My personal recommendation is to use a dice roller. If you have an iPhone, I recommend Dicenomicon ($5), dynamicDICE ($1) or Dice Bag (free). I have heard that Pip ($1) is also pretty good. I have found that the luck associated with physically rolling dice is dissociated from digitally rolling them.
If you find that you still think that a dice roller is giving you below average rolls, try one of these solutions:
Pre-Generated Dice Roll List
This is very similar to using the dice card decks suggested in another answer, but relies on a computer rather than a deck of cards.
- Generate a list of 1,000 random integers between 1 and 20. Do not look at the individual numbers.
- Average the numbers.
- If the average is between 10.0 and 11.0, your set is not necessarily smooth, but is going to have a fairly even distribution.
- Whenever you need to roll a die, instead use the next number on the list.
- Continue to use this list until you have used all 1,000 numbers before generating a new list.
Point Buy System
- Choose an average dice roll - 9, for example. This is the roll that you get if you do not apply any points to the roll. Important rolls are not every roll - instead, they are rolls that your DM feels are important enough for it to matter if you subtracted points from them. For everything else, you roll manually.
- Whenever you need to make an important roll, you instead get to use the "average dice roll." You can accept a lower dice roll. Based off of how low you reduce your roll, you get that many points to add to future rolls. For example, 8=>1, 7=>2, 6=>3, 5=>5, 4=>7, 3=>9, 2=>12, 1=>16
- You could instead choose to use points you have accumulated to increase your roll. For example, 10=>1, 11=>2, 12=>3, 13=>5, 14=>7, 15=>9, 16=>11, 17=>14, 18=>17, 19=>20, 20=>24
Alternatively, have this system apply to every roll, but let the DM rate the importance of given rolls, with a multiplier of anywhere from 0.5 to 3. Adding or subtracting points to such rolls costs or rewards the multiplier times the cost/reward.
Point Buy Modifier System
This system is identical to the one above, but you instead modify your actual roll. This modification may be required to be before your roll or may be after it. Alternatively: It costs half as much and rewards twice as much if the change is determined before the roll rather than after.
Narration System
You are encouraged to narrate your actions in exquisite detail. Doing so (before rolling) in a way appreciated by the DM and other players grants the acting player either a +4 bonus to his/her roll (if the roll is between 8 and 13) or the ability to re-roll if the roll is below 8. Alternatively, base the bonus off of how effective your narration was.
Pure Narration System
As above, but the DM determines your success based on the quality of your narration.
I play online almost exclusively these days, using Mote. I DM two campaigns, and this particular issue comes up often enough.
First, you have to keep things interesting. Try to design combats with more than just "attack roll -> damage roll". It's not a simple task, but it's really important to make combats interesting. Add some ranged enemies, healing enemies, mages, etc. Throw in environmental hazards, or effects which occur between turns. Give the PCs a time crunch to win. Give them a reason to be concerned.
In terms of process, I have found that the method of forfeiting turns does help. If they take too long, they lose their turn. However, I found that rewarding them for taking swift actions is far more effective. I give my players a +1 circumstance bonus per five levels to any rolls they make during their turn if they begin taking actions within fifteen seconds. I call this bonus "seizing opportunities." It works phenomenally. That, plus some recommendation for the players to plan actions during someone else's turn, means that they are likely to have their plans ready and in motion within those fifteen seconds. The bonus is big enough to be worth striving for, without being so big that it's not worth sacrificing for the occasional tactical overview. Players are much more likely to confer with each other this way. This has sped up our gameplay immensely.
You can adjust the bonus, time limit, and everything else to whatever works for you. I just insist that rewarding players is a more effective motivator than punishing them. It also makes them feel good and have more fun. That is the most important thing.
EDIT: Now that I think of it, this is also a pretty effective way to speed up offline tabletop combat, for anyone else who might be doing one or the other or both.
EDIT EDIT: One other thing I should mention is that when playing through an online tabletop you usually have computing tools available. Take advantage of them! The human brain is great at lots of amazing stuff, like creativity, imagination, reason, and emotion. One thing it has a hard time with is mathematics. Fortunately, computers rock that stuff. It helps to shave more time off of a turn if you have as much action content (rolls, initiative order, etc.) hard-coded beforehand. Dice macros are a simple example.
On a non-computing basis, it also helps to have actions listed out in front of the player. I know there was a lot of hate around 4e's action system, but having a sheet of actions clearly delineated in front of the player was a huge advantage for newbies. Having some clear options in front of them helps remind them of what they can do, and thereby lets them make a decision sooner.
Naturally, you do not have to be as hard-and-fast as 4e. You can even go as simply as reminding the player of what they can do in your game, or give them some soft suggestions.
Best Answer
You have two basic choices for how to have your players roll their dice:
If you take the first option you must trust your players and accept what they tell you!
You need to be able to trust them and work with them, and they need to be able to trust you and work with you. The game experience is going to be quite toxic otherwise - I'll talk about that later. You've chosen a method which requires trust, so provide that trust.
If they are fudging, accept it for the fun value. You can't tell if they're doing it anyway. This is exactly the same as DMs who roll behind the screen: the players may suspect that sometimes the DM went easy on the nearly-dead target and fudged a miss, but the players should be able to trust the DM to make the game fun for them. That doesn't mean you need to pretend it isn't happening - just accept it and be okay with it, and continue playing.
Higher than average results are entirely possible.
They're improbable, yes, but not impossible, and doesn't suggest cheating.
Today, I played my first Pokemon TCG game with a friend. We did a lot of things that called for coin flips, and out of ~25 flips, only four were tails. It's highly improbable, but it happened.
Some players somehow roll quite well on average. I recently rolled several sixes on a series of challenges, some of which only a six would beat. We have a member here who's the opposite: he consistently rolls improbably low, with any set of dice, even with a dice tower. (It's incredible, but it's been happening for years; here's the records from one session which was played in person - no fudging occurred.)
Some don't have precision cut dice, and the common types of dice manufacturing can make vulnerable dice by accident. Ilmari explains at the beginning of his answer here. This isn't conscious cheating, it's just a flaw of the manufacturing process. You probably have slightly weighted dice!
Bottom line: If you want to use a dice rolling method requiring trust, extend that trust, or don't use that dice rolling method.
What if you can't provide that trust and accept their rolls?
If you keep going like this and can't trust your players, you'll end up relating to them as cheaters and you'll probably be irritated by their cheating - whether or not they really are. That's going to be really fun for them being treated like cheaters, and I say that with massive amounts of sarcasm. They won't be able to trust you to be impartial with them because you won't trust them. The experience will be very toxic and toxic experiences destroy groups and any fun value in games.
So, if you cannot trust them, tell them you want to switch dice rolling method.
Be straightforward and honest in telling them why: the results they're getting bugs you, and you have no way of knowing they're cheating or not, and you'd prefer to trust them but you'd rather just find a rolling method that eliminates any reason to be suspicious to begin with - that way, you can all have fun without you needing to be concerned.
Find a dice roller that lets you view each others' rolls and use it.
If you do this and the average results change, don't use that as a reason to suspect they were cheating. The success rates they were having were entirely possible, however improbable, and they may not have been cheating at all.