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They seem to represent the same idea as far as I can tell - can they be used interchangeably?

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    $\begingroup$ I don't understand in what sense you can think they are interchangeable $\endgroup$ – niandra82 Nov 12 '14 at 21:06
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    $\begingroup$ Please explain yourself better $\endgroup$ – niandra82 Nov 12 '14 at 21:15
  • $\begingroup$ Sorry for the lack of clarification; the Wikipedia page for degrees of freedom (physics & chemistry) says "In physics, a degree of freedom is an independent physical parameter in the formal description of the state of a physical system." So if you have a single monatomic gas molecule moving in 3D space, wouldn't we say the d.o.f. = # independent variables = 6 (x, y, z coordinates and velocities)? This leads me to believe they are describing the same thing - I would like to know whether this is generally the case, and if not, what an exception would be. $\endgroup$ – ajkal Nov 12 '14 at 21:21
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    $\begingroup$ Beware of different uses of the same word or phrase in different disciplines. In a physical system "degrees of freedom" is used in the same sense that "number of identifiable parameters" would be used in a statistical model. (Technically, it's the dimension of a real manifold whose points correspond to "states" of a system under investigation.) Although the concepts are related, they are not exactly the same. I suspect the thread at stats.stackexchange.com/questions/16921/… may answer this question. Does that help? $\endgroup$ – whuber Nov 12 '14 at 21:24
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    $\begingroup$ Wikipedia makes a distinction, though I can't say the article on degrees of freedom in statistics is particularly clear. One paper I did find very useful in clarifying my own notions in some of the more complex contexts was Ye's 1998 paper (which is in the references at the latter article). See also the discussions of df under regularization or smoothing in Hastie et al (Elements of Statistical Learning 2ed, which is - legitimately - available for download at the book site). $\endgroup$ – Glen_b Nov 12 '14 at 21:45
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In discipline of Statistics, the terms "Independent Variables" and "Degrees of Freedom" are two distinct concepts.

Independent variables generally refers to a set of variables (traits) that are independent from one another. For example, if we take a random sample of people and record their age, we do not expect that the age of one individual influences the age of another individual.

Per wikapedia degrees of freedom refers to "the number of values in the final calculation of a statistic that are free to vary." For example, in a situation where we are trying to calculate the variation of the age of our sample, this would be $(n-1)$.

This may not be the case in another discipline, but you would need to ask someone of that field to know for sure.

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