I agree with what the other posters have said about random number tests, but if we look at what the OP is trying to quantify I think that Boltzmann Entropy would be a better measure.
Boltzman Entropy for a deck of cards D with respect to a particular game G could be defined as:
S = natural log of (the number of distinct decks D' equivalent to D for the purposes of a round of G)
from this definition it follows that the state with the maximum entropy is the state where the arrangement of the cards was 'most typical' with respect to G, and states with a low entropy represent an atypical shuffling of the deck. Here is an analysis from another page (
http://www.cs.unm.edu/~saia/infotheory.html):
Now since all permuations have equal probability in a random deck of cards, the entropy of that deck is log52! = 225.6 bits. When we shuffle a deck of cards, that shuffle has entropy equivalent to log(52 choose 26) = 48.8 bits (we assume the deck is divided in half and a "rifle" shuffle is used). This means we should use a "rifle" shuffle 225.6/48.8 = 4.6 or 5 times on average to assure complete randomness. This computation is relatively simple because the probabilities of all events are assumed to be equal.
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In my experience the problem with bad shuffling is that it causes the appearance of card combinations similar to those in the previous round of the game, which is repetitive and therefore boring. It should be pretty simple to generate statistics about these kind of repeats, just by counting them. Say something like "Your shuffling has caused X incidents of repetition in the last Y rounds of the game, while we know that with proper shuffling the probability for this is negligible." Finding the exact probability of "negligible" means would be a worthy homework problem in combinatorics.
