The degree of freedom (df) is used in statistics and physics to define the boundaries and location of a system. In physics, df refers to the independent coordinates of a system in phase space. In statistics, df defines the shape of a population or sample. Different distributions have different definitions of df, and it changes the appearance of the distribution. It is important to remember that df is not the same for different tests and should be represented differently when drawing distributions.
The degree of freedom (df) is a more used concept in statistics and physics. In both cases it tends to define the boundaries of a system and the location or size of what is being analysed, so that it can be represented visually. The definition of df in both fields is related, but not exactly the same.
In physics, the degree of freedom positions objects or systems, and each degree refers to a location in time, space, or other measurement. Df could be used synonymously with the term coordinate, and usually means independent coordinates of the fewest. The effective degree of freedom is based on the system being described in phase space or all potential space types that a system inhabits at the same time. Every single part of the phase space that the system occupies can be considered a df, which helps to define the complete realities of the considered system.
From a statistical point of view, the degree of freedom defines the distributions of populations or samples and is encountered when people begin to study inferential statistics: hypothesis testing and confidence intervals. As with the scientific definition, df in statistics describes the shape or aspects of the sample or population depending on the data. Not all drawn representations of distributions have a degree of freedom measurement. The common standard normal distribution is not defined by degrees; instead, it will be the same bell curve in all cases.
A distribution similar to the standard normal is student-t. Student-t is defined in part by the degree of freedom in the formula n-1, where n is the sample size. This means that if the variables of the distribution were to be selected one by one, all but the last one could be chosen freely. There is no choice but to take the last one and no freedom to choose another variable at that point. So a variable is not free; it’s like having to take the last tile from a bag during a game of Scrabble® where there is no choice but to choose that letter.
Different distributions such as the F and chi-square have different definitions of degrees of freedom, and some even use more than one df in the definition. The problem becomes confusing because the df definition is related to the type of test performed and is not the same with the various parametric (parameter-based) and non-parametric (non-parameter-based) tests. Essentially, it won’t always be n-1. Goodness-of-fit or contingency table testing may use the chi-squared distribution with different df than single-variable hypothesis testing of variance or standard deviation evaluates.
What’s important to remember is that every time you use the degree of freedom to define a distribution, it changes it. It may still have some features that are immutable, but the size and appearance vary. When people draw representations of distributions, especially two of the same distributions that have different dfs, it is recommended to make them appear to be of different sizes to convey that df is not the same.
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