1Statistics::ChiSquare(3U)ser Contributed Perl DocumentatiSotnatistics::ChiSquare(3)
2
6 "Statistics::ChiSquare" - How well-distributed is your data?
7
9 use Statistics::ChiSquare;
10 print chisquare(@array_of_numbers);
12 Statistics::ChiSquare is available at a CPAN site near you.
14
16 Suppose you flip a coin 100 times, and it turns up heads 70 times. Is
17 the coin fair?
18 Suppose you roll a die 100 times, and it shows 30 sixes. Is the die
20 loaded?
21 In statistics, the chi-square test calculates how well a series of
23 numbers fits a distribution. In this module, we only test for whether
24 results fit an even distribution. It doesn't simply say "yes" or "no".
25 Instead, it gives you a confidence interval, which sets upper and lower
26 bounds on the likelihood that the variation in your data is due to
27 chance. See the examples below.
28 If you've ever studied elementary genetics, you've probably heard about
30 Gregor Mendel. He was a wacky Austrian botanist who discovered (in
31 1865) that traits could be inherited in a predictable fashion. He did
32 lots of experiments with cross breeding peas: green peas, yellow peas,
33 smooth peas, wrinkled peas. A veritable Brave New World of legumes.
34 But Mendel faked his data. A statistician by the name of R. A. Fisher
36 used the chi-square test to prove it.
37
39 chisquare
40 There's just one function in this module: chisquare(). Instead of
41 returning the bounds on the confidence interval in a tidy little two-
42 element array, it returns an English string. This was a deliberate
43 design choice---many people misinterpret chi-square results, and the
44 string helps clarify the meaning.
45 The string returned by chisquare() will always match one of these
47 patterns:
48 ".*!" (ie strings ending in an exclamation)
50 various error messages for when you supply Obviously Wrong data.
52 "There's a >\d+% chance, and a <\d+% chance, that this data is random."
54 or
56 "There's a <\d+% chance that this data is random."
58 or
60 "I can't handle \d+ choices without a better table."
62 That last one deserves a bit more explanation. The "modern" chi-square
64 test uses a table of values (based on Pearson's approximation) to avoid
65 expensive calculations. Thanks to the table, the chisquare()
66 calculation is very fast, but there are some collections of data it
67 can't handle, including any collection with more than 250 slots. So
68 you can't calculate the randomness of a 500-sided die should such an
69 insane thing ever exist.
70 You will also notice that the percentage points that have been
72 tabulated for different numbers of data points - that is, for different
73 degrees of freedom - differ. The table in Jon Orwant's original
74 version has data tabulated for 100%, 99%, 95%, 90%, 70%, 50%, 30%, 10%,
75 5%, and 1% likelihoods. Data added later by David Cantrell is
76 tabulated for 100%, 99%, 95%, 90%, 75%, 50%, 25%, 10%, 5%, and 1%
77 likelihoods.
78
80 Imagine a coin flipped 1000 times. The expected outcome is 500 heads
81 and 500 tails:
82 @coin = (500, 500);
84 print chisquare(@coin);
85 prints "There's a >90% chance, and a <100% chance, that this data is
87 random.
88 Imagine a die rolled 60 times that shows sixes just a wee bit too
90 often.
91 @die1 = (8, 7, 9, 8, 8, 20);
93 print chisquare(@die1);
94 prints "There's a >1% chance, and a <5% chance, that this data is
96 random.
97 Imagine a die rolled 600 times that shows sixes way too often.
99 @die2 = (80, 70, 90, 80, 80, 200);
101 print chisquare(@die2);
102 prints "There's a <1% chance that this data is random."
104 How random is rand()?
106 srand(time ^ $$);
108 @rands = ();
109 for ($i = 0; $i < 60000; $i++) {
110 $slot = int(rand(6));
111 $rands[$slot]++;
112 }
113 print "@rands\n";
114 print chisquare(@rands);
115 prints (on my machine)
117 10156 10041 9991 9868 10034 9910
119 There's a >10% chance, and a <50% chance, that this data is random.
120 So much for pseudorandom number generation.
122
124 Jon Orwant, Readable Publications, Inc; orwant@oreilly.com
125 Maintained and updated since October 2003 by David Cantrell,
127 david@cantrell.org.uk
128 This software is free-as-in-speech software, and may be used,
130 distributed, and modified under the terms of either the GNU General
131 Public Licence version 2 or the Artistic Licence. It's up to you which
132 one you use. The full text of the licences can be found in the files
133 GPL2.txt and ARTISTIC.txt, respectively.
134
136 Data for 31 to 250 degrees of freedom is from Wikibooks
137 <https://en.wikibooks.org/wiki/Engineering_Tables/Chi-
138 Squared_Distibution> and is covered by the Creative Commons
139 attribution-sharealike licence
140 <https://creativecommons.org/licenses/by-sa/3.0/>.
141perl v5.38.0 2023-07-21 Statistics::ChiSquare(3)