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What average measure are you applying here? Mean, median or mode?
How can someone arriving five minutes late to catch a train actually catch it? The 5 o'clock train leaves at 5 o'clock, if you get there at five minutes past you've missed it. If the train's late and the person arriving late catches it the reason isn't that they arrived late but that they were lucky because the train was running late. In this instance there's an external influence on the outcome, i.e. the train's on time departure which skews any data you might use to prove your theory.
I'm not sure that variance of train on time departure data would show a normal distribution if you used two and a half or five minutes late / early as your mean, but would be more than happy to be proven wrong. Obviously I haven't done the research and am simply challenging your rested case.
My theory is if you arrive five minutes late for 100 trains you'll miss more than you'll catch.
I also have a theory on climate change. Yes climate change is happening but it's more due to the fact that the planet is coming out of an ice age than Shell pulling oil and gas out of the North Sea. If us older folks cast our minds back to e=when we were young we can easily argue that summers were hotter and longer whilst winters were both colder at times and warmer at times and that there's no difference today. In 1976 when we had a long, hot summer followed by a harsh winter there were no climate activists around to warn of looming planetary disaster and lo and behold - no planetary disaster.
The other part of my theory on climate change is that if that bag of perfection and irrefutable climate knowledge Greta Thunberg was to be burned as a witch her carbon effect on the planet would surely be enough to counter all the fossil fuels burned in 100 years across the globe?