One way to think of degrees of freedom is how many datapoints can move independently (as in, be anything you want) while maintaining the same average. So, think of a dataset of two numbers. If you move one of the two numbers in that dataset, the other one HAS to change in order to maintain the average of the pair. Thus, this dataset would only have 1 degree of freedom as only one datapoint can move freely while the other must move in-tandem with the other.

In statistics, degrees of freedom is best understood as a way to numerically quantify the sample size in conjunction with the number of treatment groups. Bigger sample sizes increase degrees of freedom, more treatment groups reduce degrees of freedom.

It really depends on what you’re talking about; you’re going to get different answers from physicists, chemists, statisticians, etc. Physicists are probably most interested in talking about entropy or gravity. Chemists are probably more concerned with how quickly/how prolific reactions are. Statisticians are more interested in, well, statistics.

To be as generic as possible, “degrees of freedom” come up any time you’re analyzing the state of a system compared to what it *could* be. The total degrees of freedom are the range of values/properties that a system *could* posses. Take a coin flip, for example. A coin being flipped has two potential degrees of freedom (with respect to the outcome of the flip). The actual flipped coin has only one of these potential states, but when we’re assessing odds, we’re interested in a potential outcome v. the universe of possible outcomes, and degrees of freedom are a helpful way of talking about that.

Degrees of freedom are often discussed in generic and relative terms, rather than specific quantitative terms. This is because it can be quite challenging to account for all of the actual potential differences between different states. In physics, for example, even if you’re talking about something very simple (a particle floating through space) the potential degrees of freedom are quite naturally infinite if you want to account for the particle’s own properties, its trajectory, its velocity, the various sources of electromagnetic/gravitational force that might be acting on it, etc. In cases like that, you usually make some approximations/generalizations to throw out the degrees of freedom you aren’t interested in and focus on the ones that you are. In the world of statistics, you control for whatever you’re not interested in so you can focus on the rest.

>What happens if the degrees of freedom go up or down? Is there a “good” or “bad” DoF?

Entirely contextual.

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