A question about a cumulative distribution curve

  1. Hello there,

    I have a Figure from a book and some text explaining the figure and I was hoping that somebody could explain/clarify what it means.

    Here is the Figure

    http://dl.dropbox.com/u/54057365/All/pic.JPG

    Here is the text explaining the Figure:

    "The data are divided into ten bins having the same probability on the cumulative density
    function (cdf). The representative driving distances in each bin are selected having the median cumulative distribution in each bin. The selected distance of one-day driving in each bin ranges from 9.56 to 81.4 miles, thus, the one-way trip distances range
    from 4.78 to 40.71 miles."


    I'm looking for clarification on the first line.

    "The data are divided into ten bins having the same probability on the cumulative density function (cdf)"

    Does this mean that the data is divided into 10 bins according the cumulative distribution curve and these bins are 0.1, 0.2 0.3....1.0?

    My question is do they have the "same probability"? I would of though that bin 0.2 would have twice the probability of bin 0.1? And bin 0.3 would have three times the probability etc.

    Am I understanding this correctly?

    Thank you for your help

    John
     
  2. jcsd
  3. mathman

    mathman 6,477
    Science Advisor
    Gold Member

    Bin 0.2 refers to those items between 0.1 and 0.2, so it has the same probability as the first interval.
     
  4. That is what the graph appears to be showing. The bins are more often called "cells"; and the analysis is called "quantile" or percentage of the cdf; I have another thread where I am asking about the same subject. In my design, the cells/bins are of equal probability based on the cdf (erf) of the bell curve.

    It can be done either way but in this case I don't think it is like your last statement; In my algorithm it is not accumulating either; By reading the description of the text, I don't believe this is talking about bins which include previous ones. Notice: The blue line is the cumulative distribution function; and check how the right hand side of the graph's description is the cumulative value up to 1.0 (100%), Then notice how the dotted black vertical lines intersect the CDF at exactly spaced % on the right. So your graph, I suppose, is perdecile; whereas mine is per/cent/ile as I break it up into 1% bins.


    The probability density function also appears on that graph; Notice it isn't a bell curve -- this is a result of the number of samples being small; The bins are 10, so the discrete binomal that would have the same shape is a bernoulli trial with p=0.1 and q=0.9 scaled to the number of data points in the original sample.
    Notice, the smoothness of the graph appears to be "fudged" by using "median" values rather than "mean"; Even with a million values, I still see some non-smooth deviation in my percentile cells.

    See graphs, here:
    Graphs at bottom of binomial cpq thread

    I hope this helps.
    If you go looking for the thread directly, note: I made a typo in the title, it is supposed to be a cumulative p,q bernoulli distribution function (cpq, not cpk...!) that I am treating the cells as.

    If your book happens to have a good estimation formula for the pdf (the purple brown graph), when n of data points is small, I'd appreciate knowing what the name of the approximation is -- so I can look it up. Computing an exact value for the Bernouli trial is time intensive...
    Thanks, Andrew.
     
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