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Variance(Hypergeometric)

Var(X)=n⋅KN⋅N-KN⋅N-nN-1Var(X)=n⋅KN⋅N−KN⋅N−nN−1
Number of Trials
Succesful Samples
Total Amount of Samples
Tags
Statistics

[Mathematics | Probability | Statistics | Distribution] The variance of a set of numbers is the measure of how spread out they are. A variance of zero indicates that there is no variance which means all of the values are the same. Variance can never be represented with a negative number. The higher the variance the more spread out the values in the data set are.

Hypergeometric distribution is a discrete probability distribution that describes the probability of k successes in n draws without replacement from a finite population of size N containing exactly K successes. This is in contrast to the binomial distribution, which describes the probability of k successes in n draws with replacement. (See Wikipedia for a better explanation)

Variables:

  • n = Number of Trials
  • K = Successful Samples
  • N = Total Amount of Samples
  • Equation:
  • n⋅KN⋅N-KN⋅N-nN-1

Notes

The following conditions characterize the hypergeometric distribution:

  • The result of each draw can be classified into one or two categories.
  • The probability of a success changes on each draw.

There is a wide range of applications for the hypergeometric test.  A marketing analyst often uses this test to characterize the  customer base.  The test examines a set of known customers for over-representation of any specific various demographic subgroup, like  (e.g., like techno-geeks in the fifties).

A POKER EXAMPLE

In Texas Hold'em Poker, players make their hands from two cards in their hand combined with the 5 community cards on the table. The deck has 52, made of 13 cards each of four suits.

Assume a player has 2 clubs in the hand and there are 3 cards showing on the table.

Assume also that 2 of the three cards on showing on the table are clubs.

To determine the probability that one of the remaining 2 cards to be shown is a club that the player can use in a flush, the player must consider the following:

  • 4 clubs turned-up mean  9 remain hidden to the player with 2 clubs.
  • There are 47 cards the player with the two clubs in hand has not seen.

Using the hypergeometric probability calculation:

  • A hyper-geometric test  would suggest a probability that one of the next two cards turned-up is a club is about  31.6% when  k=1, n=2, K=9 and N=47.
  • A hyper-geometric test  would suggest a probability that both of the next two cards turned-up are clubs is about 3.3% when k=2, n=2, K=9 and N=47
  • A hyper-geometric test  would suggest a probability that neither of the next two cards turned-up are clubs is about  65.0% when k=0, n=2, K=9 and N=47.