Error Propagation with Log2 Concentration: Fluorescence Measurement

[PhysicsInquirer]

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?

 

[DEvens]

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?

 

[PhysicsInquirer]

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: -3.7859Percent 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!

 

[WWGD]

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:

$\frac {a}{b}=\frac {f(a)}{f(b)}$##

 

[PhysicsInquirer]

Oh yes, this makes a lot of sense. Thank you WWGD