Some Fractiles of Normal Medians

Discussion in 'Scientific Statistics Math' started by Luis A. Afonso, Dec 10, 2010.

  1. Some Fractiles of Normal Medians


    ___TABLE

    ________0.600__0.750__0.800
    ________0.950__0.975__0.990__0.995

    _n=7____0.116__0.309__0.385__
    ________0.754__0.900__1.070__1.186
    ___9____0.103__0.275__0.343__
    ________0.670__0.799__0.951__1.055
    __11____0.094__0.250__0.311__
    ________0.609__0.727__0.863__0.957
    __13____0.086__0.230__0.287__
    ________0.562__0.671__0.797__0.882
    __15____0.081__0.215__0.268__
    ________0.524__0.626__0.743__0.824
    __17____0.076__0.203__0.253__
    ________0.494__0.589__0.698__0.774


    REM "MED-G"
    CLS : PRINT : PRINT " ***** MED-G ***** ";
    PRINT " FRACTILES OF NORMAL MEDIANS "
    INPUT " all= "; all
    pi = 4 * ATN(1)
    DIM x(101), y(101), z(101), W(8001)
    FOR n = 7 TO 18 STEP 2: COLOR 7
    FOR j = 1 TO all
    RANDOMIZE TIMER
    LOCATE 7, 50
    PRINT USING "##########"; all - j
    COLOR 14
    PRINT " .600 .750 .800";
    PRINT " .950 .975 .990 .995 "
    PRINT " 0.253 0.674 0.842";
    PRINT " 1.645 1.960 2.326 2.576 "
    REM
    REM
    locc = INT(n / 2) + 1: REM THE MEDIAN location
    FOR i = 1 TO n
    aa = SQR(-2 * LOG(RND))
    x(i) = aa * 1 * COS(2 * pi * RND)
    y(i) = x(i)
    NEXT i
    REM
    FOR t = 1 TO n: v = x(t): c = 1
    FOR y = 1 TO n
    IF y(y) < v THEN c = c + 1
    NEXT y: z(c) = v: NEXT t
    REM
    wi = z(locc) + 4: wii = INT(1000 * wi +.5)
    IF wii < 0 THEN wii = 0
    IF wii > 8000 THEN wii = 8000
    W(wii) = W(wii) + 1 / all
    NEXT j
    REM
    v(1) = .6: v(2) = .75: v(3) = .8
    v(4) = .95: v(5) = .975: v(6) = .99: v(7) = .995
    FOR vi = 1 TO 7: v = v(vi): s = 0
    FOR i = 0 TO 8000
    s = s + W(i)
    IF s > v THEN GOTO 44
    NEXT i
    44 LOCATE 5 + n, -9 + 10 * vi
    PRINT USING " ##.### "; i / 1000 - 4;
    NEXT vi
    LOCATE 5 + n, 1: PRINT USING "###"; n
    FOR i = 0 TO 8000: W(i) = 0: NEXT i
    NEXT n
    END
     
    Luis A. Afonso, Dec 10, 2010
    #1
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  2. Luis A. Afonso

    Henry Guest

    I think "quantiles" is more common than "fractiles". Some of those
    numbers look slightly out, from the following using the R project:
    + qnorm(qbeta(q, halfup(s), halfup(s)))}
    0.6 0.75 0.8
    0.95 0.975 0.99 0.995
    7 0.11582476 0.3085066 0.3850679
    0.7543432 0.9000322 1.0701745 1.1865441
    9 0.10295407 0.2742042 0.3422363
    0.6701922 0.7994716 0.9503583 1.0534988
    11 0.09359376 0.2492604 0.3110927
    0.6090371 0.7264104 0.8633348 0.9568868
    13 0.08639363 0.2300752 0.2871405
    0.5620264 0.6702617 0.7964773 0.8826799
    17 0.07589013 0.2020912 0.2522059
    0.4935003 0.5884406 0.6990905 0.7746194
     
    Henry, Dec 13, 2010
    #2
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  3. Thank you very much Henry!
    Can you say me want´s the procedure to
    get analytically the quantiles (fractiles)?
    Thanks in advance!
     
    Luis A. Afonso, Dec 13, 2010
    #3
  4. Luis A. Afonso

    Henry Guest

    As an order statistic, the sample median has a Beta distribution, so you
    need to take the suitable quantile of a Beta distribution to get a value
    while you can then apply to the distribution you are taking the sample
    from (in this case normal)

    So for example with a sample size of 7, the median has to a distribution
    where 3 values are lower and 3 higher, so proportional to x^3*(1-x)^3,
    i.e. a distribution ~beta(x;4,4). The 0.600 quantile of a beta(x;4,4)
    distribution is about 0.5461, while the 0.5461 quantile of a standard
    normal distribution is about 0.1158.

    Looking at my previous post, I omitted the line for a sample size
    of 15. It should have been:
    15 0.08063362 0.2147286 0.2679817
    0.5244394 0.6253785 0.7430494 0.8233912
     
    Henry, Dec 14, 2010
    #4
  5. Thank you a lot, Henry: my theoretical education in Mathematics of Statistics has flaws, I must admit, in this case the Beta Distribution. Your explanation is clear and the example enlighten. I was a Physicist very early in my professional carrier *seduced* by Data Mining, so I often fall into this kind of situations as the present one.
    Thank you very much by your kind concern. My English, too, is barbaric, Portuguese is a Latin language, which’s very, very different compared with Germanic-English ones. All constrains to an easy communication . . .

    *Tchau*

    Luis
     
    Luis A. Afonso, Dec 14, 2010
    #5
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