# Tag Info

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The idea is just that if the uncertainties are small enough you can approximate the function by its Taylor series $$f(x_i + \delta_i) \approx f(x_i) + \sum_j \frac{\partial f(x_i)}{\partial x_j} \delta_j + \sum_{j,k} \frac{1}{2} \frac{\partial^2 f(x_i)}{\partial x_j \partial x_k} \delta_j \delta_k + \cdots.$$ If you neglect the second order terms the ...

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This problem is generally called propagation of error / uncertainty. You can google it and find a lot of info (I'd also recommend Taylor's "Introduction to Error Analysis"). Here's the gist of it, though. If you have independent measured quantities $x, y, z, \ldots$ with errors $\sigma_x, \sigma_y, \sigma_z, \ldots$, then the error on a function ...

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Jerry Schirmer's right about why solving for $r$ first is the right procedure. One way to illustrate this is to notice that with the other procedure the uncertainty could go negative, which can't be right. But the main thing I wanted to point out is that, if the measurements of $V$ and $h$ are independent, and if the "errors" mean standard deviations as ...

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You're confusing independent and dependent variables. When you propogate from uncertainties in the $x_{i}$ to some $f(x_{1},x_{2}...)$, the formula $\delta f(x_{1}...)=\sum \left|\frac{\partial f}{\partial x_{i}}\right|\delta x_{i}$ assumes that each of the $x_{i}$ is an independently measured variable and that $f$ is a dependent variable to be calculated ...

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When quoting results, there are a few good rules to follow: Avoid rounding errors in intermediate calculations. Write your error to 1 significant figure if your data set is smaller than $10^2$, 2 if it's smaller than $10^4$ etc. Write your estimate and its error with the same number of decimal places. Rules 1. and 3. are simple to understand. Rule 2. ...

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CODATA is a group that compares and combines all the most accurate experimental measurements of fundamental constants to give recommendations for best-guess values that should be used. They periodically update their values as new experiments are done. You are seeing that some wikipedia pages use old (not-updated) CODATA recommendations, while others have ...

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The formula you've specified $$\Delta k = \sqrt{(\Delta k_1)^2 + (\Delta k_2)^2}$$ is the formula to obtain error of quantity $k$, as being dependent on $k_1$ and $k_2$ according to the following expression $$k = k_1 + k_2.$$ Generally, to obtain experimental error of a dependent quantity (and the expression stated in your question), you start with ...

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You should always find an answer that is a formula, and then only apply significant figures once you get to the one final step of substituting your numbers back into the problem in place of variables. Avoid multiple intermediate steps of substituting numbers at all costs. Not only will this save your pencil a lot of work, but it will also cause your ...

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The ultimate answer is the JCGM 100:2008 guide followed by most of the metrology institutes around the world. The specific chapter on combining uncertainties is Chapter 5. Specifically, for a two-variable function $f(t_1, t_2)$ of two random variables, Eq. (16) of Section 5.2.2. gives $$\Delta f^2= \left (\frac{\partial f}{\partial t_1} \right )^2 \Delta ... 6 When you divide numbers with uncertainties, the relative uncertainties of the two numbers add in quadrature (pdf). If one of the relative uncertainties is much lower than the other, than you can ignore it. Given the wording of the problem that you quote, it appears that you can treat the radius of the rod as having a negligible uncertainty. So your reasoning ... 5 Yes, the only sensible formula for the total error is the sum in quadrature,$$ \Delta X_{\rm total} = \sqrt { \Delta X_{\rm syst}^2 + \Delta X_{\rm stat}^2 } $$The key assumption behind the validity of the formula is that the two sources of error are independent i.e. uncorrelated.$$ \langle \Delta X_{\rm syst} \Delta X_{\rm stat} \rangle = 0$$Because of ... 5 I think you're exercising an incorrect picture of statistics here - mixing the inputs and outputs. You are recording the result of a measurement, and the spread of these measurement values (we'll say they're normally distributed) is theoretically a consequence of all of the variation from all different sources. That is, every time you do it, the length of ... 5 It's not, at least not in the statistical sense. What you are doing is finding the (linearly approximated) change in y obtained by changing the inputs by their standard deviations. This is okay as a rough approximation, but you can do better with almost no extra work. If you want the actual variance and standard deviation of y, the formula is different. ... 4 That depends entirely on what you consider to be "expected range of values." When you see a value like 3.43\pm 0.04 (I will omit units for brevity), in many cases, it actually represents a normal probability distribution with a mean of 3.43 and a standard deviation of 0.04. If the 3.43\pm 0.04 is the result of an experiment, for example, then the ... 4 Here's the general derivation of the commonly used, and often (but not always) valid, uncertainty propagation formula for independent small Gaussian errors. \newcommand{\bbv}[1]{\mathbf{#1}} Consider a quantity y, calculated from measured quantities \bbv{x}$$ y + \Delta{y} = f(\bbv{x}+\Delta\bbv{x}) = ...

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