The degrees of freedom In modern statistics, they constitute a central content, however, their definition is very vaguely explained in books on the subject.
Its concept is easily understood from a geometric, algebraic and intuitive perspective.
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Geometry specifies degrees of freedom as spaces by which the summary unit of measure can vary and display different values. From an algebraic point of view, it is understood as the number of equations established using the data.
Both definitions are related to aid in the understanding of the concept, since its applications extend throughout all statistical science.
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In this article you will find:
What is known as degrees of freedom?
To understand the subject a bit more, below, I present some of the definitions found in commonly used statistics texts:
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Definitions of degrees of freedom
According to Daniel Wayne "It is the sum of the values, the deviations and individual values, with respect to their mean being equal to zero" Knowing n-1 values from the mean, the n-th value is known, automatically determined by restriction of 3 where all the values of n add up zero.
For Dawson "The degrees of freedom and their value are related to the number of opportunities in which the sample information is used."
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Last but not least Pagano understands “The degrees of freedom as the number of data free of variation when calculating a statistical test”.
What are the degrees of freedom?
The GL (degrees of freedom) is the amount of information provided by the data that can be used to estimate the unknown parameters of the population and calculate the variability of the estimates.
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This is determined according to the number of parameters of the model and the observations of the show. As the sample size increases, more information is obtained and consequently the degrees of freedom in the data increase. In case parameters are added to the model, for example, the terms in the regression equation are increased, spending information and reducing the possible degrees of freedom to estimate the variability of the appraisals of parameters.
They are also used to define a specific distribution, families of distributions, such as F, t, chi-square, it is used by GLs to specify the appropriate specific distribution for different sample sizes and different amounts of parameters in the model.
In conclusion, degrees of freedom GL refers to the number of independent values that are needed in statistical calculation, minus the number of constraints attached to the observations. That is, it is the number of values in the sample that can be freely specified, after knowing information about said sample.
Uses of degrees of freedom
The degrees of freedom they are necessarily related to the size of the sample, therefore they are used in the definition of statistical distributions to carry out hypothesis tests.
They are used when calculating the standard deviation of the sample giving a representation of the degree of dispersion by n data around the mean, and for Knowing the mean, the relationship between the data is established by adding them and dividing them by the number of them. themselves.
They are the basis for the Student's t distribution, which is used to test hypotheses of equality of the means between two groups of data.
Its use is mainly differentiated between statistics that use population parameters Y show them.
In population parameters, given that n all the values are known, the degrees of freedom will be all elements of the population "N ".
For the sample parameters, they are estimates since all the sample values are known.
Both cases allow the observations of the sample set to be random, therefore, when estimating the statistic, you can obtain different results. So the observations have full property of varying like the observations of the population set.
Understanding degrees of freedom
For a better understanding of number of degrees of freedom, it is recommended to view it as the number of dimensions in space in which a value is free to vary or move.
Each relationship is established or calculated from the data provided by the sample itself, which generates the need to modify the degrees of freedom GL if the statistic will be used in calculations futures. In this sense, degrees of freedom they remain limited to the difference that results from the amount of data and the amount of relationships established between them.
They can be estimated with the formula:
N - r
Where n is equal to the number of subjects belonging to the sample, which can beat a value.
Where r is equal to the number of subjects whose value will depend on the value of the free elements of the sample.
Finally, it is worth mentioning that, like other topics in statistics, degrees of freedom in statistics They play an important role in studies in other areas such as science and society.
