The chi-square test appears in a lot of different places. Some recent data
on Astrology, published in the
May newsletter of
the Skeptic Society, offers an interesting opportunity to show one of
these tests. In an article offering dark matter as a possible explanation of
the effects of astrology, the London Times published a list of the 1,000
richest people and their star signs. It noted a significant difference
between the number born under Gemini and the number born under Pisces. A
careful look at the full data set, though, shows that numbers observed could
easily be explained by sampling error.
There are actually 1,067 people listed here, which suggests either a
typographical error or perhaps inflation has hit the top 1,000 list. The
question is whether the distribution of star signs is uniform or not. A
non-uniform distribution could be taken as evidence for Astrology, but it
would probably need replication to be taken seriously.
If the distribution is uniform, we would expect to see about 89 people in
each star sign. We compare the expected count to the observed count using the
Chi-square goodness of fit statistic, X2:
where Oi are the observed counts (110, 104, 95, ..., 73) and Ei
are the expected counts (88.92 for each star sign).
The table below shows the calculation of
You should compare this statistic to a Chi-squared distribution with 11
degrees of freedom. The 95th percentile of this distribution is 19.7. Since
your test statistic is smaller than 19.7, you would accept the null
hypothesis and conclude that the distribution is uniform. Actually, you
should state the conclusion more cautiously: there is insufficient
evidence for a non-uniform distribution of star sign among the richest 1,000
people. You can also compute a p-value for this test statistic. In Microsoft
Excel, the function CHIDIST(13.57,11) produces a p-value of 0.258.
Now if you just took the largest group, Gemini, and compared it to the
smallest group, Pisces, you would indeed get a statistically significant
differrence:
Compare this test statistic to a Chi-squared distribution with 1 degree of
freedom, where the 95th percentile is 3.84. The p-value (CHIDIST(7.48,1)) is
0.0062.but there was no reason to believe a priori that Gemini would have a
surplus of rich people compared to Pisces. Here's what the astrology.com web
site says about Gemini:
Gemini is the third Sign of the Zodiac, and those born under this Sign
will be quick to tell you all about it. That's because they love to talk!
It's not just idle chatter with these folks, either. The driving force
behind a Gemini's conversation is their mind. The Gemini-born are
intellectually inclined, forever probing people and places in search of
information. The more information a Gemini collects, the better. Sharing
that information later on with those they love is also a lot of fun, for
Geminis are supremely interested in developing their relationships.
Dalliances with these folks are always enjoyable, since Geminis are bright,
quick-witted and the proverbial life of the party. Even though their
intellectual minds can rationalize forever and a day, Geminis also have a
surplus of imagination waiting to be tapped. Can a Gemini be boring? Never!
http://www.astrology.com/allaboutyou/sunsigns/gemini.html
and about Pisces
Pisces is the twelfth Sign of the Zodiac, and it is also the final
Sign in the Zodiacal cycle. Hence, this Sign brings together many of the
characteristics of the eleven Signs that have come before it. Pisceans,
however, are happiest keeping many of these qualities under wraps. These
folks are selfless, spiritual and very focused on their inner journey. They
also place great weight on what they are feeling. Yes, feelings define
Pisceans, and it's not uncommon for them to feel their own burdens (and
joys) as well as those of others. The intuition of the Pisces-born is
highly-evolved. Many people associate Pisceans with dreams and secrets, and
it's a fair association, since those born under this Sign feel comfortable
in an illusory world.
http://www.astrology.com/allaboutyou/sunsigns/pisces.html
If you tried very hard, you could possibly infer some hints that Geminis
are richer because they are smarter, and Pisces are poorer because of their
selfless nature. I don't see anything in the description that would make me
think so, and it is easy to develop post hoc rationalizations. Unless you had
a hypothesis developed prior to data collection, you need to make some sort
of adjustment.
The simplest adjustment is a Bonferroni
correction. This adjusts the p-value by multiplying it by the number of
possible comparisons that could be made. With 12 Astrology signs, there are
12*11/2=66 possible comparisons. The adjusted p-value would be 0.41 which is
not statistically significant. Other adjustments are actually better than
Bonferroni here, but they would lead to the same conclusion.
This is hardly a definitive study, but it does tend to support most of the
other research on Astrology. The most prominent claim in support of Astrology
is the Mars Effect, which was a claim by Michel Gauquelin in 1955 that
prominent sports figures were more likely to be born at times when the planet
Mars was in a certain position in the sky. A careful study of this is
actually quite difficult because birth dates and times are not quite
uniformly distributed and because it is hard to define what a prominent sport
figure is. Jan Willem Nienhuys published a
critical review of the Mars effect
that appeared in the November/December 1997 issue of
Skeptical Inquirer.
A good resource about Astrology is in the Skeptic's Dictionary website:
http://www.skepdic.com/astrolgy.html.
