MathBench > Statistical Tests

Goodness of Fit Tests

One last thing: degrees of freedom

woman chasing fish with net

When doing a chi-square goodness of fit test, there is one last wrinkle to iron out, called degrees of freedom.

When I told you that 42 out of 100 sickdays were on Mondays or Fridays, you automatically knew that 58 had to be in the middle of the week, right? I was "free" to specify how many were on Monday/Friday, but then I was NOT "free" to decide how many were on non-Monday/Friday. So we say that, in this problem, there is only 1 degree of freedom.


Say you flip a coin 100 times. If we want to do a chi-square test to determine whether a coin is fair (lands equally on heads and tails), how many degrees of freedom would the test have?

 

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I think I have the answer: There are two variables here -- number of
heads and number of tails. But only 1 is free to vary -- once I tell you
how many heads there were, you know how many tails there were,
or vice versa.


It is possible to do chi-square tests using more than 2 variables. For example, let's say I got data on how many sickdays fell on EACH of the five weekdays:

day observed expected
mon 22 20
tues 19 20
wed 19 20
thurs 20 20
fri 20 20

We could do a chi-square test to check whether the distribution of sickdays matched our expectations for ALL FIVE weekdays

How many degrees of freedom would this test have?

 

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I think I have the answer: Once I know how many sickdays occurred
on 4 of the 5 days, the fifth day is no longer "free" to vary.
Therefore there are only 4 degrees of freedom.