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<blockquote data-quote="buzz" data-source="post: 511320" data-attributes="member: 6777">This discussion is prompted by a thread from GR's Mutants &amp; Masterminds forum that I participated in and <a href="http://villa.lakes.com/JamesStarlight/LinearVsNonLinear.html" target="_blank">an article by James L. R. Beach</a>.The PHB and DMG make some references to the idea of an &quot;average roll&quot; on 1d20. E.g., the PHB states on p.58: &quot;On average, Devis will roll 10 or 11 on the d20...&quot;Now, by &quot;on average,&quot; I assume that the authors mean &quot;most frequently.&quot; I.e., they are talking about a frequency distribution or mode, implying that a 10 or 11 will show up most often on a 1d20 roll.Since a 1d20 roll is a flat distribution, with all results being equally likely, we know that this statement is incorrect. Devis' player is no more likely to roll a 10 or 11 than they are to roll a 1 or a 20.The most common meaning of &quot;average&quot; is arithmetic mean. That is, over N number of rolls, it is the sum of the results divided by N. With a d20, this will give us an arithmetic mean of 10.5. However, this doesn't really tell us anything about the probability that a result will occur.What I'm curious about is, are the 3e designer's aware that this statement is erroneous? I suspect they are, as the Take 10 and 20 rules seem to be prompted by this lack of an &quot;average roll.&quot; Without them, low-level PCs and commoners are going to botch even mundane skill checks (DC 5) more than is acceptable.The article cited above assumes that the designers are perhaps unaware of this, and suggests using a 2d10 roll instead. 2d10 provides a bit of a curve, actually producing an &quot;average&quot; roll of 10 or 11.My assumption is that the statements, being so few and far between, are simply erroneous, and we can count on the fact that Tweet, Williams, and Cook all know better. Still, I wonder.N.B.: I am not a math or statistics whiz. I just found this kind of curious and started reading up on the math involved.</blockquote>

[QUOTE="buzz, post: 511320, member: 6777"] This discussion is prompted by a thread from GR's [i]Mutants & Masterminds[/i] forum that I participated in and [URL=http://villa.lakes.com/JamesStarlight/LinearVsNonLinear.html]an article by James L. R. Beach[/URL]. The PHB and DMG make some references to the idea of an "average roll" on 1d20. E.g., the PHB states on p.58: "On average, Devis will roll 10 or 11 on the d20..." Now, by "on average," I assume that the authors mean "most frequently." I.e., they are talking about a frequency distribution or [i]mode[/i], implying that a 10 or 11 will show up most often on a 1d20 roll. Since a 1d20 roll is a flat distribution, with all results being equally likely, we know that this statement is incorrect. Devis' player is no more likely to roll a 10 or 11 than they are to roll a 1 or a 20. The most common meaning of "average" is [i]arithmetic mean[/i]. That is, over N number of rolls, it is the sum of the results divided by N. With a d20, this will give us an arithmetic mean of 10.5. However, this doesn't really tell us anything about the [i]probability[/i] that a result will occur. What I'm curious about is, are the 3e designer's [i]aware[/i] that this statement is erroneous? I suspect they are, as the Take 10 and 20 rules seem to be prompted by this lack of an "average roll." Without them, low-level PCs and commoners are going to botch even mundane skill checks (DC 5) more than is acceptable. The article cited above assumes that the designers are perhaps unaware of this, and suggests using a 2d10 roll instead. 2d10 provides a bit of a curve, actually producing an "average" roll of 10 or 11. My assumption is that the statements, being so few and far between, are simply erroneous, and we can count on the fact that Tweet, Williams, and Cook all know better. Still, I wonder. N.B.: I am not a math or statistics whiz. I just found this kind of curious and started reading up on the math involved. [/QUOTE]

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