Explain Degrees of freedom.

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Degrees of freedom (DoF) in statistics and mathematics refer to the number of independent values or quantities that can be assigned to a statistical distribution or a mechanical system, without violating any constraints. The concept is used in various fields, including physics, engineering, and statistics, with slightly different interpretations in each.

In physics and engineering, degrees of freedom describe the number of independent motions a system or body can undergo. For example, a rigid body moving in three-dimensional space has six degrees of freedom: three translational (movement along the x, y, and z axes) and three rotational (rotation about the x, y, and z axes).

In statistics, degrees of freedom often relate to the number of independent pieces of information available to estimate another parameter. For instance, in a sample of ( n ) values, the degrees of freedom for estimating the population variance is typically ( n – 1 ). This is because one degree of freedom is lost by using the sample mean as an estimate of the population mean. The concept is crucial in hypothesis testing and in determining the distribution of various test statistics, such as the chi-square and t-distributions.

Understanding degrees of freedom helps in accurately modeling systems and in making correct inferences in statistical analysis. It ensures that the variability and constraints of the system or dataset are appropriately accounted for.