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4 minor typo
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  • Adding some constant to the roll, with the value of that constant increasing as level increases. Suppose you have some number of dice of some number of sides. Taken together, these dice will have some mean value. Adding a constant to the die roll will shift the mean value by that amount. Variance is constant for this mechanic!

  • Similarly, multiplying the roll by some constant greater than 1 works, given that the constant increases as level increases. Multiplying your dice roll by a factor greater than 1 will increase its mean value by that factor (e.g., for a factor of 10, 10×1d6 has a mean of 10 × 3.5 = 35). HoeverHowever, this also increases variance.

  • Likewise, raising a roll to some power greater than 1 works, given that the constant increases as level increases. However, raising the dice roll by a power greater than 1 increases the mean value of the sum of of the average of each die's values raised to that power (e.g, for a power of 2, 2d6^2 has a mean of [(1 + 4 + 9 + 16 +25 + 36) ÷ 6] + [(1 + 4 + 9 + 16 +25 + 36) ÷ 6] = 30.3). This increases variance.

  • Increasing the number of dice will increase the mean value for a dice roll by the mean value of each die added (where the mean value of a die is simply the average of all the values the die can take, so for 1d6 (1 + 2 + 3 + 4 + 5 + 6) ÷ 6 = 3.5, for a first 6 primes cube (2 + 3 + 5 + 7 + 11 + 13) ÷ 6 = 6.83, etc.).

How to insureensure a bell curve at all skill levels: Central Limit Theorem to the rescue! The sampling distribution of the mean value of a bunch of independent random variables approximates the normal distribution as the number of those independent variables increases. Translation: the more dice (any size) you use, the closer your dice roll's distribution will be to a normal, and the closer it's distribution will get to a smooth bell-like curve. With three dice, you start to get pretty bell-like, so depending on how smooth you want your distribution, decide on how many dice to use. Caveat emptorCaveat emptor: having players roll 30 dice may not be everyone's cup of tea.

  • Adding some constant to the roll, with the value of that constant increasing as level increases. Suppose you have some number of dice of some number of sides. Taken together, these dice will have some mean value. Adding a constant to the die roll will shift the mean value by that amount. Variance is constant for this mechanic!

  • Similarly, multiplying the roll by some constant greater than 1 works, given that the constant increases as level increases. Multiplying your dice roll by a factor greater than 1 will increase its mean value by that factor (e.g., for a factor of 10, 10×1d6 has a mean of 10 × 3.5 = 35). Hoever, this also increases variance.

  • Likewise, raising a roll to some power greater than 1 works, given that the constant increases as level increases. However, raising the dice roll by a power greater than 1 increases the mean value of the sum of of the average of each die's values raised to that power (e.g, for a power of 2, 2d6^2 has a mean of [(1 + 4 + 9 + 16 +25 + 36) ÷ 6] + [(1 + 4 + 9 + 16 +25 + 36) ÷ 6] = 30.3). This increases variance.

  • Increasing the number of dice will increase the mean value for a dice roll by the mean value of each die added (where the mean value of a die is simply the average of all the values the die can take, so for 1d6 (1 + 2 + 3 + 4 + 5 + 6) ÷ 6 = 3.5, for a first 6 primes cube (2 + 3 + 5 + 7 + 11 + 13) ÷ 6 = 6.83, etc.).

How to insure a bell curve at all skill levels: Central Limit Theorem to the rescue! The sampling distribution of the mean value of a bunch of independent random variables approximates the normal distribution as the number of those independent variables increases. Translation: the more dice (any size) you use, the closer your dice roll's distribution will be to a normal, and the closer it's distribution will get to a smooth bell-like curve. With three dice, you start to get pretty bell-like, so depending on how smooth you want your distribution, decide on how many dice to use. Caveat emptor: having players roll 30 dice may not be everyone's cup of tea.

  • Adding some constant to the roll, with the value of that constant increasing as level increases. Suppose you have some number of dice of some number of sides. Taken together, these dice will have some mean value. Adding a constant to the die roll will shift the mean value by that amount. Variance is constant for this mechanic!

  • Similarly, multiplying the roll by some constant greater than 1 works, given that the constant increases as level increases. Multiplying your dice roll by a factor greater than 1 will increase its mean value by that factor (e.g., for a factor of 10, 10×1d6 has a mean of 10 × 3.5 = 35). However, this also increases variance.

  • Likewise, raising a roll to some power greater than 1 works, given that the constant increases as level increases. However, raising the dice roll by a power greater than 1 increases the mean value of the sum of of the average of each die's values raised to that power (e.g, for a power of 2, 2d6^2 has a mean of [(1 + 4 + 9 + 16 +25 + 36) ÷ 6] + [(1 + 4 + 9 + 16 +25 + 36) ÷ 6] = 30.3). This increases variance.

  • Increasing the number of dice will increase the mean value for a dice roll by the mean value of each die added (where the mean value of a die is simply the average of all the values the die can take, so for 1d6 (1 + 2 + 3 + 4 + 5 + 6) ÷ 6 = 3.5, for a first 6 primes cube (2 + 3 + 5 + 7 + 11 + 13) ÷ 6 = 6.83, etc.).

How to ensure a bell curve at all skill levels: Central Limit Theorem to the rescue! The sampling distribution of the mean value of a bunch of independent random variables approximates the normal distribution as the number of those independent variables increases. Translation: the more dice (any size) you use, the closer your dice roll's distribution will be to a normal, and the closer it's distribution will get to a smooth bell-like curve. With three dice, you start to get pretty bell-like, so depending on how smooth you want your distribution, decide on how many dice to use. Caveat emptor: having players roll 30 dice may not be everyone's cup of tea.

3 Rollback to Revision 1 - "Insure" is not a spelling error, it is a legit spelling variant
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How to ensureinsure a bell curve at all skill levels: Central Limit Theorem to the rescue! The sampling distribution of the mean value of a bunch of independent random variables approximates the normal distribution as the number of those independent variables increases. Translation: the more dice (any size) you use, the closer your dice roll's distribution will be to a normal, and the closer it's distribution will get to a smooth bell-like curve. With three dice, you start to get pretty bell-like, so depending on how smooth you want your distribution, decide on how many dice to use. Caveat emptor: having players roll 30 dice may not be everyone's cup of tea.

  • Dividing the roll by some constant greater than 1 will decrease the variance, if the constant increases as level increases. However, this will also (1) decrease the mean value of the dice roll by the same divisor, and (2) require calculators, and possibly (3) rounding rules for fractional values.

  • If you want the variance to take a specific value at each level, then you need to divide the roll by the square root of the variance of the distribution of your roll (let me know in comments if you want this... it will require a lot of mathbe mathy, but it is doable). This quotient (roll ÷ square-root of variance of distribution of roll) will have a variance equal to exactly 1 no matter what. So if you then want variance to be X at such and such level, you simply multiply the quotient by X.

  • Going toGonna leave n-th roots out of this for the sake of simplicity. :)

  • Reducing the size of the dice you are rolling works, if die reductions decrease as level increases. However, this will also decrease the mean value of the dice roll. If you are using non-digital dice, the number of die sizes may also present difficulties (e.g., if your game requires 20 levels, and you want the die-size to change each level). One way around this is by allow different sizes of dice in the same roll (e.g., at level 1 there's a pool of d10s, but at level 2 there's 1d8 plus one less number of d10s as at level 2, and so forth).

How to ensure a bell curve at all skill levels: Central Limit Theorem to the rescue! The sampling distribution of the mean value of a bunch of independent random variables approximates the normal distribution as the number of those independent variables increases. Translation: the more dice (any size) you use, the closer your dice roll's distribution will be to a normal, and the closer it's distribution will get to a smooth bell-like curve. With three dice, you start to get pretty bell-like, so depending on how smooth you want your distribution, decide on how many dice to use. Caveat emptor: having players roll 30 dice may not be everyone's cup of tea.

  • Dividing the roll by some constant greater than 1 will decrease the variance, if the constant increases as level increases. However, this will also (1) decrease the mean value of the dice roll by the same divisor, and (2) require calculators, and possibly (3) rounding rules for fractional values.

  • If you want the variance to take a specific value at each level, then you need to divide the roll by the square root of the variance of the distribution of your roll (let me know in comments if you want this... it will require a lot of math, but it is doable). This quotient (roll ÷ square-root of variance of distribution of roll) will have a variance equal to exactly 1 no matter what. So if you then want variance to be X at such and such level, you simply multiply the quotient by X.

  • Going to leave n-th roots out of this for the sake of simplicity. :)

  • Reducing the size of the dice you are rolling works, if die reductions decrease as level increases. However, this will also decrease the mean value of the dice roll. If you are using non-digital dice, the number of die sizes may also present difficulties (e.g., if your game requires 20 levels, and you want the die-size to change each level). One way around this is by allow different sizes of dice in the same roll (e.g., at level 1 there's a pool of d10s, but at level 2 there's 1d8 plus one less number of d10s as at level 2, and so forth).

How to insure a bell curve at all skill levels: Central Limit Theorem to the rescue! The sampling distribution of the mean value of a bunch of independent random variables approximates the normal distribution as the number of those independent variables increases. Translation: the more dice (any size) you use, the closer your dice roll's distribution will be to a normal, and the closer it's distribution will get to a smooth bell-like curve. With three dice, you start to get pretty bell-like, so depending on how smooth you want your distribution, decide on how many dice to use. Caveat emptor: having players roll 30 dice may not be everyone's cup of tea.

  • Dividing the roll by some constant greater than 1 will decrease the variance, if the constant increases as level increases. However, this will also (1) decrease the mean value of the dice roll by the same divisor, and (2) require calculators, and possibly (3) rounding rules for fractional values.

  • If you want the variance to take a specific value at each level, then you need to divide the roll by the square root of the variance of the distribution of your roll (let me know in comments if you want this... it will be mathy, but it is doable). This quotient (roll ÷ square-root of variance of distribution of roll) will have a variance equal to exactly 1 no matter what. So if you then want variance to be X at such and such level, you simply multiply the quotient by X.

  • Gonna leave n-th roots out of this for the sake of simplicity. :)

  • Reducing the size of the dice you are rolling works, if die reductions decrease as level increases. However, this will also decrease the mean value of the dice roll. If you are using non-digital dice, the number of die sizes may also present difficulties (e.g., if your game requires 20 levels, and you want the die-size to change each level). One way around this is by allow different sizes of dice in the same roll (e.g., at level 1 there's a pool of d10s, but at level 2 there's 1d8 plus one less number of d10s as at level 2, and so forth).

2 I really just wanted to correct "ensure" but that's not enough editing, so I tweaked some extra grammar.
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How to insureensure a bell curve at all skill levels: Central Limit Theorem to the rescue! The sampling distribution of the mean value of a bunch of independent random variables approximates the normal distribution as the number of those independent variables increases. Translation: the more dice (any size) you use, the closer your dice roll's distribution will be to a normal, and the closer it's distribution will get to a smooth bell-like curve. With three dice, you start to get pretty bell-like, so depending on how smooth you want your distribution, decide on how many dice to use. Caveat emptor: having players roll 30 dice may not be everyone's cup of tea.

  • Dividing the roll by some constant greater than 1 will decrease the variance, if the constant increases as level increases. However, this will also (1) decrease the mean value of the dice roll by the same divisor, and (2) require calculators, and possibly (3) rounding rules for fractional values.

  • If you want the variance to take a specific value at each level, then you need to divide the roll by the square root of the variance of the distribution of your roll (let me know in comments if you want this... it will be mathyrequire a lot of math, but it is doable). This quotient (roll ÷ square-root of variance of distribution of roll) will have a variance equal to exactly 1 no matter what. So if you then want variance to be X at such and such level, you simply multiply the quotient by X.

  • GonnaGoing to leave n-th roots out of this for the sake of simplicity. :)

  • Reducing the size of the dice you are rolling works, if die reductions decrease as level increases. However, this will also decrease the mean value of the dice roll. If you are using non-digital dice, the number of die sizes may also present difficulties (e.g., if your game requires 20 levels, and you want the die-size to change each level). One way around this is by allow different sizes of dice in the same roll (e.g., at level 1 there's a pool of d10s, but at level 2 there's 1d8 plus one less number of d10s as at level 2, and so forth).

How to insure a bell curve at all skill levels: Central Limit Theorem to the rescue! The sampling distribution of the mean value of a bunch of independent random variables approximates the normal distribution as the number of those independent variables increases. Translation: the more dice (any size) you use, the closer your dice roll's distribution will be to a normal, and the closer it's distribution will get to a smooth bell-like curve. With three dice, you start to get pretty bell-like, so depending on how smooth you want your distribution, decide on how many dice to use. Caveat emptor: having players roll 30 dice may not be everyone's cup of tea.

  • Dividing the roll by some constant greater than 1 will decrease the variance, if the constant increases as level increases. However, this will also (1) decrease the mean value of the dice roll by the same divisor, and (2) require calculators, and possibly (3) rounding rules for fractional values.

  • If you want the variance to take a specific value at each level, then you need to divide the roll by the square root of the variance of the distribution of your roll (let me know in comments if you want this... it will be mathy, but it is doable). This quotient (roll ÷ square-root of variance of distribution of roll) will have a variance equal to exactly 1 no matter what. So if you then want variance to be X at such and such level, you simply multiply the quotient by X.

  • Gonna leave n-th roots out of this for the sake of simplicity. :)

  • Reducing the size of the dice you are rolling works, if die reductions decrease as level increases. However, this will also decrease the mean value of the dice roll. If you are using non-digital dice, the number of die sizes may also present difficulties (e.g., if your game requires 20 levels, and you want the die-size to change each level). One way around this is by allow different sizes of dice in the same roll (e.g., at level 1 there's a pool of d10s, but at level 2 there's 1d8 plus one less number of d10s as at level 2, and so forth).

How to ensure a bell curve at all skill levels: Central Limit Theorem to the rescue! The sampling distribution of the mean value of a bunch of independent random variables approximates the normal distribution as the number of those independent variables increases. Translation: the more dice (any size) you use, the closer your dice roll's distribution will be to a normal, and the closer it's distribution will get to a smooth bell-like curve. With three dice, you start to get pretty bell-like, so depending on how smooth you want your distribution, decide on how many dice to use. Caveat emptor: having players roll 30 dice may not be everyone's cup of tea.

  • Dividing the roll by some constant greater than 1 will decrease the variance, if the constant increases as level increases. However, this will also (1) decrease the mean value of the dice roll by the same divisor, and (2) require calculators, and possibly (3) rounding rules for fractional values.

  • If you want the variance to take a specific value at each level, then you need to divide the roll by the square root of the variance of the distribution of your roll (let me know in comments if you want this... it will require a lot of math, but it is doable). This quotient (roll ÷ square-root of variance of distribution of roll) will have a variance equal to exactly 1 no matter what. So if you then want variance to be X at such and such level, you simply multiply the quotient by X.

  • Going to leave n-th roots out of this for the sake of simplicity. :)

  • Reducing the size of the dice you are rolling works, if die reductions decrease as level increases. However, this will also decrease the mean value of the dice roll. If you are using non-digital dice, the number of die sizes may also present difficulties (e.g., if your game requires 20 levels, and you want the die-size to change each level). One way around this is by allow different sizes of dice in the same roll (e.g., at level 1 there's a pool of d10s, but at level 2 there's 1d8 plus one less number of d10s as at level 2, and so forth).

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