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The average value of throws of perfect dice would be 3.5, not 3, so if you get just 3 something's severely wrong with your dice.
However, the statistical methods with which you evaluate whether a source of random numbers is really random are a bit more involved than just taking the average. During any sequence of random numbers, you can find subsequences that look suspiciously non-random (for example, three or four equal numbers, or 1-2-3-4-5-6, or your birthdate if it happens to consist only of digits 1-6).
The probability of such subsequences can be calculated, and there are methods to determine the probability that a given sequence of numbers comes from a truly random source or a somewhat skewed one.
You household dice will most likely be somewhat skewed as they are produced primarily with cost in mind, and not perfect fairness. Casino dice are made to higher standards, and when they are rolled properly, should yield good random numbers.
However, I think there are techniques to compute usable random number sequences even if your source has slightly skewed values.

Onkel Paul