not that big... 

Although no statistical tomfoolery can help you understand data you don't already understand, you should be able to detect when you don't understand it.
Return a third time to my IQ example. This time, everyone has 100 IQ, but how much is genes and how much is not-sledgehammer varies. Some get 100 for free, others have adamantium skulls that aren't damaged even if they do encounter great blunt force. You can imagine a level line rotating between 20/80 for everyone, to 40/60 for some and 10/90 for others, all the way to 100/0 and 0/100 as in my second example. You can clearly calculate the angle this line is at and get a scalar measure of how apple-orange your sample is. Maybe this procedure even already exists and has a name...but if so, it's clearly not used by most statisticians.