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Error propagation log2

  1. Jun 15, 2015 #1
    Hi there,

    I have a quick question to report some numbers on an experiment. I made measurements of fluorescence in a titration of a chemical. The titrations were 1:2 serial dilutions so I report each fluorescence as a function of the log2 concentration:

    concentration chemical x: 1 , 0.5, 0.25
    reported concentration chemical x (log2): 0, -1,-2
    fluorescence measurement: 5, 10, 25

    I’m interested in reporting the concentrations of a chemical that lead to a specific fluorescence. So, let’s say I want to report when the fluorescence reaches 10, in this case that would be -1.

    So keeping that in mind: When I make error propagation calculations should I use the log2 or the linear measurements?

    Sometimes I need to interpolate to get the reported concentrations. For instance, when I want to get the concentration that leads to fluorescence 15. Does that change anything for the calculations?
     
  2. jcsd
  3. Jun 15, 2015 #2

    DEvens

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    Education Advisor
    Gold Member

    Welcome to the forum.

    It depends on context.

    Generically, you should report what you measured, and the error in your measurement. If you measured x then you should report x and the error in x. If you measured log2 of x, then you should report log2 of x and the error in log2 of x.

    If you are calculating something based on the concentration, then if you are using x in the calculation you should use the uncertainty in x. If you are using log2 of x in the calculation, then you should use the error in log2 of x.

    Does this help?
     
  4. Jun 15, 2015 #3
    Thanks for your reply. So if I understand correctly, as long as I don't mix logs and linear measurements I should be all set for what I'm reporting.

    I'm still a confused though when I want to compute different metrics, like the error propagation or the percent difference ( i.e. , difference/average)

    So for instance if I have the following replicate measurements and I want to compute the percent difference:

    Replicate 1: 0.0625
    Replicate 2: 0.0725

    Replicate 1 in log2: -4
    Replicate 2 in log2: -3.7859


    Percent difference in linear: (0.0725-0.0625) / ( (0.0725+0.0625)/2)=
    =0.1481
    =14.8%

    Percent difference in log2: abs(-4-(-3.7859))/((-4+-3.7859)/2)
    =0.0550
    =5.5%

    Why are these two numbers different when I'm reporting a percentage? Shouldn't this percentage be the same since I'm only changing the base but not the values of the measurements?

    Thanks again!
     
  5. Jun 15, 2015 #4

    WWGD

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    Science Advisor
    Gold Member

    Non-constant functions do not, in general, preserve ratios, e.g., : ## \sqrt {\frac {81}{4}}=4.5 \neq \frac {81}{4}=20.25 ##. I think this has to see with the derivative of these functions not being constant, i.e., only when the function ##f ##, as below, is linear.

    Basically, few functions preserve ratios, i.e., few functions satisfy:

    ## \frac {a}{b}=\frac {f(a)}{f(b)} ####
     
    Last edited: Jun 15, 2015
  6. Jun 16, 2015 #5
    Oh yes, this makes a lot of sense. Thank you WWGD
     
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