What techniques exist to determine the "randomness" of a sample?
For instance, say I have data from a series of $1200$ six-sided dice rolls. If the results were
1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, ...
1, 1, 1, ..., 2, 2, 2, ..., 3, 3, 3, ...
The confidence of randomness would be quite low.
Is there a formula where I can input the sequence of outcomes and get back a number that corresponds to the likelihood of randomness?
UPDATE: awkward's answer was the most helpful. Some Googling turned up these two helpful resources:
Statistical analysis of Random.org - an overview of the statistical analyses used to evaluate the random numbers generated by the website, www.random.org
Random Number Generation Software - a NIST-funded project that provides a discussion on tests that can be used against random number generators, as well as a free software package for running said tests.