# One last thing: degrees of freedom

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?

(To make this problem interactive, turn on javascript!)

- I need a hint ... :If I tell you the number of heads, do you also know the number of tails?

- ...another hint ... : How many variables are "free" to vary?

#### 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?

(To make this problem interactive, turn on javascript!)

- I need a hint ... : There are 5 weekdays -- how many of those am I "free"

to specify data for?

- ...another hint ... : If I knew that there were 20 sickdays each on Monday

through Thursday, is Friday still "free" to vary?

#### 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.

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