In statistics we often have to deal with the “degrees of freedom” in a calculation.  Where do those come from?  What does that mean?  I pondered over that for a long time before “the light bulb went on."

Consider:

  Suppose I asked you to fill out the above line of boxes (a vector, actually) with the numbers from 1 to 9, using each number only once.  How many ways are there to do that?   Well…a lot.  We could do the math, if we were so inclined, but I'm not, and right now I really don’t care about that, anyway.  It’s just that there’s a whole lot of ways for you to do it.

  Ok, now suppose we had this one:

...and, as before, I ask you to finish filling out the vector with numbers from 1 to 9 and we have the same caveat that each number can be used only once…how many ways are there to complete this vector, Victor?  One.  That’s it…one.  And that’s your basic concept of degrees of freedom. 

Under the conditions I just gave, I had choices about which numbers I could decide to put in boxes #1 thru #8.  But when we got to box #9, you were no longer free to make your choice—the value was constrained by the conditions—you had to put a 9 there. 

  Now an example that's a little more complicated, but a little more realistic—here’s some totally made up data about blood type and eye color:

…if I told you the above totals for blood type and eye color had been found in a particular sample of folks, could you tell me how to fill out the blank spots in the table to break down the frequencies of each blood type/eye color combination?  No, there's too much data missing from the table.

  Ok, what if I helped you out just a little bit?

  Well, this isn’t much better than the original, but at least you could deduce from this incomplete table that there must be 60 people with blue eyes and type A blood in this group (210 total A - 150 green A = 60 blue A). 

So I’ll help out with some more numbers…

  When you look closely at the table now, you’ll find I’ve actually provided you with all the information you really need in order to fill out the rest of the blank spots...sort of like doing a puzzle.

  Thus in the above table, the degrees of freedom are 3—that is, once I fill in values for three boxes--any three of the empty boxes--then, given the constraints of the totals in the margins, the rest are automatically determined—we no longer have freedom to "choose" their values--the die has been cast, the moving finger has writ and moved on. 

  Anyway, this works out to be: degrees of freedom = (#rows – 1) x (#columns – 1) = (2 – 1) x (4 -1) = 1 x 3 = 3.

  If you like to do puzzles, does this remind you of...something?  I suspect it does.

  Hmmm.  But there aren’t (9 -1) x (9 – 1) = 64 degrees of freedom in Sudoku because of the additional condition that the marked 3 x 3 squares are also constrained. The computations are way beyond my ken, but Prof. Gary McGuire of Dublin College says that one must have at least 17 “clues” to solve a 9 x 9 Sudoku, and that 16 just won’t do it. 

  See: http://www.math.ie/McGuire_V1.pdf   

  I'm sure he will be glad to know that I have no reason to doubt him.

Now, you may ask "why the interest in these statistical things?"

For one thing, I really like doing statistics.  In my PhD program I actually have a declared minor in Statistics--not many PhD programs have real "minors," but mine does.

And, not to be too coy, it also may have something to do with dental age assessment.  Stay tuned.....