Chi-squared test in layman's terms:
Answer the question: Do the results from a sample line up close enough to what is expected?
Here's an example. Let's look at 1300 cards at random and record the number of each rank. We would expect 100 Aces, 100 Kings, ... , down to 100 deuces. With any sample, it would be incredibly coincidental to get that exact distribution. Here is how the test would work. Let's say the sample looked like this:
Ace 103
King 93
Queen 96
Jack 111
Ten 116
Nine 93
Eight 96
Seven 92
Six 98
Five 95
Four 103
Three 95
Two 109
What we do for each of the above frequency numbers would be to see a measure of how far each number is from 100 (the expectation). For Aces, that calculation is (103 - 100)^2 / 100 = 0.09; for Tens, it would be (116 - 100)^2 / 100 = 2.56; for Sevens it would be (92 - 100)^2 / 100 = 0.64. Do this calculation for each of the 13 ranks and sum the results you get 7.24. Since this number is smaller than what is called a critical value for this particular test, 21.03; this result would not reject the idea that the ranks were selected randomly with equal probability fo reach rank.
In order to explain how the number 21.03 was chosen, we need to go beyond a layman's explanation.
Here is a different example:
Ace 83
King 85
Queen 102
Jack 75
Ten 107
Nine 99
Eight 87
Seven 113
Six 100
Five 105
Four 120
Three 121
Two 103
In this one, the sum of the calculations works out so that it fails the chi-squared test; i.e., it is greater than 21.03.
Of course, if I was doing this for a regulatory agency, like tech58 originally said, I would use a sample size higher than 1300, though we might stop shy of 1.7 bazillion.
And we would likely measure more than just the counts of the ranks of the cards.
I don't have any particular knowledge of how various jurisdictions actually do their auditing. There are lots more tests for randomness that are likely employed. You could look up Marsaglia Diehard tests if you were interested in seeing some examples of other tests.
Also, there actually is a similar question answered in the Ask the Experts section of the Pro Training feature on this website in case anyone is interested in doing additional exploration.
