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Thread: 10 point scale and central limit theorem

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    10 point scale and central limit theorem




    I want to do a comparative analysis between two means of groups. I want to hear opinions on two things.

    1. I asked participants to rate me something on a 10point scale with only text points of reference the beggining and the end of the scale. E.G. do you like tomatoes, rate it from 1 - 10 when 1 is hate it and 10 is super like it. Values should be equidistant and thus i can treat it as an interval scale.

    2. I am expecting that the data won't be normally distributed due to the nature of the problem. Can i however invoke the central limit theorem in order to say that the sampling distribution is going to be the normal? My sample will have at least 50 cases on each group.

    Based on the literature i should be able to claim both of these things but i want to see what the community has to say.

    Note: my sample is not a random but a volunteer sample. The central limit requires a random sample but since this is an experimental design where participants are assigned randomly to each group and i am interested in the sampling distribution of each group, i am hoping that this won't be a problem.

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    Re: 10 point scale and central limit theorem

    It probably should be good enough but one thing you could do is some bootstrapping to see if the distribution is good enough.

    Code: 
    # Making some fake data
    a <- rpois(50, 6)
    a[a>=10] <- 10
    
    # Takes a random sample the same size as 'a' from 'a'
    # with replacement, then calculates the mean.
    # Does this 10000 times
    samp <- replicate(10000, mean(sample(a, replace = TRUE)))
    # Examine the distribution.  'samp' should approximate
    # the true sampling distribution so we can use this
    # to assess whether the distribution is close to normal.
    hist(samp)
    It sounds like your situation is a little more complex since you have different groups but you should be able to use a similar idea to approximate the sampling distribution for whatever statistic you're looking at.
    I don't have emotions and sometimes that makes me very sad.

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    micdhack (03-31-2012)

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    Re: 10 point scale and central limit theorem

    Quote Originally Posted by Dason View Post
    It probably should be good enough but one thing you could do is some bootstrapping to see if the distribution is good enough.

    Code: 
    # Making some fake data
    a <- rpois(50, 6)
    a[a>=10] <- 10
    
    # Takes a random sample the same size as 'a' from 'a'
    # with replacement, then calculates the mean.
    # Does this 10000 times
    samp <- replicate(10000, mean(sample(a, replace = TRUE)))
    # Examine the distribution.  'samp' should approximate
    # the true sampling distribution so we can use this
    # to assess whether the distribution is close to normal.
    hist(samp)
    It sounds like your situation is a little more complex since you have different groups but you should be able to use a similar idea to approximate the sampling distribution for whatever statistic you're looking at.
    Great idea! I haven't even thought of bootstrapping to assess sampling distribution. I already tested your code in R and works like a charm. Thank you very much! :-)

  5. #4
    josecamoessilva
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    Re: 10 point scale and central limit theorem

    deleted by creator
    Last edited by josecamoessilva; 04-03-2012 at 12:13 AM.

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    Re: 10 point scale and central limit theorem

    Quote Originally Posted by josecamoessilva View Post
    The Central Limit Theorem (there are several CLTs, in fact) is not about the sample distribution, it's about the distribution of the sample MEAN. What these theorems say is that the sample mean converges in distribution to a Normal centered around the population mean.

    The sample distribution should converge to the population distribution, which will be discrete (since the scale is discrete) and bounded (between 1 and 10), two characteristics that are most decidedly not Normal.

    Regards,
    JCS
    Good catch! Except that we were talking about the mean the whole time. What did you think we were talking about? Also note that we were talking about the sampling distribution (which is the distribution of the test statistic - in this case the sample mean) - not the "sample distribution".

    Quote Originally Posted by micdhack View Post
    I want to do a comparative analysis between two means of groups.
    Quote Originally Posted by Dason View Post
    ...
    ...
    samp <- replicate(10000, mean(sample(a, replace = TRUE)))
    ...
    I don't have emotions and sometimes that makes me very sad.

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    Re: 10 point scale and central limit theorem

    By the way I found your tumblr...

    I really think you're the one with the misconception. Or at least you need to learn about the term "sampling distribution" because that's what we were talking about here and it's what most people who talk about the CLT bring up. Maybe I've just had a bad day but finding that tumblr and not even being able to leave a comment to tell you how very wrong YOU actually were was slightly disappointing.
    I don't have emotions and sometimes that makes me very sad.

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    Re: 10 point scale and central limit theorem


    I think i missed the discussion

    I agree with you Dason, but it is a common question that i also receive from my students. They recently asked me what about age? that is not normally distributed and probably positively skewed.

    I think the problem was created because people insisted on testing data and their normal distribution without anyone making a reference to the sampling distribution and that this essential what we are interested in.

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