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(n-1) degrees of freedom

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  • #38325

    Dr. Chidi
    Participant

    I have taught Statistics for many years, but I still could not explain the physical interpretation on the (n-1) degrees of freedom when dealing with the ki-square-statistics or such topics. I would be grateful if you help me explain the physical meaning of this statistical expression. Thanks.
     
    Dr. Chidi

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    #114581

    Mikel
    Member

    Dear Doc,
    You may want to start with the more basic issue of how to spell CHI before tackling the more esoteric question of degrees of freedom (hint – you may want to read what your stats book has to say about it and make sure that book also knows how to spell CHI)

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    #114749

    Dr. Chidi
    Participant

    I still need help understanding the physical or statistical interpretation of (n-1) degrees of freedom. Does any one out there know it? Is this community shy to tackle this problem?
     
    Of course, we all know that for a sample size of n, there is (n-1) degrees of freedom. Please explain to me the physical intepretation of this. I need more than this. Thanks
     
    CAC

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    #114752

    Peppe
    Participant

    Dr. Chidi,  the concept of degrees of freedom, from a general point of view are the number of parameters that must be fix so that a system is exactly defined (see the beam examples in construction eng that have 3 way to move). So for statistics n=data to study and the tie are the variable under study.
    So, for unidimensional analysis you have dof=n-1 (number of data – number of variable), for bidimensional analysis you have dof=n-2.
    This is the explanation I know. Hope this help.
    Rgs, Peppe

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    #114753

    TCJ
    Member

    If there are four chairs in a room (N=4) and you sit in 1.  How may options, degrees of freedom, are left? N-1.

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    #114754

    Darth
    Participant

    Here is a physical way to demonstrate to a class.  I take a handful of markers and walk around the class.  I tell the first student to take any marker he/she wants.  I tell the second student to take any marker he/she wants….and so on.  When I get to the last marker in my hand, I ask the last student to pick any one he/she wants.  They look confused because they have no choice or “freedom” to pick one since there is only one left.  Therefore, of the 10 markers, how many times did someone have the “freedom” to pick one…..n-1.  Then I go to the flip chart or board and put up 10 numbers.  I give them the average of the 10 numbers.  Now I ask, “Given the average, how many of the 10 numbers do I need to know to be able to know the value of them all?”  Of course, if I know the average and 9 of the numbers, I can calculate the value of the 10th.  That last number has no “freedom” to vary.  Therefore, nine or n-1 of the values have the freedom to take on a value but the tenth does not.  Hope this helps.

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    #114755

    Doc
    Participant

    n people have just finished eating at a Chinese restaurant and the waiter gives them n fortune cookies on a single plate in the center of the table.
    One person picks up the plate, chooses one of the cookies, and then passes the plate to the next person.
    That person chooses one of the remaining cookies, and then passes the plate to the next person.
    How many “choices” are there?
    The answer is (n-1) since the last person to receive the place has no choice. They are forced to take the remaining cookie.
    Its the same when estimating the chi-square statistic. If you know the sum of the n observed data, then the data themselves are “over-defined”. You only need (n-1) data values to be able to determine the value of the last one.
    It is the very same reason we use (n-1) df for the sample mean. Though a different statistic, it also uses the sum of the n observed data.

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    #114757

    Mikel
    Member

    What if I had a friend and they sat in one two? Would it then be n-2?
    Pretty bad explanation.

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    #114764

    Darth
    Participant

    Stan, in my example, if you took two markers I would have hit you up side your head.

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    #114766

    TCJ
    Member

    Stan,
    How helpful was your suggestion?  Your post add no value to the discussion. The person wanted a simple explanation and that is what I provided. 

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    #114767

    PANOS
    Participant

    ****

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    #114768

    Mikel
    Member

    TCJ,
    The value of my post was to tell you that your explanation was not helpful because it is wrong.

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    #114776

    Savage
    Participant

    I just read about this in my stats book last night – so I’ll take a stab at explaining this too. 
    This is a reason for why using n-1 is a good idea, and not so much an in depth discussion of the statistics behind n-1:
    If you have a finite population, it has a mean, mu.  All of the values in the population are used to calculate mu.  However, in most cases we don’t have the whole population, so we must use a sample to calculate the sample mean, x bar.  Because only the values in the sample are used to calculate x bar, it only estimates the population mean, mu.  Also, when calculating the variance of the sample, it only estimates the true variance of the population.  Because, in the sample, the values will vary more closely around the sample mean, naturally, than than they will the population mean, it is wise to divide by n-1 to calculate the sample variance rather than n, as you would for the population variance.  By decreasing the denominator by a value of 1, it builds in a little protection and increases the sample variance. 
    I’m afraid I failed miserably at explaining this but at least I tried!

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    #114777

    Darth
    Participant

    Check this out and you will physically see what’s up with the n-1, especially in the context of the std. dev.
    http://www.uvm.edu/~dhowell/SeeingStatisticsApplets/N-1.html

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    #114778

    Mikel
    Member

    Hey, its beer thirty here at the farm. Go have a shot of tequila and forget this statistical nonsense.

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    #114786

    Darth
    Participant

    Stan, just picked up your email….GREAT NEWS, CONGRATULATIONS.  Just what the world needs.

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    #135140

    Layla
    Participant

    From what I have read from you all and from different people I understood that the N-1 has two meanings: 
    One:  To increase the accuracy of the population estimation by increasing the sample variance through the decreasing of the denominator by one (N-1); Here I have a question: Why should we increase the variance if we know that by increasing the sample size the variance decreases so that when it reaches the population size it decreases to its most? So by any means by having a smaller sample the variance will be larger than the population sample!!  This is like the case  of z-test when we use the sigma divided by root N in order to test the hypothesis where the more the N size is the less the estimated sigma becomes!!!
    Second: Some explained that the idea of degrees of freedom stems from the fact that the last participant doesnt have any degree freedom of his choice of the last observation:  Why do we calculate from the beginning the mean of all sample observations and divide by a N-1 size? It is the same as in population: if we calculate the mean and then choose the last observation we will have no choice as well but we just divide by N. Someone talked about that sometimes researchers use two degrees of freedom: Tell me more about it!
    I am not sure if those two explanations exist at the same time or not!
    I hope that I have made my point clear and I want someone to provide me with explanation to what I have asked!
                                                                                    Layla!

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    #181565

    Cletus
    Participant

    Instead of giving the man a hard time about his grammer why don’t you just answer the question.  Unless of course you don’t know yourself.

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    #181568

    Stevo
    Member

    Cletus,
    You are responding to a post that is 4 years old.  I’m not saying you’re wrong in you observation, but 4 years….
    Stevo

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    #181570

    Darth
    Participant

    Jessica, someone is messing with you and challenging your attempt at bringing civility to the Forum. Of course, you now his/her address and can take appropriate action.

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    #181627

    Craig
    Participant

    What if there are 10 beers to choose, but the last 2 are bud lights? Does the 9th person have a freedom of choice? The 10th certainly doesn’t!
    99 degrees of freedom on the wall, 99 degrees of freedom, you take one down and pass it around, 98 degrees of freedom on the wall.
    98 degrees of freedom on the wall, 98 degrees of freedom, you take one down…….

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    #181634

    Gary Cone
    Participant

    That must be some good coffee you are having at 5:55 in the am.

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