# What is Uncertainty propagation: Definition and 17 Discussions

In statistics, propagation of uncertainty (or propagation of error) is the effect of variables' uncertainties (or errors, more specifically random errors) on the uncertainty of a function based on them. When the variables are the values of experimental measurements they have uncertainties due to measurement limitations (e.g., instrument precision) which propagate due to the combination of variables in the function.
The uncertainty u can be expressed in a number of ways.
It may be defined by the absolute error Δx. Uncertainties can also be defined by the relative error (Δx)/x, which is usually written as a percentage.
Most commonly, the uncertainty on a quantity is quantified in terms of the standard deviation, σ, which is the positive square root of the variance. The value of a quantity and its error are then expressed as an interval x ± u. If the statistical probability distribution of the variable is known or can be assumed, it is possible to derive confidence limits to describe the region within which the true value of the variable may be found. For example, the 68% confidence limits for a one-dimensional variable belonging to a normal distribution are approximately ± one standard deviation σ from the central value x, which means that the region x ± σ will cover the true value in roughly 68% of cases.
If the uncertainties are correlated then covariance must be taken into account. Correlation can arise from two different sources. First, the measurement errors may be correlated. Second, when the underlying values are correlated across a population, the uncertainties in the group averages will be correlated.

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1. ### I Alternate expressions for the uncertainty propagation

I have seen that there are two different formulas that we can use when calculating the propagation of uncertainty in a measurement. If ##X=f(A, B, C, \ldots)## is the quantity whose uncertainty we want to estimate, which depends on the quantities ##A,B,C,...##, then we could calculate the...
2. ### I Propagation of uncertainty with some constants

Hi, I have a value ##100 \pm 0.1## and a function ##\frac{V}{E} = \frac{1}{\sqrt{1 + (\omega r c)^2}}## and I would like to find the uncertainty. Where r = 1000 and c = ##5 \cdot 10^{-8}## are constants. However, I'm not sure to understand how. Here's what I think and did. Since I multiply the...
3. ### I Uncertainty in measurements of the thickness of a pile of pellets

Howdie! We have been playing around with melting and molding HDPE pellets recently. After that, we measured their diameter and thickiness 5 times each to get an uncertainty. In our experiments we put one pellet between gamma-source and detector and measure its attenuation. After that we place...
4. ### B Combining uncertainties (error propagation)

Hi! I was wondering how would I calculate the uncertainty of a value that is calculated using both multiplication and division? For example, with something like: Q = mc(T2 - T1) I'm not sure what to do with the uncertainties for T2 - T1 uncertainty = Δm/m + Δc/c + (ΔT1 + ΔT2)/(T1 + T2) Or...
5. ### Calculating Uncertainty: Where Am I Going Wrong?

So the only part of this question I am having an issue with is the uncertainty part in part e). I have included the whole question as reference. So to derive the uncertainty in ff I used the uncertainty equation outlined above but the issue is that when I propagate the uncertainty I end up with...
6. ### Time Uncertainty Propagation - Stoke's Law & Average Time

Homework Statement [/B] So basically I am calculating the terminal velocity for a small sphere falling in a measuring cylinder filled with glycerine. The distance traveled is 20 cm (0.20 m), and I have conducted 3 trials for each temperature. I have measured the displacement of the ball using...
7. ### Uncertainty of the function? My answer is a bit different

Homework Statement Homework EquationsThe Attempt at a Solution The answer is different though,
8. ### A How to combine correlated vector-valued estimates

I'd need to combine several vector-valued estimates of a physical quantity in order to obtain a better estimate with less uncertainty. As in the scalar case, the weighted mean of multiple estimates can provide a maximum likelihood estimate. For independent estimates we simply replace the...
9. ### A Uncertainty Propagation of Complex Functions

Suppose I have some observables \alpha, \beta, \gamma whose central values and uncertainties \sigma_{\alpha}, \sigma_{\beta}, \sigma_{\gamma} are known. Define a function f(\alpha, \beta, \gamma) which has both real and complex parts. How do I do standard error propagation when imaginary...
10. ### A Uncertainty Propagation in Coupled Oscillator

I am a senior physics and mathematics major, and this is my last semester. As a result, I am taking advanced physics lab, which feels more like a grad school experiment than an undergrad. One of the labs deals with the modal analysis of three spring-mass systems placed vertically as shown in the...
11. ### B Simple error propagation questions

Hi, I'm looking at an Italian high-school physics textbook. The subject is uncertainty propagation, and the target is 9th grade students. The book is allegedly by J.S. Walker, but I'm not sure how much it was "redacted" by the Italian editor. I am a little puzzled by two rules that are stated...
12. ### What is the Total Uncertainty in Measuring Force with a 0-10 lbf Load Cell?

Say I have a 0-10 lbf load cell that can measure the force it takes to lift an object. The load cell is accurate to 1% of the full scale. I take 5 measurements and get the following readings: 5.2, 5.1, 4.9, 5.0, & 4.8, all in lbf. Now I am asked to give the mean with the associated...
13. ### Uncertainty Propagation for the Slope of a Line of Best Fit

Hi guys, So I'm writing up a physics lab and I have a bunch of data points. All of these data points have both x and y error bars. The relationship between x and y is linear and so I've made a line of best fit using Python passing through the data. Now the slope of that line of best fit...
14. ### Uncertainty propagation for Normalising Signals

I am measuring an average signal through a range of filters and compute the standard deviation of that signal over a certain range on my image plate. Now I want to normalise all of the signals to an arbitrary filter signal - does the standard equation of uncertainty propagation hold...
15. ### Uncertainty Propagation Argument

A while back, one of my undergraduate physics professors gave an argument for why the uncertainty in a function or quantity F is given by \Delta F = \sqrt{^{N}_{i-1}\sum(\frac{\partial F}{\partial x_{i}})^{2}(\Delta x_{i})^{2}} He argued to think of a right triangle and think of...
16. ### Uncertainty propagation visible light spectrum

Homework Statement I have conducted an experiment which attempts to calculate the range of the visible light spectrum. Basically white light was shined through a diffraction grating (300 lines/mm) and diffraction theory is applied to calculate the wavelength. So, here are the variables...
17. ### How Do You Calculate Uncertainty in the Function f(x,z)=z/x?

Homework Statement Calculate "f" and its uncertainty, watch the units, show all work. Homework Equations f(x,z)= z/x x=100.5(+ or -) 3.8 cm y=71(+ or -) 1 s The Attempt at a Solution ok so i know that to find uncertainty i have to use the equation delta f(x,y)=df(x,y)/dx...