# I Degrees of Freedom

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1. Jun 14, 2017

### lola2000

I am trying to understand how to decide the number of degrees of freedom when calculating a chi-squared and p value.

I have the data:

England:
people with no pets = 665
people with 1 pet = 976
people with 2+ pets = 913

Scotland
people with no pets = 313
people with 1 pet = 527
people with 2+ pets = 506

Wales
people with no pets =302
people with 1 pet = 440
people with 2+ pets = 358

I've calculated the expected frequency and therefore the (observed - expected)^2 / expected for each cell but im stuck with degrees of freedom

One thing ive found says dof = (rows-1 ) * (col -1 ) which would = 2 * 2 = 4
another thing says dof = number of cells - number of restrictions = 9 - 2 = 7
where number of restrictions is number of things you are categorising by

2. Jun 14, 2017

### andrewkirk

Both methods are correct and give the same answer, which is 4.
To apply the 'number of restrictions' method we need to count the restrictions carefully. There are five restrictions, being:
• items in column 1 must add to column total 1
• items in column 2 must add to column total 2
• items in column 3 must add to column total 3
• items in row 1 must add to row total 1
• items in row 2 must add to row total 2
We don't need a restriction on row 3 because it is already implied by the five restrictions above, as
total row 3 = tot col 1 + tot col 2 + tot col 3 - tot row 1 - tot row 2

I suggest sticking with the (r-1) x (c-1) mnemonic, as it's easier to remember.

3. Jun 14, 2017

### lola2000

Thanks so much for the help. I see where I went wrong with the number of restrictions now

4. Jun 18, 2017

### lola2000

Similar question but how would this hold true for a situation where you have a random number generator producing numbers between 0 and 9 and you are counting their frequency? So you would have a table of one row and 10 columns.

Degrees of freedom is 9?

5. Jun 18, 2017

### andrewkirk

Yes.

6. Jun 20, 2017

### chiro

Hey lola2000.

The degrees of freedom is a function of the number of independent parameters in the model in addition to how many independent observations exist within the sample.

If you can understand this then you will be able to get an arbitrary value for the degrees of freedom of any statistic.

You use D = I - P where I is the number of independent sample points and P is the number of independent parameters being assessed.

Different test statistics use different rules to get it but the idea above is central to that of all statistics.