It's maths. It's not "your formula". You just got the same result by dividing two numbers. That video is actually really simplified since it doesn't prove the expressions used, in other words it doesn't get to them by any means of mathematical deduction, there's nothing that proves your calculus to be correct without this background, it's just dividing two numbers essentially and the result happens to be correct since, you said it yourself, you don't actually understand the distribution well.
Using a formula that the video explains you can get the actual variance with some simple calculus and you don't have to run a program to "bruteforce" the result.





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