What does 'swingy' mean, in the context of tabletop roleplaying games?
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Although I mostly agree with @Brian's answer of "Highly Variable," swingy isn't used just to indicate the standard deviation of a single die roll.
Yes, you can call a 1d20 more "swingy" than a 3d6 mechanic, but the term is also used on a larger scale, as in "swingy" combat is combat where overall results can be highly variable. This can be from a number of different reasons and swinginess of individual mechanics isn't necessarily part of it.
For example, a system where you roll group initiative and PCs are not super durable. If everyone on one side gets to go before everyone on the other side, and there's no admonitions that "it really happens at the same time" (Alternity does this to mitigate the effect), you get a swingier system, because there is a huge benefit in going first - you can often eliminate the other side's heavy hitter/mage/priest/whatever is relevant a priori and then the rest of the combat is very one-sided. So even if the initiative roll itself isn't "swingy" mechanics wise, so much depends on it that the overall outcomes are swingy. So high standard deviation, but not just of single rolls but of total results.
It could also mean a game system that is good for swingers, like Theatrix Ironwood.
I use the term to mean "highly variable." Specifically, I consider a 1d12 (avg 6.5) to be swingy whereas the roughly equivalent 2d6 (avg 7) are not swingy, because they follow a bell curve rather than an equal probability of falling to either extreme like the d12.
It has a subtext of "unreliable" because of its variable nature, though that's not true of all cases.
Swingy is in general used to define some form of number generation with a wide variance in outcomes. A good example would be damage done by weapons in D&D. In 3.5D&D a Greatsword does 2d6 while a Greataxe does 1d12, thus the axe is more swingy. The sword is overall more consistent in doing damage and as a bonus has a minimum of 2 damage instead of 1. A single die will be more swingy then numerous dice because the chance of being any of the outcomes is equal unlike with multiple dice which will tend to have a bell curve.
I have heard this in more than RPGs, and it generally means that something can either be really powerful or really weak.
Ex: Something that does 1d20 damage is more swingy than something that does 2d10 damage is more swingy than something that does 3d6 damage.
Ex: Something that can kill 1d10 enemies and will knock you unconscious. Killing 10 enemies is really powerful, but killing 1 of 5 enemies and knocking you unconscious is really weak.
If you need more clarification, ask and I'll give more examples.