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Thread: Median with "tied values"

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    Median with "tied values"




    Hi all,

    My first stats course was almost 20 years ago. I'm starting again and have come across a calculation for median that I've never seen before and I can't find an example of it on the web, so I want to make sure I'm not missing something.

    If there are more than one raw score at the median, my instructor is suggesting the following. So, if you have 1,2,2,3,4,4,4,4,4,5 that the median is actually 3.7.
    To get this, you multiply the lower real limit of the "tied" interval by (0.5n-#scores below the "tied" value)/number of "tied scores"
    So, in this data set, the fraction is (0.5 x 10 - 4)/5. You then add that to 3.5

    Is this accepted? Thanks!

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    Re: Median with "tied values"

    I... don't understand? If you have more than one raw score at the median why are you trying to change it away from that number? Literally every variation on how to calculate the median I've ever seen would call the median of that dataset 4.

    Why are you adding 3.5? Where does that come from?

    Where did you see this example?

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    Re: Median with "tied values"

    Quote Originally Posted by Dason View Post
    I... don't understand? If you have more than one raw score at the median why are you trying to change it away from that number? Literally every variation on how to calculate the median I've ever seen would call the median of that dataset 4.

    Why are you adding 3.5? Where does that come from?

    Where did you see this example?
    Thanks for the quick reply, and now I don't feel like such an idiot. I agree, I would have said the median is 4. The 3.5 is the lower real limit of the "tied value". This is from my stats book - Statistics for the Behavioural Sciences - Gravetter (8th edition). Does anybody do this in the real world?

    Thanks again.

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    Re: Median with "tied values"

    I've personally never seen that done in practice. Does it say anything about it being a specialized estimator for a certain type of distribution?

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    Re: Median with "tied values"

    yeah. i've never seen it done that way either.

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    Re: Median with "tied values"

    I've taught from the G&W book a couple times and both times I told the students not to mess with that real limit crap. I've never seen it anywhere else and frankly I don't see what the utility of it is.
    “In God we trust. All others must bring data.”
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    Re: Median with "tied values"

    Do you know what the theoretical basis for it is?

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    Re: Median with "tied values"

    I have just search that book on the web.

    The idea from the book is simple - for those integer valued data, we first construct the histogram (bar chart) with bin width 1, and try to draw a line to divided the area of the histogram into half. So you will see such interpolation. And it is just an analog from the continuity correction.

    I do not know whether which estimate is better (although this one is not well known). Many quantiles estimators have different interpolations as well. But I guess both estimators are consistent anyway.

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    Re: Median with "tied values"

    I had a question about bootstrapping with medians and BGM linked me to this post.

    So, out of interest, I bootstrapped your sample 100,000 times to generate 100,000 medians from random samples of your data, each of equal sample size to your data (i.e. it picks random numbers from your original dataset to generate 100,000 sets of 10 data points). The software then works out the mean of the 100,000 medians generated to work out the bias of the median (defined as the difference between the sample median and the mean of the 100,000 medians).

    My results were:

    Median: 4 (as we know)
    Bias: -0.363

    So adding that to the median would give you 3.637

    This is very close to the 3.7 that your instructor suggested. I suspect, therefore, that the interpolation he suggested is trying to yield greater information from the general spread of the data, which the bootstrapping procedure also seems to be doing.

    I may be talking rubbish but it's interesting how similar the results from these two separate techniques are, so there must be something in it....

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    Re: Median with "tied values"

    Normally with an even number of values the median is the mid point between the two middle values. So if you have 1 3 4 5 it would be 3.5. With odd number of values it would be the middle value. For 1 3 5 it would be 3. This is pretty standard stuff
    "Very few theories have been abandoned because they were found to be invalid on the basis of empirical evidence...." Spanos, 1995

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    Re: Median with "tied values"

    Quote Originally Posted by noetsi View Post
    Normally with an even number of values the median is the mid point between the two middle values. So if you have 1 3 4 5 it would be 3.5. With odd number of values it would be the middle value. For 1 3 5 it would be 3. This is pretty standard stuff
    And that view has already been covered in the thread. BGM and SiBorg77 both bring up good points though. I think the question that needs to be asked is "why not do it a different way?". A lot of times we can find better ways for certain situations. Plus we're dealing with a quantile estimator here. Sure the median is a standard quantile that typically gets estimated the same way but there are LOTS of ways to calculate quantiles in general.

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    Re: Median with "tied values"

    We do it the same way for the same reason most things get done in the real world. Because that is the solution, or proper way of doing things, agreed on by the majority of experts. You are a radical at heart Dason
    "Very few theories have been abandoned because they were found to be invalid on the basis of empirical evidence...." Spanos, 1995

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    Re: Median with "tied values"

    Oh really? Can you explain to me why it is the proper way of doing things? Do you know why it's the best way or if there are conditions in which the "accepted" way wouldn't be proper? I'm sorry but I hate the attitude that "this is the way to do it because it's the way everybody does it". I'm not saying the standard way of estimating the median is bad necessarily - but why do you think it should be the only way to do things.

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    Re: Median with "tied values"

    I already did. Because its in the standard text and taught in classes. That makes it reality. In, the non-academic world, reality is socially created not absolute. What authority agrees on is reality. Now we are in my research expertise.
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    Re: Median with "tied values"


    Quote Originally Posted by noetsi View Post
    I already did. Because its in the standard text and taught in classes. That makes it reality. In, the non-academic world, reality is socially created not absolute. What authority agrees on is reality. Now we are in my research expertise.
    And yet we're talking about an academic problem. One in which we can define some sort of goal and can objectively decide which method is better under some set of assumptions when we want to pick between two.

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