NAME

Statistics::ChiSquare - How random is your data?

SYNOPSIS

    use Statistics::Chisquare;

    print chisquare(@array_of_numbers);

Statistics::ChiSquare is available at a CPAN site near you.

DESCRIPTION

Suppose you flip a coin 100 times, and it turns up heads 70 times. Is the coin fair?

Suppose you roll a die 100 times, and it shows 30 sixes. Is the die loaded?

In statistics, the chi-square test calculates "how random" a series of numbers is. But it doesn't simply say "yes" or "no". Instead, it gives you a confidence interval, which sets upper and lower bounds on the likelihood that the variation in your data is due to chance. See the examples below.

If you've ever studied elementary genetics, you've probably heard about Georg Mendel. He was a wacky Austrian botanist who discovered (in 1865) that traits could be inherited in a predictable fashion. He did lots of experiments with cross breeding peas: green peas, yellow peas, smooth peas, wrinkled peas. A veritable Brave New World of legumes.

But Mendel faked his data. A statistician by the name of R. A. Fisher used the chi-square test to prove it.

There's just one function in this module: chisquare(). Instead of returning the bounds on the confidence interval in a tidy little two-element array, it returns an English string. This was a deliberate design choice---many people misinterpret chi-square results, and the string helps clarify the meaning.

The string returned by chisquare() will always match one of these patterns:

  "There's a >\d+% chance, and a <\d+% chance, that this data is random."

or

  "There's a <\d+% chance that this data is random."

or

  "I can't handle \d+ choices without a better table."

That last one deserves a bit more explanation. The "modern" chi-square test uses a table of values (based on Pearson's approximation) to avoid expensive calculations. Thanks to the table, the chisquare() calculation is very fast, but there are some collections of data it can't handle, including any collection with more than 21 slots. So you can't calculate the randomness of a 30-sided die.

EXAMPLES

Imagine a coin flipped 1000 times. The most likely outcome is 500 heads and 500 tails:

  @coin = (500, 500);
  print chisquare(@coin);

prints "There's a >90% chance, and a <100% chance, that this data is random.

Imagine a die rolled 60 times that shows sixes just a wee bit too often.

  @die1  = (8, 7, 9, 8, 8, 20);
  print chisquare(@die1);

prints "There's a >1% chance, and a <5% chance, that this data is random.

Imagine a die rolled 600 times that shows sixes way too often.

  @die2  = (80, 70, 90, 80, 80, 200);
  print chisquare(@die2);

prints "There's a <1% chance that this data is random."

How random is rand()?

  srand(time ^ $$);
  @rands = ();
  for ($i = 0; $i < 60000; $i++) {
      $slot = int(rand(6));
      $rands[$slot]++;
  }
  print "@rands\n";
  print chisquare(@rands);

    

prints (on my machine)

  10156 10041 9991 9868 10034 9910
  There's a >10% chance, and a <50% chance, that this data is random.

So much for pseudorandom number generation.

AUTHOR

Jon Orwant

MIT Media Laboratory

orwant@media.mit.edu

cpanm Statistics::ChiSquare

perl -MCPAN -e shell
install Statistics::ChiSquare