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Message: <blockquote data-quote="EzekielRaiden" data-source="post: 8737587" data-attributes="member: 6790260">The "formally correct" way--that is, the one which correctly captures all the statistics with no simplification--is that you need to expand out each result into the number of faces on the die, and then replace the N faces that show 1 with [1,...,N]. With d4, that looks like this: d{1,2,3,4,2,2,2,2,3,3,3,3,4,4,4,4}. Alternatively, you can think of it as becoming an N^2 sided die, where you have a single 1, and then the remaining values appear N+1 times. You'll get the same average value if you just enter 1d{(N+1)/2,2,3,...,N} but you won't get the same overall statistical behavior.Having crunched those numbers, I can now say that the ceiling of the change appears to be 0.5, as noted above. That is, we get:d4 average improves from 2.5 to 2.88; SD falls from 1.12 to 0.93d6 average improves from 3.5 to 3.92; SD falls from 1.71 to 1.48d8 average improves from 4.5 to 4.94; SD falls from 2.29 to 2.05d10 average improves from 5.5 to 5.95; SD falls from 2.87 to 2.62d12 average improves from 6.5 to 6.96; SD falls from 3.45 to 3.19It's rather tedious to calculate anything beyond d12 and pretty much irrelevant so I'm not going to do the full, formal numbers for any higher die values (and sure as heck not d20). If you like data visualization, you can look at plots for these with <a href="https://anydice.com/program/2a957" target="_blank">this AnyDice program</a>.But we can certainly say that, while the increase in the average always goes up, it pretty well appears to be bounded from above by 0.5, and thus the percentage increase is always lower for larger dice. This might not be true for broader ranges, e.g. if you could reroll 1 or 2 this way, the benefit might grow rather than shrink. (We could imagine, frex, if every value less than the average could be rerolled once, then the average benefit surely should not shrink with higher die values.)</blockquote>

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