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Interval Data

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  • #48362

    Dillon
    Participant

    By definition, is Interval data always continuous or can it be discrete too?  For example, Interval data is defined as where the interval between measurements has meaning.  Therefore, one could argue that the number of children is Interval data since obviously the difference between 2 and 4 children is the same as between 6 and 8, and if one had 4 children, one would have twice as much as another with 2 children.  However, “number of children” is discrete since one cannot have 2.4 children.  I believe Interval data can be continuous too, but a friend does not believe so.  Who owes whom one month’s salary!?

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    #162793

    annon
    Participant

    The type of data you are dealing with remains constant, but attribute data can be analyzed using statisitical tools designed for continuous data sets with good results if several conditions exist. 
    For example, you will hear debates about the validity of treating Likert scales (5 or 10 usually) as continous data if certain conditions are present in the responses and in the scale design. 
    My understanding is that it is technically possible and yields fine analytic results occasionally, but the majority of situations don`t lend themselves to a successful application.
    But verify this opinion. 

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    #162794

    Regression
    Participant

    The trick is as follows: Stevens (1947) defined measurement as “the assignment of numbers to objects according to a rule”. Thus, as long as you can assign a number according to a rule, this is called a measurement.
    The scale levels are defined by the mathematical operations that you can perform on the numbers. So, the key question is: “What type of mathematical operation can you perform on the number that you assign to an object at the interval level of measurement?”. With interval data you can add and subtract numbers and conduct multiplications and divisions of ratios of differences between your numbers. (100 degrees + 10 degress) = 105 degrees. (100 degrees – 10 degrees)/2 = 95 degrees. The difference between 95 degrees and 100 degrees is the same as the difference between 100 degrees and 105 degrees. You cannot perform this mathematical operation on the number of children: (1 child + 2 children)/2 = 1.5 children?
    So to your question, # of children is not an interval scale. It is count data. Your category is: Child vs. No-child. This is what makes it discrete.

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    #162795

    Dr. Scott
    Participant

    Doug,
    You jumped pages in the book here I think. The “number” of children is Poisson distributed distribution, not continuous or interval.
    Now, if you are talking about a survey scale, then while the data is truly not continuous, it can be assumed as such due to the central limit theorem. This is why many have gone from Likert’s original 5 point scale to a 7 point scale in an attempt to make it more continuous.
    I have no idea if this addresses your question, but feel free to ask me again if not.
    Cordially,
    Dr. Scott
     

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    #162822

    Dillon
    Participant

    I’m not sure I agree with your argument.  However, first let me restate the question: Can interval data be discrete?  I agree the number of children is discrete—Poisson distribution and not binominal like you suggest—child vs. no child.  Also, one can have an average of count data.  The Census Bureau reports the average number of children per household—2.2 children per household.  I agree, one will NEVER find a household with 2.2 children, but that’s not to say the average cannot be computed because, as I’m suggesting, it is discrete, but also interval data.  An average is also computed with the Poisson distribution to determine lambda.  No one would ever dispute the number of deaths due to a horse kick in the Prussian army could every be anything other than an integer when Poisson developed this distribution; however, to determine the probability of an event(s), one needs to compute the average to determine lambda.  So my question still is: Can interval data be discrete?

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    #162826

    Regression
    Participant

    Yes, interval data can be discrete, but discrete data cannot be interval. You can always go to a lower scale level from a higher scale level. You can classify a rank: High, Mid, Low as discrete categories and as a rank order. But you cannot rank Los Angeles, Miami, and New York unless you have an underlying attribute that you measure. The cities themselves are discrete. But you can rank them based on crime rates in a certain order.

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    #162877

    Regression
    Participant

    Doug,
    To fully answer your question: There are two positions in regards to the treatment of data as interval and/or discrete for purposes of statistical testing. This has caused a lot of confusion and continues to do so:
    The one position is that of Steven which I outlined before. In essence, the mathematical operation that can be meaningfully performed on the numbers determines the scale level. The scale level then determines if a parametric or non-parametric test is to be performed. This has become the official position in psychometric theory (see any text book).
    The alternative position was proposed by Lord. He states that the “numbers don’t care where they come from”. For purposes of statistical testing, the scaling level is irrelevant. Thus, discrete level data can be used in parametric statistical testing.
    So to your points: You don’t owe each other money. You are just proponents of two different theories about the usage of numbers in parametric statistical testing. The discussion about this topic was heated at one point, but has pretty much calmed down. Those who prefer Steven will continue to use the scale level as the determining factor in running parametric statistical test. Those who prefer Lord will simply ignore the scale level because “the number does not care where it comes from”. … you can also hand both paychecks to me :-).

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    #163366

    Deanb
    Participant

    Doug,Anytime I see extended dialog on the use of statistical methods on Likert scale data I get concerned the forest is being missed for the trees.If the Likert survey inherently contains a robust relationship, then any of the statistical methods posted above will reveal this relationship. If the relationship is too sensitive to the method, then it is not robust enough to consider IMHO. Sharper statistical methods, in this instance, simply are not capable of turning aggregated general relationships into specific and actionable root causes any more than inspection can turn a Buick into a Corvette.This is why survey data of this type should be used as a general pointer for targeting deeper investigation, and not used conclusively by itself. The important issue is not the absolute correctness of the statistical method, but how you utilize the knowledge of robust survey relationships to dig deeper. Surveys are like SPC. They can tell you something is happening in aggregate, but you usually must go walking around to find the actionable whys.Good luck.

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    #163368

    Brandon
    Participant

    Deanb – what a shame that would be to turn a nice, big, quiet Buick into a Corvette. Thank goodness that can’t happen.

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    #163377

    Deanb
    Participant

    Brandon. I am a Buick man myself too, but in this six sigma business there is always a hot dog or two who threaten to turn our love boats into speed boats. there is no substitute for love.Cheers

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