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degrees of freedom

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  • #29542

    PB
    Participant

    Need some help…  I have been perpetually confused by the concept of degrees of freedom and the best way to explain it.  Perhaps I am not alone.  I’ve search for decent explanations that could be passed on to GBs but have never found one that worked.  Given that “n-1” and other df references appears everywhere…I know I need to find an acceptable explanation of how and why it’s important.  Taking it as a given just doesn’t cut it.  Would appreciate some input.  Thanks
    PB
     

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    #75952

    Breizh
    Participant

    PB-
    You could find entire chapters devoted to the topic, but the simplest and best explanation for GBs that I have seen is this: degrees of freedom are the equivalent of currency in statistics – you earn a degree of freedom for every data point you collect, and you spend a degree of freedom for each parameter you estimate. Since you ususally need to spend 1 just to calcultae the mean, you then are left with n-1 (total data points “n” – 1 spent on calculating the mean).

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    #75955

    James A.
    Participant

    PB,
    Mathematician I’m not, but this works OK for me:-
    You are a traveller in search of knowledge, each day you need to move from one place to another.  Luckily for you, it takes no time at all to travel the distance, so you only ever have to decide where you want be, just to be there.
    Each day you start with a list of ‘n’ options of where you can go.  This includes where you are now, but being adventurous, you never do that, so although it’s important to know where you are – you only have ‘n-1’ options available out of the total population of data.
    It’s importance is in the value of knowing what happens in the whole population of data that is being considered.
    Does this help – or merely confuse even further?  Is there a mathematical type out there who can improve on this?
    Regards
    James A.

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    #75956

    James A
    Participant

    Breizh,
    Sorry, just read your posting – it works better than mine.
    James A

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    #75979

    PB
    Participant

    Breizh,
    I like it! thanks for the help – I will use this in an upcoming GB class. Thanks.
    PB

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    #76143

    Lee Ronald
    Participant

    Another way of thinking about degrees of freedom is to image that you ask 100 people to pick a number between -infinity and infinity so that the mean of all the numbers is exactly zero.  99 of the people can pick any number they want, but the 100th person HAS to pick a specific number.  He is not “free” to pick what he wants.  Thus the degrees of freedom to determine the mean for 100 samples is n-1 or 99.
    That’s how I learn to grasp it.

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    #76156

    Ronny
    Member

    The degrees of freedom are related to the sum of squares ( SS)
    SS= sum[ (yi-ybar)^2] when i is from 1 to n
     Since,  sum[yi – ybar] = 0 when i is from 1 to n , consists the sum of squares of n elements. These elements are not all independent. As a result, only n-1 elements are independent, implying that SS has n-1 degrees of freedom. That’s the mathemathical explanation.
     

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    #76174

    Gabilan
    Participant

    Thanks all because this is one of the concepts that I hadn’t clear for myself and I always was afraid to teach DOE ’cause of  a misunderstanding of the concepts involved.

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    #89360

    Akhilesh Gulati
    Participant

    Perhaps a physical depiction will help clarify this
    concept.
    Take a piece of rope and hold the two ends.
    Now twirl it (like you did in high/junior school
    gym) so that it starts moving in the middle (like a
    skipping rope). You had two nodes and when
    you twirl it you find that the rope is able to move
    at one point only. It has the freedom to move at
    one point only! It has two nodes and one
    degree of freedom. Now ask someone to hold it
    at the center point thus creating another node;
    you now have 3 nodes. If you try twirling it now,
    you find that the rope can be twirled at 2 points.
    Three nodes, two degrees of freedom!! This might be a real basic explanation and
    mathematicians might disagree with me, but it
    beats any other way of explaining Degrees of
    Freedom.I was also asked a question along these lines
    at one of my training sessions, while we were
    looking at Regression. n-1 crops up there as
    well. Again, as I explained the basis of
    Regression, drawing physical lines and why we
    use ‘distance squared’ I showed them how we
    draw one line between two points, two lines
    between three points, four lines between five
    points . . . and the point seemed to hit home.I hope this helps. I too am always on the
    lookout for better ways to explain concepts.

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    #90343

    Liz Cavanaugh
    Participant

    Thank you Akhilesh Gulati and others for your explanations on degrees of freedom.  My question now turns to “what do you do with the number?”  Is a high number better than a low number?  Do I look for a range that it should be within?  Is it to be compared to other values for the test? 
    Thank you.

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    #90925

    kosatka
    Participant

    ok..i have this kind of quetion: you have got double spherical pendulum. How many degrees of freedom it has got? I think that four…but I am not sure…
    can you tell me and write a way you find it out?
    Nina

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    #93198

    Akhilesh Gulati
    Participant

    You could think of Degrees of Freedom as the amount of
    information available for the estimation of variability. For
    example, the degrees of freedom to estimate variability from
    one observation would be zero, i.e., it is impossible to
    estimate variability. But for each observation after the first,
    you get one addtional degree of freedom to estimate variance.
    Hope this helps.

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    #93403

    Shaun X. Li
    Member

    Each data point adds one DOF. Each constrain (or ‘relationship’ or ‘equation’ such as ybar=sum(yi)) reduce one DOF.  Find the data points number, then minus the constrains number, you get the number of freedom.
    e.g.: for SSt, it contains all n data points, with one constrain (ybar), therefore, DOF = n-1
    Constrains and combination of constrains always hold one DOF.
    e.g.:  (ybar+std.dev) has 1 DOF.
     
    That’s my understanding.
     
     
     

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    #93409

    Mark Almeter
    Participant

    The best explanation I’ve heard for understanding degrees of freedom in various statistical calculations is as follows:
    Degrees of freedom of  n – 1 is required when taking a sample from a population because when taking the limited size sample, you have only a very slight chance of picking the extreme data values of the population.  In order to accomodate for this, you subtract the value of 1 (n – 1) in order to inflate the std deviation and make the standard deviation calulation more adequately represent the parent population. 
    Notice that in the calculation of the population std deviation, the sum of squares is divided by the value N and not n – 1.  This is because all values in the poplulation (including extreme values) are taken into account.
     
    Just my 2 cents…..
     
    Mark

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    #93410

    Andy Urquhart
    Participant

    Another explanation …
    If I have five numbers and I know the average; I’m only free to choose four numbers.

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    #99888

    Whitehurst
    Participant

    I can calculate a mean, variance and so on in statistics. I can develop a process to determine the mode and median, i.e. I can program that in a computer and get the answer. How would I write a program to determine the degrees of freedom? It seems impossible and thus seems arbitrary….

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    #99902

    In the example I mentioned, to calculate the degrees of freedom just subtract 1 from the sample size. I believe the justification is self-evident. If that doesn’t satisfy your curiousity, then I apologise for my limitations.

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    #102094

    Bobbie
    Participant

    I have an excellent BB Coach that is able to explain things to me so that I can understand them.  The way he explained Degrees of Freedom to me is as follows:   You are given 4 checkers and you are told to place a checker in each corner of the board.  For the first placement you have 4 choices (degrees of freedom); for the second you have 3, for the third placement you only have 2 degrees of freedom and for the 4th you have none so this works out.  n-1   if n is 4, then you only have 3 degrees of freedom.
     
    Hope this helps.

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    #113969

    Tesfu Isak
    Member

    Dear
    Can you please tell me the exact meannig of dgree of freedom in statistical analysis I know that it is n-1but I want to know the exact in terpretation and how it infulence ,coorelate and affect to the data ?
    Thank u
     

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    #114131

    Lin
    Participant

    Hi, obviously this analogy makes sense in some way, but the real meaning of variance is some kind of “averaging concept” by which we calculate the average variation of a population. Especially, when we use sample mean to estimate the population mean, we use n-1 instead of n as a denomenator; but when we calculate population mean by using up every subject in a population, we will use N instead of n (sample size) – 1. If we talk about degree of freedom in this sense, what is the proper explanation of it? it is the essence of this term.

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    #114136

    Peppe
    Participant

    If you have  A + B = 30 and you define A = 10  then you can have just one degree of fredom (n-1), because B can be only 20.
    If you have A + B + C =30 and A is always =10,  now you have 2 degree of freedom (B, C) because B and C can be many different values to satisfy the equation.
    And so on …
    Apologies for the simplicity.
    Rgs, Peppe

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    #117167

    Jeff
    Participant

    Bobbie, that is the best description of degrees of freedom that I have seen on this board.  My BB coach taught the same lucid way with a different example.  It is also important to note that the neccesity to subtract 1 from n becomes negligible as n increases in size where n > 30.

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    #120484

    Orfu
    Participant

    I like your explanation Andy. It’s simple and seems solid. Tank you.

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    #120495

    Thanks Orfu …  I first heard this explanation from a statistician who taught a Taguchi Methods course.
    Dr. Taguchi believes that there is a similarity between Anova and Fourier Analysis – but, as yet, I can’t see it!!!
    Cheers,
    Andy

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    #120498

    Lomax
    Participant

    Another simple way to understand dof is :
    If one were to break a stick into 3 pieces of different lengths… he/she would have only 2 chances to do that…(in order words the degree of freedom to determine the outcome(length of 3 pieces) is governed only by 2 chances

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    #120709

    Johncox
    Participant

    It helped me!! thank you

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    #122002

    Brian Ford
    Participant

    How can i use Kurtosis in a statement?

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    #122021

    A.S.
    Participant

    It is obvious that Degrees of freedom is significant only when we have small sample size.If we increase the no of samples,there will be less impact on our standard deviation calculation due to degrees of freedom.
    Is there any guidelines / thumb rule when we have to consider the dF.some thing like sample vs population ratio etc

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    #122097

    Z
    Member

    i hv not heard of any relation between the sample and population to consider DF.
    However, as mentioned earlier, when n>30, the DF do not make a difference in the calculation of Std Dev. Try to plot
    (1/(n-1)) – (1/n), and you will see this difference tends to ZERO (0.00064) at n=30.
    This difference is the difference in the variance of the population which is calculated using N and not (N-1), and the variance estimated from the sample using (n-1).
    Hence, the DF will not be important for calculating the Std Dev, but will be more important in further tools. Higher DF would always be better as this will give you enough liberty to find the effects of blocks etc, which then do not need to be confounded with some interaction terms, due to scarcity of data points and hence, low DF.

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    #122099

    A.S.
    Participant

    Please let me have more light on this statement
    “but will be more important in further tools”.
    Please let me have clarity on significance of DF.Where it is more important apart from basic S.D calculation
     

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    #122106

    MAK
    Participant

    Do we have proof mentioned in any book what you are saying.
    Regards,
     

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    #127963

    sangal v k
    Member

    dear sir
    i am woring in the field of process analysis. please send me some literature about this.
    thaknig you
    v k sangal
    Sr Lect.
    Dr KNMIET, Modinagar, INDIA
     

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    #127969

    Nitish
    Participant

    Dear Mr.Sangal,
    Quality Control describes numerous methods for monitoring the quality of a production process. However, once a process is under control the question arises, “to what extent does the long-term performance of the process comply with engineering requirements or managerial goals?” For example, to return to our piston ring example, how many of the piston rings that we are using fall within the design specification limits? In more general terms, the question is, “how capable is our process (or supplier) in terms of producing items within the specification limits?” Most of the procedures and indices recently introduced to the US by Ford Motor Company (Kane, 1986). They allow us to summarize the process capability in terms of meaningful percentages and indices.
    The computation and interpretation of process capability indices will first be discussed for the normal distribution case. If the distribution of the quality characteristic of interest does not follow the normal distribution, modified capability indices can be computed based on the percentiles of a fitted non-normal distribution. Process Analysis  allows you to fit various specific non-normal distributions (e.g., Weibull, log-normal, Beta, Gamma, etc.) as well as general non-normal distributions by moments.
    Order of business. Note that it makes little sense to examine the process capability if the process is not in control. If the means of successively taken samples fluctuate widely, or are clearly off the target specification, then those quality problems should be addressed first. Therefore, the first step toward a high-quality process is to bring the process under control, using the charting techniques available in vaious quality control modules.
    Thank you
    Nitish

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    #128136

    SixSigmaGuy
    Participant

    I thought this was a GREAT discussion!!  But for me, it just perpetuated my confusion. :-(.  The examples were very helpful, but I still don’t understand how to explain what the purpose of DFs is and WHY it exists.  Could I say it was a bias applied to my results when I have small sample sizes to account for error in the estimations?
    The question came up in my GB classes and I need an answer that says why we have DFs and what the words “Degree” and “Freedom” have to do with it.
    While we are at it, can someone explain the 1.5 sigma shift?  Just kidding. :-)
    Thanks!!

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    #128878

    Dickson
    Participant

    May I know what is the meaning of penalities of degree of freedom?

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    #134554

    Jonathan
    Participant

    I wonder if you can help me with the Degrees of Freedom tables that appear in my SSBB memory jogger and other stat books. At the end of the SSBB memory jogger there are 3 tables that refer to DOF. The tables are “Probability Points of t Distribution with v (???) Degrees of Freedom”, “Ordinates of t Distribution with v Degrees of Freedom”, and ” X2 Distribution with v Degrees of Freedom”. I understand that the reference on the x axis reference to the confident level (95% ….). Why do people have tables for degrees of freedom? Don’t you only need (n-1) for DOF as the standard?

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    #134556

    Craig
    Participant

    Degrees of freedom computations are straightforward. Whether you are doing a t-test, one way ANOVA, or calculating degrees of freedom for an interaction term. The reason you see them on the tables is to allow you to determine the critical vaue of the test statistic, given your sample size and required confidence level. Most stats training these days will only teach you to look at p-values, and they skip over this critical aspect. Try doing your hypothesis test without looking at the p-value and see if you make the right decision. (Reject Ho or Fail to reject Ho based on comparing critical F to calculated F)

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    #134557

    Darth
    Participant

    The different distributions use different d.f. to define them.  n-1 is one of them.  There is also one that is (rows-1) and (columns -1) and others.

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    #134558

    Mr IAM
    Participant

    No, Df  is used to determine the critical value (table value) for the tests, which you then compare to the calculated value.
    The tables do not give you Df, they give you the critical values based on the Df for the given test.  If you do not perform tests by hand – you don’t really need to use these tables.  Your software will give you a P-Value that is calculated based partially off of numbers in these tables.
     

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    #148256

    narupv
    Participant

    thank you LIN your message helped me alot to come out of confussion

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    #154206

    Dianna
    Participant

    not really sure but i neeed some help to  plz post a anwser for me if u click on this link

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    #154207

    qualitycolorado
    Participant

    Dianna,Thanks for your posting. Prior to your posting, the most recent reply in this thread was way back in March of 2006, so it is not clear what you need help with — please clarify.
    Best regards,
    QualityColorado

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    #163742

    Fredo
    Participant

    So the definitions help me understand the concept here, but can anyone outline a real world example where this concept is necessary for a black belt? Where would you need to identify how many df in a practical problem? Thank you!

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    #164039

    madiha
    Participant

    i understand the concept of degree of freedom, but please explain it to me further by answering these questions:
    we have a set of six numbers by rolling a die six times: 2, 3, 3, 4, 6, 6. This set of numbers (let’s call them ‘Set A’) has how many degrees of freedom?
    i think the answer should be 5, as we have no of values, i.e. n=6 so df=n-1=5
    if we converts these six numbers to deviations from the mean. The resulting set of six deviation scores (let’s call them ‘Set B’) has how many degrees of freedom? well i am confused about it, please help.
    the say we abtain another set of six numbers by rolling a die six times. This set of numbers (let’s call them ‘Set C’) have the same sum as the number is Set A. Explain how many degrees of freedom there are in Set C? Again i think the degree of freedom for this should be 5. please let me knowif i am wrong also explain, what should be degree of freedom for deviation scores

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    #164805

    Tobias
    Member

    A very good explanation, also on why it is called degrees of “freedom” can be found athttp://web.archive.org/web/20060706151302/http://seamonkey.ed.asu.edu/~alex/computer/sas/df.htmlRegards,
    Tobias

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    #172229

    TonyK
    Member

    I’m grateful to all for their explanations, but if truth be told I’d have to say that on a deeper level not one person on this thread really understands the concept of “degrees of freedom.” Here’s the best explanation I could find, so far: http://courses.ncssm.edu/math/Stat_Inst/PDFS/DFWalker.pdf

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