What is Error propagation: Definition and 92 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.

View More On Wikipedia.org
  1. A

    I Formula for the propagation of complex errors

    If I have 2 measurements ##x = (3.0 ± 0.1), y = (-2.0 ± 0.1)## and want to calculate how the error propagates when calculating a function from those values this formula should be used: ##f(x, y) = f(x, y) ± \sqrt {(\frac{\partial f}{\partial x}*\Delta x)^2+(\frac{\partial f}{\partial y}*\Delta...
  2. TheMercury79

    Error propagation leads to different uncertainties, which do I choose?

    I understand how to compute and propagate errors but have trouble with conceptualizing all things put together. I have performed an experiment to determine a value for some quantity. This quantity depend on two variables. The first one depend in turn on some other quantities as well but I think...
  3. E

    I Error Propagation in Measurements

    I was imagining trying to construct a rectangle of area ##A = xy## If we give a symmetric error to each dimension ##\epsilon_x, \epsilon_y## $$ A + \Delta A = ( x \pm \epsilon_x )( y \pm \epsilon_y )$$ Expanding the RHS and dividing through by ##A## $$ \frac{\Delta A}{ A} = \pm...
  4. M

    I Error propagation and symmetric errors

    Hello! I am a bit confused about how to interpret symmetric error when doing error propagation. For example, if I have ##E = \frac{mv^2}{2}##, and I do error propagation I get ##\frac{dE}{E} = 2\frac{dv}{v}##. Which implies that if I have v being normally distributed, and hence having a...
  5. A

    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...
  6. Arman777

    I Error propagation of a variable for an integral

    I have an integral that depends on two parameters ##a\pm\delta a## and ##b\pm \delta b##. I am doing this integral numerically and no python function can calculate the integral with uncertainties. So I have calculated the integral for each min, max values of a and b. As a result I have obtained...
  7. Cheesycheese213

    Propagation of uncertainty with tangent function?

    For a lab, I needed to calculate the uncertainty of a refractive index that was found using Snell's law. I found an equation online for propagation of error for any general function, which was I thought that since my equation was I could just get rid of the variable y, and have After...
  8. Cheesycheese213

    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...
  9. T

    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...
  10. E

    Combining Multiple Rules for Error Propagation

    Homework Statement An object of mass m=2.3±0.1kg moves at a speed of v=1.25±0.03m/s. Calculate the kinetic energy (K=1/2mv2) of the object and estimate the uncertainty δK? Homework Equations - Addition error propagation--> z = x + y and the Limit error--> δz = δx + δy -...
  11. S

    I Error propagation and standard deviation

    Hey there, First time on this forums, looking forward to some interesting discussions :) I am currently trying grasp the concepts of error propagation and standard deviation in relation to experimental physics. I have some data set and i want to determine the difference between the measured and...
  12. CricK0es

    How Should Error Be Calculated When Determining Angles from Voltage Maxima?

    Homework Statement I have taken 5 sets of readings of voltage against angle for an experiment to determine specific rotation. (I'm using the maxima to determine the phase shift) So, having averaged these values and determined the maximum, I can then trace back to the angle at which this...
  13. NihalRi

    Error propagation when dividing by exact number

    Homework Statement Some Background - We are calculating the amount of acetylsalicylic acid in a sample using spectrophotometry. We were told to make sure to include the error in our answer. So first to calculate the moles of acetylsalicylic acid in a measured mass. 0.1620 ± 0.0005g measured...
  14. CricK0es

    Using STdev wrong or apparatus error?

    Homework Statement I have 3 (n) measurements of the radius of a capillary tube, and I'm wondering whether I should take the STDev of these values and then divide by sqrt(n-1) to obtain a standard error, and use this as my error on the mean... or Do I propagate the apparatus error through and...
  15. Hi Im Paul

    Graphing Data in Excel: Error Bar Confusion

    I'm currently writing a paper over an experiment I did for class and I have a very stupid question over I need clarification on before I continue. I am using excel to graph out data, and I have determined that the error in my Length is .001 m. However, I am graphing the inverse of length to...
  16. EastWindBreaks

    Error propagation, is this correct?

    Homework Statement given E is constant, find the uncertainty in the angular frequency, ω. can someone please check my work? Homework EquationsThe Attempt at a Solution
  17. andresfirman

    How to calculate the propagation error for a tricky Eq

    Hello Please help me, my function is; alpha = [ i1.g2 / (i2.g1) - 1 ] / ( t1-t2 ) I will have to measure i1, i2, g1, g2, t1 and t2 them I made classical error porpagation, but I don't know if this is ok. How is the proper way to calculate the propagation error for alpha?
  18. J

    I Error Propagation in Transcendental Equation

    Hey guys, I'm in a class where we're learning about waveguides, and without going into too much depth, we often solve an equation $$ \tan{(\kappa (\frac{a}{2}))} = \frac{\gamma}{\kappa} $$ for ##\kappa## numerically since there isn't an analytic solution for ##\kappa##. I'm doing a project...
  19. S

    I Error Propagation (Percentage) - sin(x)^2 / x^2

    Hey, I'm trying to propagate my percentage errors through some hefty equations and come up on a bit of snag: I've got a percentage error for x and know how to deal with it for trig functions and powers, however since both errors are from the same source: y = sin(x)^2 / x^2 Should I just...
  20. H

    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...
  21. H

    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...
  22. F

    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...
  23. F

    Error propagation (tape shorter than measured length)

    Homework Statement The problem is about calculating the error on the area of a rectangular field. What is known is that the sides of the rectangle are 120 m and 180 m, and they have been measured with a 10 m measuring tape. The tape has a sensitivity of 2 cm.Homework Equations Since the area...
  24. E

    Error propagation - inverse of an error

    Homework Statement Let t = f(g, h, A, Δm, Γ, r). For t = 2 s, the propagated error is σ = 0.02 s. Can the error of 1/t2 be simply determined using the known error in t = (2 ± 0.02) s, or must the variance formula (with all the partial derivatives and errors of each dependent variable) be...
  25. O

    Finding relative error for radial acceleration

    Homework Statement I'm trying to find da/a to calculate the relative possible error in the radial acceleration. The equation I have to derive from is a = 4π²n²rt ⁻² (it cannot be a = v²/r). I'm not really sure how to find da since it has 3 variables? Homework Equations a = 4π²n²rt ⁻² da/a =...
  26. K

    Calculating the error in <x^2> from the error in <x> (Molecular Dynamics)

    Hi, does anyone know of an easy way to calculate the error in <x^2> from the error in <x>? I am running a molecular dynamics simulation and trying to work out the error in the fluctuation of kinetic energy <dEk> = <3/2NT^2> - <3/2NT>^2 from the error in <T>. Thanks in advance
  27. B

    I How to Estimate Uncertainty for a Physical Quantity with Dependent Variables?

    I have a physical quantity A defined as ##A=(74.5 B^2*(M+N))^{1/3}## where B, M, N and relative uncertainties are given. And M and N are dependent on B. Could you show me how to calculate and estimation for the uncertainty on A? Thanks a lot
  28. T

    I Error propagation of exponentials

    I am just wondering why there is a discrepancy between two different methods for error propagation. For example, if you have ## Q = (a)(b)(c) ## then the relative error in Q is simply the square root of the sum of the squares of each of the terms being multiplied together, correct? But what if...
  29. T

    How does error propagate in a complex equation involving averages and variances?

    Homework Statement Hello, I have the following operation that I want to perform: f=\frac{\bar{X}}{100-\sum \bar{Y}_j}*K \bar{X} and \bar{Y} are averages with variances S_{X}^2 and S_{Y_j}^2 and K is a constant. How will the error propagate? Homework Equations According to...
  30. B

    Error propagation and significant digits

    Moderator's note: Thread moved to homework section. Thus no template. I have an exercise in which I have to calculate the Area from the following measurements: L = 22.1 ± 0.1 cm W = 7.3 ± 0.1 cm Of course, A = W * L = 161.33 but since I have a measurement with just 2 significant digits the...
  31. R

    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...
  32. C

    Error Propagation in Mass Flow Rates

    I tried posting this question in this forum a couple of weeks ago, but didn't get an answer to my question. I'm going to try posting it again using the formatting template so that it is hopefully clearer. I am also not sure if this is the right forum to be posting this in. It is a problem I ran...
  33. AstroKeith

    Finding Uncertainty Using Upper/Lower Bound

    Hello, I'm working on a lab report and am having a bit of trouble when it comes to figuring out uncertainty. Trial 1 Acceleration: 0.93 ± 0.14 m/s^2 Trial 2 Acceleration: 0.83 ± 0.35 m/s^2 Trial 3 Acceleration: 0.93 ± 0.14 m/s^2 I have three values listed above and and wanted to find the...
  34. C

    Error Propagation - Reconciling Two Approaches

    Hi, I am trying to find the error propagated by calculating the sum of a set of mass flow rates collected over the same length of time. The sum of mass flow rates can be calculated with two approaches, since the collection time is the same for all of them. Approach (1) is adding up all of the...
  35. L

    Error propagation with dependent variables

    Homework Statement Based on Microdosimetry theory, trying to figure out error propagation for a lot of quantities that are produced from radiation spectra. I am having trouble finding information on how to calculate and propagate errors when the quantities in my equations are not independent...
  36. N

    Error Propagation: x/(y-z) Uncertainty

    Homework Statement Suppose you measure three numbers as follows: Homework Equations x= 200. +-2. y= 50. +-2. z= 40. +-2. where the three uncertainties are independent and random. Use step-by-step propagation to find the quantity q= x/(y-z) with its uncertainty. The Attempt at a...
  37. I

    Need clarification on sig-figs and propagation of error.

    Hello, I have a question asking me to find the volume of a rectangular prism. The dimensions are as follows: x = 20 ± 0.2 cm, y = 30 ± 0.2 cm, z = 70 ± 0.4 cm I am asked to report the answer with the correct number of significant figures and include the error. What I have so far: V = xyz =...
  38. T

    Error propagation when using modulus operator

    Sorry if the answer is very simple, but I just had a question regarding error propagation when using a modulo operator in intermediate steps. For example, I have ## \theta = arctan(\frac {A}{B}) ## and then I do ## \theta ## % ##2\pi## (modulo ##2\pi##). This gives me an answer between ##...
  39. P

    Error Propagation with Log2 Concentration: Fluorescence Measurement

    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...
  40. P

    Benefits of Joining the PF Community as a Biologist

    Hi everybody, Looking forward to be part of this community! A biologist here trying to become more math savvy and willing to help others too.Thanks,
  41. Aristotle

    Finding the Uncertainties in Frequency with Given Capacitor and Inductor Values

    Homework Statement I am given a frequency value of 95 GHz (9.5x10^10 Hz), C= 25 F, L=1.12x10^(-25) H. The question is to find the uncertainties in frequency by taking account of inductor being 5% accurate & capacitor being 8% accurate. Homework Equations I believe this is the correct formula...
  42. I

    Error propagation - partial derivative?

    I am getting a little confused on which error propagation to use: I am looking to calculate the error in B*Cos(θ) , for the vertical axis of a williamson hall plot. where B is fwhm of a peak with it's own error and cos of the bragg angle I am unsure of whether i need to use partial derivative...
  43. F

    Error propagation for a sum of means

    I have a = {a1, a2, .., a1000}, where this set forms a distribution of photoelectrons (pe) seen by a particular photomultiplier tube (pmt) over 1000 repeated events. I then have N sets of these (N pmts), each containing 1000 pe values which I believe are indeed random and independent. So a, b...
  44. P

    Error Propagation: Dividing By 20 in Pendulum Timing

    Homework Statement suppose i measure the time t for 20 oscillations fro a pendulum. the period is T. Homework Equations Since T = t / 20 delta T = delta t right? The Attempt at a Solution since the 20 is a numerical value, it does not come in the error propagation, does not? when i used the...
  45. 5

    Error Propagation for Tube Volume

    Homework Statement You have a cylindrical tube for which you know the length is 16±0.1 cm and the radius is 8±0.1 cm, what is the error of the volume? Homework Equations The Attempt at a Solution V = π*82*16 = 3216.99 cm But since the r term is squared, must we account for its error twice...
  46. S

    Error propagation for non-normal errors

    I have several measurements taken over a time series. Each data point has a standard error value. I need to sum up the data points, and determine the error associated with that sum. The error values across the time series are non-normal, so I'm assuming that I can't use the usual error...
  47. N

    How can ISO 11929 norm help combine Poisson errors in low counting statistics?

    I have low counting stats and need to subtract background, account for efficiency, and divide by volume. How do I combine the asymmetrical (Poisson) errors?
  48. J

    How to calculate error propagation for several measurements?

    I'm having trouble with error propagation analysis. When you make a single measurement of several variables, say (x,y,z) and you calculate a function f(x,y,z), you just have to apply the common formula of error propagation: $$\sigma_f(x,y,z)=\sqrt{\left| \frac{\partial f}{\partial x} \right|...
  49. S

    Error Propagation and Uncertainties in Parameters

    There is a quantity (W) that I would like to calculate that is, ultimately, a function of parameters that I can measure directly (a and b), W = W(a, b) But I cannot measure a and b perfectly, there will be some uncertainty in these measurements. This uncertainty will propagate into my...