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Why the Name “Degrees of Freedom” in Statistics?

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  • #246010

    PlsKISS
    Participant

    I saw this was asked before and the answer was way off and involved a ‘discussion’ with intensity so let’s start with:

    “If you hear hoofbeats behind you, expect a horse, not a zebra”  accreditation here https://quoteinvestigator.com/2017/11/26/zebras/

    My question is statistically based not six sigma based ( though when I read the explanation for Six Sigma DF I was also confused.)

    Why is the df on the Critical Values of t tables called degrees of freedom?

    I found this:

    Statisticians use the terms “degrees of freedom” to describe the number of values in the final calculation of a statistic that are free to vary. Consider, for example the statistic s². To calculate the s² of a random sample, we must first calculate the mean of that sample and then compute the sum of the several squared deviations from that mean. While there will be n such squared deviations only (n – 1) of them are, in fact, free to assume any value whatsoever. This is because the final squared deviation from the mean must include the one value of X such that the sum of all the Xs divided by n will equal the obtained mean of the sample. All of the other (n – 1) squared deviations from the mean can, theoretically, have any values whatsoever. For these reasons, the statistic s² is said to have only (n – 1) degrees of freedom. http://www.animatedsoftware.com/statglos/sgdegree.htm

    But that doesn’t really help me understand. I understood the answer could vary. Does that mean it could be a range? Does that mean if it is Tuesday and raining in the summer expect houseflies? How can you have several squared deviations of mean? Mean is mean! Square it and it is just (mean)(mean) it is a fixed number from a set of numerical data.  Right?

     

     

    Though I got to tell you the SixSigma DF makes about as much sense as the df in the Critical Values of t table

    Six Sigma(DF)

    Definition of Degrees of Freedom (DF):

    (#rows – 1)(#cols – 1)

    https://www.isixsigma.com/dictionary/degrees-of-freedom-df/

    Thank you,

    PlsKISS

     

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    #246013

    Robert Butler
    Participant

    The short answer to your question is there isn’t a short answer to your question.  The explanation provided by Wikipedia is a good starting place but you will have to take some time to not only carefully read and understand what is being said but also put pencil to paper and work through some math.

    I’ve spent most of my professional career as an engineering statistician and biostatistician. I always keep paper and pencil handy when I’m reading a technical article and I take the time to convert what I think I have read into mathematical expressions.  I find when I do this I not only gain a better understanding of what I’ve read but, if I’ve misunderstood what was written, I find it is easier to see that I have misunderstood. This, in turn, helps me on my way to a correct understanding.

    Perhaps if you had written out, in mathematical form, the highlighted section of the quote in your initial post you would have realized your question “How can you have several squared deviations of mean? Square it and it is just (mean)(mean) …?” is incorrect. The highlighted segment does not say “squared deviations of mean” it says “several squared deviations FROM that mean”  In mathematical terms it is saying (x(i) – mean)*(x(i) – mean) where x(i) is an individual measurement. The difference between the previous squared expression and what you wrote is all the difference in the world.

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    #246080

    Fausto Galetto
    Participant

    @PlsKISS

    You wrote: Statisticians use the terms “degrees of freedom” to describe the number of values in the final calculation of a statistic that are free to vary.

    This is explanation is due to (whom I call) “normal drugged statisticians” (they are the people that have their “statistical” knowledge based on the Normal Distributed data).

    I refer, as you did, to ””a statistic”” … IF the data are exponentially distributed, the Confidence Interval of the Mean depends on the “degrees of freedom” AND has nothing to do with ””the number of values … that are free to vary””

     

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