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Error analysis

  1. Apr 5, 2005 #1
    For a function f(x,y): The error is:

    [tex] \Delta f(x,y) = \sqrt{(\frac{df}{dx})^2+(\frac{df}{dy})^2} [/tex]

    Is this a form of the approximation in algebraic error determination:

    [tex] \Delta f(x,y) = \sqrt{f(x+\Delta x,y) + f(x,y+\Delta y)} [/tex]?

    Now I was trying to do this in my bio lab for genetics and probability distribution. We had 27 samples, predicted 75% (20.25) to be purple and 25% (6.75) to be white. We observed that 23 were purple and 4 were white, so to calculate the deviation I used the above function and got:

    [tex] \frac{df}{dx} = \frac{23-20.25}{20.25}, \frac{df}{dy} = \frac{4-6.75}{6.75} [/tex]

    *For some reason latex doesnt want to show it, but I plugged in my numbers into the first equation above.

    So the observed sample was 18.44% off?

    They gave us an equation to find the deviation of a population from the expected value which is similar to the one above:

    [tex] \chi^2 = {\frac{(Obs_p - Exp_p)}{Exp_p}^2+\frac{(Obs_w - Exp_w)}{Exp_w}^2} [/tex]

    [tex] \chi^2 = {\frac{(2.75)}{20.25}^2+\frac{(2.75)}{6.75}^2} = 1.494[/tex]

    It claims that "If the value for chi squared is less than or equal to 3.841, then your sample is within the expected range."

    Is this a standard deviation?

    This is pretty similar to the first equation, except we're squaring the numerator instead of the whole fraction. Did I get my error equation wrong, or are these truly diferent and unlinked?
    Last edited: Apr 5, 2005
  2. jcsd
  3. Apr 5, 2005 #2
    [tex] \Delta f(x,y) = \sqrt{\left(\frac{\partial f}{\partial x}\right)^2+\left(\frac{\partial f}{\partial y}\right)^2} [/tex]

    isn't quite right. It's

    [tex] \Delta f(x,y) = \sqrt{\left(\frac{\partial f}{\partial x}\Delta x\right)^2+\left(\frac{\partial f}{\partial y}\Delta y\right)^2}[/tex]

    this error formula is indeed just the standard deviation of [itex]f[/itex], assuming [itex]\Delta x[/itex] and [itex]\Delta y[/itex] are standard deviations of the respective variables, and that [itex]x[/itex] and [itex]y[/itex] are independent.
    Last edited: Apr 5, 2005
  4. Apr 5, 2005 #3
    Your corrected formula is the standard deviation of f? Or are you talking about my Chi formula? Are they the same?
  5. Apr 5, 2005 #4
    The equation I posted gives the standard deviation of [itex]f[/itex]. Your chi formula is an example of the chi square test for goodness of fit of experimental data to a certainly distribution (the one that the expected values come from). The best online resource that I found to explain it (and it's not very good...) is here: http://www.stat.yale.edu/Courses/1997-98/101/chigf.htm
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