Statistics::ChiSquare - How well-distributed is your data?


    use Statistics::ChiSquare;

    print chisquare(@array_of_numbers);

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


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 well a series of numbers fits a distribution. In this module, we only test for whether results fit an even distribution. 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 Gregor 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:

  ".*!" (ie strings ending in an exclamation)

various error messages for when you supply Obviously Wrong data.

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


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


  "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 250 slots. So you can't calculate the randomness of a 500-sided die should such an insane thing ever exist.

You will also notice that the percentage points that have been tabulated for different numbers of data points - that is, for different degrees of freedom - differ. The table in Jon Orwant's original version has data tabulated for 100%, 99%, 95%, 90%, 70%, 50%, 30%, 10%, 5%, and 1% likelihoods. Data added later by David Cantrell is tabulated for 100%, 99%, 95%, 90%, 75%, 50%, 25%, 10%, 5%, and 1% likelihoods.


Imagine a coin flipped 1000 times. The expected 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));
  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.


Jon Orwant, Readable Publications, Inc;

Maintained and updated since October 2003 by David Cantrell,

This software is free-as-in-speech software, and may be used, distributed, and modified under the terms of either the GNU General Public Licence version 2 or the Artistic Licence. It's up to you which one you use. The full text of the licences can be found in the files GPL2.txt and ARTISTIC.txt, respectively.


Data for 31 to 250 degrees of freedom is from Wikibooks and is covered by the Creative Commons attribution-sharealike licence.