Understanding the Proper Uncertainty Formula to Use

In summary, the article discusses the importance of selecting the correct uncertainty formula in scientific measurements. It outlines various types of uncertainties, including systematic and random errors, and emphasizes the need to apply appropriate statistical methods based on the nature of the data. The text also highlights common pitfalls in calculating uncertainty and provides guidance on how to ensure accurate and reliable results by using the right formulas and approaches tailored to specific experimental conditions.
  • #1
TRB8985
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Homework Statement
A rectangle is 7.60 +/- 0.01 cm long and 1.90 +/- cm wide. Find the area of the rectangle and the uncertainty in the area.
Relevant Equations
See post for provided formulas.
Good evening,

I'm running into a little confusion on the second part of this problem due to finding two different formulas for calculating the uncertainty in multiplied quantities.

The way that I was taught was something like this.

If ##z = x \cdot y##, then: $$\dfrac{\delta z}{z} = \dfrac{\delta x}{x} + \dfrac{\delta y}{y}$$ Where ##\delta x##, ##\delta y##, and ##\delta z## represent the absolute uncertainties in ##x##, ##y##, and ##z## respectively.

With this approach, the uncertainty in the average area of this rectangle (14.4 cm) would be: $$\delta A = (14.4\: \text{cm}) \Big (\dfrac{0.01\: \text{cm}}{7.90\: \text{cm}} + \dfrac{0.01\:\text{cm}}{1.90\:\text{cm}} \Big) = 0.1\:\text{cm} $$ This matches what's listed in my textbook as the solution.

However, when looking a bit more into uncertainties online, I found this:

Statistics tells us that if the uncertainties are independent of one another, the uncertainty in a product is obtained by: $$\delta z = |z| \sqrt{\Big(\dfrac{\delta x}{x} \Big)^2 + \Big(\dfrac{\delta y}{y} \Big)^2 } $$ It's certainly true that the length and width of this rectangle are independent, and if I use the expression above, I still end up with the same answer. So then.. why the need for two formulas?
 
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  • #2
Indeed the formula to use depends on whether you consider the errors independent or dependent. In this case, the side lengths are presumably from two independent length measurements (although there could technically be a systematic part as well such as using the same ruler, which may have a slightly faulty grading etc).

The reason you are getting the same result is that the relative error on the short side is so much larger that it completely dominates the result.
 
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  • #3
TRB8985 said:
Homework Statement: A rectangle is 7.60 +/- 0.01 cm long and 1.90 +/- cm wide. Find the area of the rectangle and the uncertainty in the area.
Relevant Equations: See post for provided formulas.
Please check the units of area in your answer.
 
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  • #4
renormalize said:
Please check the units of area in your answer.
Agreed! Looks like I missed the square there in ##\delta A## at the end.

Thank you both, have a great weekend.
 
  • #5
It depends what matters to you.

An engineer specifying manufacturing tolerances has to worry about worst case. If the bolt has diameter ##r_b\pm \Delta r_b## and the hole has diameter ##r_h\pm \Delta r_h## then it had better be that ##r_b+ \Delta r_b<r_h- \Delta r_h##.
The first formula you quote takes that view. It assumes that the two errors reinforce.

The statistical version considers that the errors could mitigate each other. It interprets the input uncertainties as proportional to standard deviations and computes (to the same proportion) the standard deviation of the result. This is certainly more appropriate when many uncertainties are combined. If the masses of certain coins each have standard deviation 0.1g, independently, then the s.d. of the mass of 100 of them is 1g, as given by the second formula, not 10g as given by the first.
 

FAQ: Understanding the Proper Uncertainty Formula to Use

What is uncertainty in scientific measurements?

Uncertainty in scientific measurements refers to the doubt that exists about the result of any measurement. It quantifies the range within which the true value is asserted to lie, considering possible errors and variabilities in the measurement process.

How do I determine which uncertainty formula to use?

The choice of uncertainty formula depends on the type of measurement and the sources of error. For single measurements, you might use absolute or relative uncertainty. For multiple measurements, statistical methods such as standard deviation or standard error are appropriate. When combining uncertainties from different measurements, you often use propagation of uncertainty formulas.

What is the difference between absolute and relative uncertainty?

Absolute uncertainty provides the margin of error in the same units as the measurement itself, indicating the range around the measured value. Relative uncertainty, on the other hand, is the ratio of the absolute uncertainty to the measured value, often expressed as a percentage, providing a sense of the size of the error relative to the measurement.

How do I propagate uncertainties in calculations?

When you perform calculations with measured values, you need to propagate the uncertainties to find the overall uncertainty of the result. For addition and subtraction, you add the absolute uncertainties. For multiplication and division, you add the relative uncertainties. For more complex functions, you use partial derivatives to find how each variable's uncertainty contributes to the overall uncertainty.

What is the standard deviation and how is it used in uncertainty analysis?

The standard deviation is a measure of the spread of a set of values around the mean. In uncertainty analysis, it is used to quantify the variability in repeated measurements. A smaller standard deviation indicates more precise measurements, while a larger standard deviation indicates more variability and hence greater uncertainty.

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