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  • 00:00

    [Degrees of Freedom]

  • 00:01

    DANIEL LITTLE: In this video, I willdiscuss the statistical concept of degrees of freedomand provide some simple examples whichdemonstrate what degrees of freedom areand why they're necessary for statistics.The term degrees of freedom was initiallyused in physics, where it referredto the number of independent parametersthat define the state of a physical system.

  • 00:24

    DANIEL LITTLE [continued]: To give you one simple example, youcan imagine a car which moves around a track, justa simple track like this, without any side-to-side motionas it goes around.So if the car starts here and continues around the track,the only parameter which we need to knowis the distance traveled by the car.

  • 00:45

    DANIEL LITTLE [continued]: If we know that distance, then wecan determine the car's position on the track.So in this particular example, there's onlya single degree of freedom.[Single Degree of Freedom System]Here's another example of a very simple physical systemfor which there's only a single degree of freedom.This simple pendulum swings back and forth,but it doesn't exhibit any other type

  • 01:05

    DANIEL LITTLE [continued]: of motion, just back and forth along the same line.The state of the pendulum can be describedby the angle-- which I've demarcated by this Greek symboltheta-- of the pendulum.So the angle of the pendulum and the vertical linetells you exactly where the position of the pendulum is.

  • 01:28

    DANIEL LITTLE [continued]: So in this particular example, there'sonly a single degree of freedom describingthe position of the pendulum.[Two Degrees of Freedom System]Now we can make things more complicatedby inserting a spring along the length of the pendulum.If we did this, now as the pendulum moves back and forththis way, it's also moving up and downas it swings back and forth along its arc.

  • 01:52

    DANIEL LITTLE [continued]: So for this particular example, thereare two degrees of freedom that we have.We have, in fact, one degree of freedomwhich describes the position along the positionfrom the vertical-- so the angle-- and another degree

  • 02:14

    DANIEL LITTLE [continued]: of freedom describing the height as it bouncesup and down along that spring.So there are two degrees of freedomin this particular system.[Degrees of Freedom]In statistics, the term degrees of freedomrefers to the number of values in a calculation of a statisticlike a t-test, like a t-statistic, or an F-statistic,and an ANOVA, that are free to vary.

  • 02:36

    DANIEL LITTLE [continued]: So let me give you an example of a very simple, intuitiveexample of what I mean by degrees of freedomin statistics.[Degrees of Freedom: An intuitive example]Let's start by the assumption that we've observed some dataand we've got a set of numbers representing that data.Here, those numbers are just 4, 9, 8, 10, and 7.

  • 02:56

    DANIEL LITTLE [continued]: We can take those numbers and compute the mean.The mean of these five numbers is 7.6.What we can also do is, if I fix one of the numbers,I can still allow all of the other numbersto vary and still come out with the same mean.So in this particular example, I can fix the number 10,but all of my other numbers in the second data set

  • 03:18

    DANIEL LITTLE [continued]: are different from where they were in the first data set.But if I compute the mean of this second set of numbers,it still has a mean of 7.6.So consequently, I have a total of four degreesof freedom for both of these sets of numbers.[Degrees of Freedom]More generally, I need to ensure that I

  • 03:39

    DANIEL LITTLE [continued]: fix one number for each estimated parameter thatgoes into the statistical test.In the previous example, I had five data points.I had four degrees of freedom.I have to fix one of those numberswhen I'm computing the mean.For an ANOVA, this refers to the group means and the total meanacross all of the groups.And I use these means in the computations

  • 04:01

    DANIEL LITTLE [continued]: of the sums of squares, which you can learn more about if youwatch the one-way ANOVA video.Even more generally, you can consideran ANOVA to be a linear model.So by linear model, I simply mean that it'ssomething analogous to a line.So if you draw a line on a graph, that is of the equation

  • 04:24

    DANIEL LITTLE [continued]: y equals mx plus b, which you learned in high schoolgeometry.An ANOVA can be considered a more complicated versionof the same type of system, where now youwould have particular coefficients for each

  • 04:45

    DANIEL LITTLE [continued]: of your independent variables that you were considering.But the equation would still be linearbecause you're simply taking eachof your independent variables, multiplying itby some coefficient, then adding them all up.Now because you can think of the ANOVA as a linear model,degrees of freedom tells you how many

  • 05:06

    DANIEL LITTLE [continued]: components you need to know before your vectorrepresenting your linear model is fully determined.So that's a very high level definitionof what degrees of freedom actually means.So to summarize, I've provided youwith a few examples of what is meant

  • 05:27

    DANIEL LITTLE [continued]: by the term degrees of freedom.Generally, each statistical test that you conduct-- for example,t-test, ANOVA, chi-square, and so on--will have their own method for specifyingtheir degrees of freedom.Each of those methods will be specific and uniqueto the statistical test that you're using.This video provides some of the reasoning

  • 05:48

    DANIEL LITTLE [continued]: for what each of these different types of degrees of freedommean and what each of them have in common.The purpose of the degrees of freedom in a statistical testis to take into account the size of the sample and the numberof conditions being compared.Thanks.

Video Info

Series Name: Statistics for Psychology

Episode: 3

Publisher: University of Melbourne

Publication Year: 2014

Video Type:Tutorial

Methods: Degrees of freedom, Analysis of variance

Keywords: mathematical concepts; physics

Segment Info

Segment Num.: 1

Persons Discussed:

Events Discussed:



In this third installment of his series on statistics for psychology, Professor Daniel Little demonstrates the degrees of freedom principle and shows how it can be applied in research statistics.

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Degrees of Freedom

In this third installment of his series on statistics for psychology, Professor Daniel Little demonstrates the degrees of freedom principle and shows how it can be applied in research statistics.

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