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Message: <blockquote data-quote="Blue Orange" data-source="post: 8220067" data-attributes="member: 7025997">All good points.I'd add that it's beneficial to take average HP when going up a level since (I believe) it rounds up, so it will get you 0.5 extra HP on average. A few statistics points:The law of large numbers states that experimentally obtained sample averages converge to the population mean over time. (The exception is certain pathological distributions like the Cauchy or Pareto with alpha < 1, but those don't apply to dice rolls, which are discrete uniform. Though I'd love to see a monster that does Cauchy damage!) This means that the result of rolling 20d6 will be pretty close to 70...a lot more than 2d6 will be close to 7, relatively speaking.The central limit theorem states that adding independent, identically distributed variables (like a large number of dice) converges to a normal distribution (the Gaussian 'bell curve') over time. 20d6 looks a lot like a normal distribution, 2d6 less so. Distributions for 2d6 and 1d6 are pretty easy to work out, but after that they get increasingly complicated. In general for a Gaussian, 68% of values will be within 1 standard deviation of the mean, 94% will be within 2 standard deviations of the mean. Die rolls aren't perfectly Gaussian, but they get closer to Gaussian the more dice you have. The mean of ndx is of course n*0.5(x+1), but the standard deviation is square root ((1/12)(n(x^2-1))). What that means in practice is that 4d6 only varies twice as much as 1d6.Also, the fewer dice you have, the more likely extreme values are, relatively speaking. 4d6 and 4*(1d6) have the same mean, but you are more likely to get a roll of 4 or 24 by rolling 1d6 and multiplying the result by 4 than by rolling 4d6.</blockquote>

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