How many
degrees of freedom do you have in your life?
Well, if you can’t answer that, can you say how many constraints you’re
subject to? The number of constraints
would tell you how many degrees of freedom you subtract from the original or
unconstrained state you started off in. But
freedom versus constraint is a difficult thing to get a hold of in the realm of
human activity, so let’s go to the realm of physics.
I first
encountered degrees of freedom and constraints in a two-semester junior-level
Newtonian mechanics course I took at Hendrix College, in the 1978-79 academic
year. Richard Rolleigh was the
professor, and a good one. I was not
that good of a student, except in the first three months of the course. The textbook is a pretty good one: Mechanics,
by Keith R. Symon. Chapter 9 in the book
discusses generalized coordinates, Lagrange’s equations, systems subject to
constraints, constants of the motion and ignorable coordinates, Hamilton’s
equations, and Liouville’s theorem.
In section
9.4, Symon says this about degrees of freedom:
“The number of independent ways in which a mechanical system can move
without violating any constraints which may be imposed is called the number of degrees of freedom of the system. … For
example, a single particle moving in space has three degrees of freedom, but if
it is constrained to move along a certain curve, it only has one. A system of N free particles has 3N
degrees of freedom, a rigid body has 6 degrees of freedom (3 translational and
3 rotational), and a rigid body constrained to rotate about an axis has one
degree of freedom.”
The concepts of constraints and degrees of freedom are also used in
statistics. Let’s see what professors
Box, Hunter, and Hunter (BH&H) have to say on the subject. First jog your memory and recall that each deviation for a sample is the difference
in an observational value y and the sample
average ŷ. (This symbol is ‘y-hat’,
which in physics is used to indicate the
unit vector in the y
direction, but the symbol set on my computer doesn’t seem to have ‘y-bar’, so
I’ll use ŷ instead.) The deviation for a population is the
difference in y and the mean η. That is, in the case of a sample average, the
deviation is y – ŷ, and in the case of the population mean, the deviation is y – η. Add up the deviations for a given sample or a
given population and what do you get?
That’s right, zero.
Here’s the
BH&H paragraph that discusses why n-1
is used for calculating the sample standard deviation, but first they have to
define the variance:
“The
deviations of n observations from
their sample average must sum to
zero. This requirement, that Σ(y-ŷ) = 0,
constitutes a linear constraint on the deviations or residuals y1 – ŷ, y2 – ŷ, …, yn
– ŷ used for calculating s2
= Σ(y- ŷ)2/(n-1). It
implies that any n-1 of them completely
determine the other [one]. The n
residuals y- ŷ [and hence their sum
of squares Σ(y- ŷ)2 and
the sample variance, Σ(y- ŷ)2/(n-1)] are therefore
said to have n-1 degrees of freedom. In this book the number of degrees of freedom
is denoted by the Greek letter ν
(nu).”
Allright! So in the People Aren’t Perfect lab, we have ν = n – 1 = 10 – 1 = 9 degrees of
freedom. BH&H say this: “The loss of one degree of freedom is
associated with the need to replace the unknown population mean η by ŷ,
the sample average derived from the data.”
Yep, and as
you just noticed, there’s this thing called the variance, which the standard
deviation comes from by taking the square root.
The population variance is σ2
= Σ(y-η)2/N, and
its standard deviation is σ =[Σ(y-η)2/N]1/2. The sample variance is s2 = Σ(y-ŷ)2/(n-1) and
its standard deviation is s = [Σ(y-ŷ)2/(n-1)]1/2.
Would you believe this is the same expression as the one at the bottom of page 1 of the People Are Not Perfect* lab handout? Different symbology is the only difference.
*Neither are machines! [But what about equations?]
Would you believe this is the same expression as the one at the bottom of page 1 of the People Are Not Perfect* lab handout? Different symbology is the only difference.
*Neither are machines! [But what about equations?]
