Degrees of Freedom

3 Key excerpts on "Degrees of Freedom"

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  eBook - ePub

  Statistics from A to Z

  Confusing Concepts Clarified

  • Andrew A. Jawlik(Author)
  • 2016(Publication Date)
  • Wiley

    (Publisher)

  Degrees of Freedom

  Summary of Keys to Understanding

  1. Degrees of Freedom (symbol: df or the Greek letter ν) is a way of adjusting for the additional error introduced when one Statistic is used to calculate another.

  1. The effect of the adjustment gets smaller for larger Sample Sizes.

  1. The individual members of some families of Distributions are specified by their df.

  1. Formulas for Degrees of Freedom vary by the Statistics and the test they are used in

  Statistic	df	Explanation
  ANOVA: Mean Sum of Squares Within (MSW)	N − k	N: total # of all data points k: # of groups
  ANOVA: Mean Sum of Squares Between (MSB)	k − 1
  χ2	n − 1	n: Sample Size
  χ2 test for Goodness of Fit	n − 1	n: # of categories
  χ2 test for Independence	(r − 1)(c − 1)	# of rows and columns
  χ2 test for Variance	n − 1	n: Sample Size
  F	n1 − 1 and n2 − 1	n1 and n2 : Sizes of the 2 Samples
  t	n − 1	n: Sample Size
  1-Sample t-test, and Paired t-test	n − 1
  2 (Independent)-Sample t-test	n1 + n2 − 2	n1 and n2 : Sizes of the 2 Samples

  Explanation

  1. Degrees of Freedom (symbol: df or the Greek letter ν) is a way of adjusting for the additional error introduced when one Statistic is used to calculate another.

  Notation: Degrees of Freedom is often abbreviated as “df” in text; the Greek letter nu (ν) is commonly used in formulas or equations.

  A Statistic is a numerical property of a Sample, for example, the Sample Mean or Sample Variance. A Statistic is an estimate of the corresponding property (“Parameter”) in the Population or Process from which the Sample was drawn. Being an estimate, it will likely not have the exact same value as its corresponding population Parameter. The difference is the error in the estimation

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  eBook - ePub

  Doing Research in Political Science

  An Introduction to Comparative Methods and Statistics

  • Paul Pennings, Hans Keman, Jan Kleinnijenhuis(Authors)
  • 2005(Publication Date)
  • SAGE Publications Ltd

    (Publisher)

  F-distribution, two Degrees of Freedom are required: one for the numerator and one for the denominator. Degrees of Freedom are derived from sample sizes, and refer to the sample size diminished by the number of units that have to be offered to calculate the test statistic. Degrees of Freedom designate the number of units that are still freely available once the required test statistic has been computed from the sample data.

  Let us consider the difference between two sample means. To compute a mean, you need at least one unit from the sample. To compare two sample means, two units have to be offered beforehand. The Degrees of Freedom for the t-distribution of the difference of sample means when variances are known amounts therefore to n − 2. As another example, the t-distribution of a regression coefficient might be used. At least two units of analysis are required to pin down a regression line since a straight line is denned by two points. Therefore the Degrees of Freedom for the t-distribution of the regression coefficient statistic also amount to n – 2.

  To compute a variance at least one observation is required. The χ2 -distribution of the sample variance therefore has n − 1 Degrees of Freedom. Chi-square also applies to the comparison of r samples from a nominal distribution with c values. The Degrees of Freedom for the χ2 -distribution of differences between samples from a nominal variable amount to (r − l)(c − 1). This number corresponds precisely with the number of different percentage differences which can be calculated from one cross-table. One conception of a cross-table of r rows and c columns is that r samples were drawn from a nominal variable with c values. To test whether these r samples might have been drawn from the same nominal distribution, one may choose one sample as the base rate and compare the other r − 1 samples with it. To be able to tell something about a sample distribution of a nominal variable, one value of the nominal variable is required as the base rate category, thus leaving c − 1 columns freely available. Altogether this results in (r − l)(c

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  eBook - ePub

  Introduction to Mechanical Vibrations

  • Ronald J. Anderson(Author)
  • 2020(Publication Date)
  • Wiley

    (Publisher)

  7 Systems with More than One Degree of Freedom

  A degree of freedom is a coordinate, either translational or rotational, required to describe the motion of a mass in a system. We could have, for example, the problem of describing the general motion of a rigid body constrained to move on a plane. A complete specification of its motion would require two orthogonal translational coordinates to locate its center of mass plus a single rotational coordinate to specify its orientation on the plane. There are therefore three Degrees of Freedom for the body. If we released the constraint that it must move on a plane, we would need to deal with a rigid body moving in three‐dimensional space and would find ourselves with six Degrees of Freedom (three translations and three rotations).

  Alternatively, we might be dealing with a system that has multiple bodies, each with a single degree of freedom so that we need as many Degrees of Freedom as there are bodies.

  In any case, we will now be dealing with NDOF systems – systems with N Degrees of Freedom.

  We start with a discussion of systems with two Degrees of Freedom. These systems are relatively easy to consider and have all of the properties of systems with any number of Degrees of Freedom.

  7.1 2DOF Undamped Free Vibrations – Modeling

  We start with a 2DOF system without damping as shown in Figure 7.1 . There are two masses, and , that are constrained to have one‐dimensional, vertical, motion only. We can define the two Degrees of Freedom as and , the vertical motions away from the equilibrium positions of the masses. The effects of gravity can be ignored because they are completely canceled out by the preloads in the springs when the system is in equilibrium. As with 1DOF systems, we are only concerned with things that change when we move away from equilibrium and gravitational forces and preloads in springs don't change.

  Figure 7.2

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