Now that you PARTLY understand it, go read on a stats page just how true it is. The mistake you’re still making is that you’re centering your bell curve on X=0 again. The only way the curve would be centered on X=0 would be if the person’s average number of consecutive free throws is 0. That would mean that he was equally likely to miss and to hit, and he would, once in some astronomical number, hit ten in a row, and he would miss ten in a row equally often. But why would his average have to be zero? If his average was 1, then the curve would be centered on X=1, and he would make 11 in a row with the same frequency that his missed 9 in a row. His average does not have to be an integer. It could 1.687. Then the curve would be centered on 1.687, but there would be no numbers that he hit in a row with equal frequency because 10 in a row is 8.313 from his average. Going 8.313 in the opposite direction gives you -6.626 and it’s impossible to miss 6.686 times. What you can do is use the curve to determine what percentage of the time he would miss 7 in a row, and it would not be the same as the percentage of times that he made 10 in a row. It’s not so wrong. It’s so true. You’re just as likely to underperform as you are to over perform, but that does not mean that your performance centers on zero. You might be just as likely to miss two in a row as you are to hit 5 in a row, and an NBA player might be just as likely to hit 27 in a row as to hit 23 in a row. Notice that you missed on the left portion of your bell curve, but the NBA player is still hitting on the left portion of his bell curve. Stop placing the center of the bell curve at x=0.
I don’t think that one is equally likely to over perform as to under perform, though, in many examples. Agree that skill level shifts the peak left and right. Agree that there will be outcomes on both sides. But disagree that the “shape” of the outcomes is, or is close to symmetrical, (my understanding of your understanding of normal distribution” being that deviations vary similarly side to side).
I originally picked the example of free throw strings where a person could reliably (half the engagements) make X number of shots, 5, as an example because it seemed broadly relatable.
If half the efforts of shots ended at 5, the other 50% of efforts ended at 0, 1, 2, 3, or 4, on the left and 6, or higher on the right.
I have understood you to be saying that normal distribution splits the other 50% - the half that were “not 5 consecutive free throws” on both sides of 5, and symmetrically looking.
This is where I’d differ. Yes, I think normal distribution could split it that way, I just don’t think normal distribution applies to the example, or to many other elements of accuracy shooting.
I firmly believe from observation, in a left side bias in numerosity of outcomes, generally when the subject matter involves a test that is hard. Is this illogical? Naturally that line above, you are just as likely to over and underperform, got my attention.
Tyler seemed to immediately agree the half of efforts that did not end in 5, did not split “evenly,” meaning that 25% ended in 6, or better than 6, and 25% ended in 4 or worse. As I wrote, I predicted that far more of the endings that were not 5, were left side. Now, the “consecutive shots” of my example may turn out to be where you (hopefully) concur another layer of complication to unpack lies, but it wasn’t accidentally chosen. (BR for example, a single errant, misfire flier renders any improvement from all remaining shots impossible. It can only get worse, never better and one might as well stop and wait to start over, - I suppose if it were the first, you could just aim at it, but not after that).
In certain kinds of performance, as with batting outcomes, it is simply easier to fall short. The batting outcomes example strongly supports my contention. I’ll grant the other can also happen, for example a well placed golf handicap ought to split the over and under outcomes.
It is perhaps not “easier” for a scale to err heavier than lighter, or a bullet to vary longer than shorter, or more shots to miss right than left, and so on, and in this regard, I don’t take issue in principle with symmetry statistically occurring, in a great many other things.
. Without reading the whole thing and specifically answering the question, yes, I do statistical tests in my testing and do use the 95% probability standard as a threshold for whether the data is likely due to chance or not. I get so many questions about stats that I made a “simple stats” video trying to explain this in as simple terms as I can muster/the type of words I used when I taught stats. The 95% standard is the industry standard and used all the time in various paradigms of research. For example, if you are taking medication for a chronic medical illness, cancer, or any other major medical illness, the research that was conducted to substantiate the effectiveness of that medication very likely used that standard when comparing active medication to placebo on medical outcome variables. There are other types of stats that do not even use probability and instead use “best fit” but those stats are usually used in different types of applications. I plan to use the “best fit” statistical approach when analyzing the tuner testing data after I capture data in various atmospheric conditions. For now, I have only used the 95% standard because I was directly comparing high and low barometric pressure conditions. Once I have more data on temperature and humidity over an appreciable amount of time and with sufficient data, I will use the “best fit” method to analyze the relationships among all variables including tuner settings. Anyway, I’m happy to bore all of you with this whole statistic mumbo jumbo 
