Propagation of error: exponents

In summary, the conversation discusses the propagation of error of exponents and the standard method of representing uncertainty for products. The speaker explains that the quadrature method only applies to uncertainties that are independent of each other, and gives examples to illustrate this. They also recommend a book for further understanding of experimental uncertainty.
  • #1
AndrewBworth
3
1
Hi all. I have been trying to understand propagation of error of exponents. Most an. Chem textbooks I see say y = a^x, sy/y = (sa/a)*x. But say y = a*b, then sy/y = ((sa/a)^2 + (sb/b)^2)^.5 . if a = b then sy/y= (2*(sa/a)^2)^.5 = 2^.5*abs(sa/a). This shows the rule y=a^x, sy/y= x^.5*abs(sa/a).
 
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  • #2
I think I understand your question. You are asking why the standard way of representing uncertainty in a product (the so-called quadrature method of addition) does not apply to powers.

The most straightforward way to explain this that I've seen may satisfy you. Recall that the quadrature method only applies to uncertainties that are independent of each other. Take a simple example of measuring two sides of a rectangle - the uncertainty in your first measurement is independent of the uncertainty of the second measurement. It is therefore preferable to state the uncertainty in the area of the rectangle using the quadrature method.

Contrast this with measuring only one side of a square and calculating its area. In this circumstance the calculation involves multiplying the same value together and the uncertainties are clearly no longer independent; the quadrature method is not justifiable in this circumstance.
 
  • #3
Thank you! I guess I have a thing or two to learn about statistics.
 
  • #4
You're welcome. One book that I found extremely useful for the basics of dealing with experimental uncertainty is John R. Taylor's An Introduction to Error Analysis. I would definitely recommend checking it out. I ended up buying a copy.
 
  • #5
We also have a statistics subforum for this kind of questions.
 

1. What is the propagation of error in terms of exponents?

The propagation of error in terms of exponents is a method used to calculate the uncertainty or error in a quantity that is calculated using exponents. It takes into account the uncertainties in the measured values that are used in the calculation.

2. How is the propagation of error calculated for exponents?

The propagation of error for exponents is calculated using the formula Δy/y = nΔx/x, where Δy is the uncertainty in the calculated value, y is the calculated value, n is the exponent, Δx is the uncertainty in the measured value, and x is the measured value.

3. What are the assumptions made when using the propagation of error for exponents?

There are three main assumptions made when using the propagation of error for exponents: (1) the uncertainties in the measured values are independent, (2) the uncertainties are normally distributed, and (3) the uncertainties are small compared to the measured values.

4. Can the propagation of error for exponents be applied to any type of exponent?

Yes, the propagation of error for exponents can be applied to any type of exponent, whether it is a positive, negative, or fractional exponent.

5. How does the propagation of error for exponents affect the final calculated value?

The propagation of error for exponents can increase the uncertainty in the final calculated value. This is because the uncertainties in the measured values are being multiplied by the exponent, which can magnify the effect of the uncertainties on the final result.

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