What are the statistical implications of doubling damage on crit instead of doubling the dice rolled?

Question

I've been playing with a group that frequently allows players to double the value of dice rolled for crits (and other things) rather than rolling double the dice. Example, someone crits with 2d6 and rolls for 8 damage, which they then double for 16 total crit damage, rather than rolling 4d6.

This mostly just bugs me on principle, but I was curious what doubling the value rather than doubling the dice does mathematically. Does it actually make a difference? Is there a greater chance to hit extreme ends of the range of values (low and high)? Does the amount and type of dice create greater inconsistencies between the two scenarios?

Answer 1

Given the example of (2d6)×2 (henceforth referred to as 'Doubled Damage') vs (4d6) (referred to as 'Doubled Dice'):

When you double the damage rolled instead of doubling the dice rolled, you create a more evenly distributed curve. Using either method, you have the best odds of rolling the average damage for the dice you are using but in the doubled damage you are far more likely (16.7%) to roll the average than when rolling double the dice (11.3%).

You can also see in the below diagram that when you double the damage rather than the number of dice rolled, you have a much higher chance of rolling the maximum or minimum damage possible (2.78%) compared to almost no chance at all (0.08%) on doubled dice. Doubling the damage rolled also has the effect of eliminating all possible odd-number results.

The standard deviation from the mean in a doubled damage scenario is 4.83, whereas when you double the dice rolled the std dev is only 3.42 points from the mean. To put this into perspective, it means that when you double the rolled damage you are more likely to land in the range of 14 ± 4.83, whereas when you roll the doubled dice, you are more likely to land in a tighter range of 14 ± 3.42.

What does this mean? The larger the standard deviation, the more distributed your data are. A bigger std dev (relative to the range of the sample) means that your data are more distributed across your sample, while a smaller std dev means you will see a steeper curve, with results more closely grouped in the middle of the data spectrum.

The end result: damage results are more varied by doubling the damage rolled with increased odds of rolling either min or max damage when compared to doubling the number of dice.

You can reproduce the following table on anydice.com by inputting:

output 4d6 named "Doubled Dice"
output (2d6)*2 named "Doubled Damage"

and selecting the default Table view, Normal data options.

The differences become more apparent when we graph the numbers together along a curve. Note that there are no yellow nodes on odd numbers — this is because you cannot have an odd-numbered result when doubling the damage (any number multiplied by two results in an even number).

The chart below can be reproduced by using the two anydice functions above and selecting the graph view, normal data options.

In this chart, the yellow nodes represent the possible outcomes of Doubled Damage, while the black nodes represent the standard method of Doubled Dice. When compared this way we can see that the odds for rolling any even number increase greatly, and that overall the damage output of Doubled Damage is more evenly distributed (it deviates more from the mean, creating a nice even pyramid-shaped distribution) than that of Doubled Dice (which has a much more normalized bell curve).

One more graph to really send the point home. In this example, we are comparing a simple 1d6 damage critical:

Anydice.com code for this table:

output 2d6 named "Doubled Dice"
output (1d6)*2 named "Doubled Damage"

From this table, it becomes apparent that your damage result varies greatly when doubling the damage (though the actual number of possible results is cut in half). For each possible result, we have the same odds, as expected — a 1d6 has the same 16.667% chance of turning up any given number, and that doesn't change when we double the results of the 1d6 roll.

On the other hand, it is apparent that rolling Doubled Dice results in a much smaller standard deviation from the mean result of 2d6.

An interesting side note: the distribution for 2d6 is in fact the same as rolling 2d6 and doubling the result — 2d6 has 11 possible outcomes, which is the same number as the possible outcomes of (2d6)×2.

Answer 2

@LegendaryDude has provided a solid answer about the effect on the numbers, however, what is more important is what those numbers mean in play.

A larger standard deviation means that there will be more extreme results, in a sense, while there is overall no change to the average, individual rolls are "more random".

Randomness hurts PCs more than monsters since over the course of an adventure or campaign the PC is subject to far more attacks than any individual monster. This is because the PCs take part in every fight; the monster generally in only one - the one where the PCs kill it, or, more rarely, it kills the PCs. Because the PCs receive more attacks, there are many more opportunities for them to suffer the extreme effects; the wrong extreme effect at the wrong time will kill them.

To illustrate, I have considered a 2d6+0 attacking a first level fighter with 12hp.

This anydice script shows the results of a single critical.

By doubling the total instead of the number of dice the fighter has an increased chance of staying on his feet, up from 23.92% to 27.78% - this is a result of the increased chance of extremely low results. However, he also has an increased chance of being killed outright, up from 0.08% to 2.78% an almost 35x increase. I have also included the variant on p.247 of the DMG where you take the average damage for monsters and only roll the critical die - this has even less variance and, as expected, further increases the chance of consciousness (16.67%) but eliminates the chance of death.