61

Are random errors necessarily gaussian? Errors are very often Gaussian, but not always. Here are some physical systems where random fluctuations (or "errors" if you're in a context with the thing that's varying constitutes an error) are not Gaussian: The distribution of times between clicks in a photodetector exposed to light is an exponential distribution....


57

First, distributions are not always bell-shaped. A very important set of distributions decrease from a maximum at $x=0$, such as the exponential distribution (delay times until a random event such as a radioactive decay) or power-laws (size distributions of randomly fragmenting objects, earthquakes, ore grade, and many other things). Stable distributions ...


48

The comment Samuel Weir makes on the fine structure constant is pretty close to an answer. For electromagnetic transitions of the nucleus, these would change if the fine structure constant changed over time. Yet spectral data on distant sources indicates no such change. The atomic transitions would change their energies and we would observe photons from ...


40

Can the Second Law of Thermodynamics / Entropy override Newton's Laws? No. In the example given, every particle obeys Newton's laws. There is no particle that is not obeying $F=ma$. From the example below, it seems that there is an underlying "Force" behind the second law of thermodynamics which drives it and which is more powerful than any other law and ...


33

There are various questions that one would have to answer, if one wished to claim that there had been large changes in decay rates over geological time. Here is what I think might be the best experiment to prove this claim. Without using radiological evidence, one can deduce that the Earth is at least a billion years old by counting annual sedimentation ...


32

The surprising answer is that nothing triggers it. In quantum mechanics all we can talk about is the probabilities of various events happening: whether they actually happen in a given period is truly random. There is no secret mechanism which we could find which controls whether an event happens or not. Well, there are, really, three-and-a-half ...


31

The basic point here is that we don't "know" anything about "the real world". All we have is a model of the world, and some measure of how well the model matches what we observe. Of course, you can construct an entirely consistent model which says "an invisible, unobservable entity created everything I have ever observed one second before I was born, and ...


17

‘Bell curve’ often refers to a Gaussian distribution. That distribution is so common that it’s also called the normal distribution. It’s very common because it emerges any time you’re looking at the sum of many things from a single distribution: I.e. lots of tiny fluctuations which, under the Central Limit Theorem, add up to a Gaussian distribution. ...


15

The reason is probably the central limit theorem: When you add lots of independent random variables, their sum will form a normal distribution, irrespective of their individual probability distributions. This makes normal distributions a pretty good guess if you do not have information about the origin of the error or if you have multiple sources of error. ...


15

Averaging destroys information. Do it as late as is practical in your analysis. A commenter points out that, if you are making many position/velocity measurements of the same object as it moves once, a simple linear regression is more robust as a velocity estimator than an ensemble of point-by-point analyses.


14

Short version $\newcommand{\ket}[1]{\lvert #1 \rangle}\newcommand{\Ket}[1]{\left| #1 \right>} % $Because you can use beamsplitters to split a coherent sate into a tensorial product of many independent low photon number coherent states. Longer version If you send $\ket{\alpha}$ on a beamsplitter of transmission coefficient $t$ and reflection coefficient ...


13

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 ...


12

Geant is a framework---which means that you use it to build applications that simulate the detector and physics you are interested in. The simulation can include all of physics and the complete detector including electronics and trigger (i.e. you can write your simulation so that it output a data file that looks just like the one you are going to get from ...


12

Actually a paper recently came out, and highlighted in Popular Science, discussing using fermionic field concepts to model crowd avoidance at Netflix. You can imagine that the same concept could be used to consider in any situation where there are large numbers of people competing for limited preferred items. Update Now that we have a few minutes, rather ...


12

You might like this 110-page paper by me and Alex Arkhipov, which is all about a quantum bosonic analogue of Galton's board (we even use the same graphic you did -- see Section 1.1!). In particular, we gave strong evidence that such a board (with an arbitrary configuration of "pegs," and with multiple entry points for the "balls") is exponentially hard even ...


12

Here is how I interpret what happened: You used Excel to compute the coefficients of the Gaussian that best describe the data: mean $\mu$, standard deviation $\sigma$, and magnitude $A$ for a curve $$Y=Ae^{-(x-\mu)^2/2\sigma^2}$$ Then you evaluated that function at a number of X values. Since the X values are not symmetrical about the calculated mean, you ...


12

There is no restriction of QM avoiding this problem even if these states appear to be a bit weird since they have no "preferred" spatial localization, but in principle they cannot be excluded. I stress that we are discussing about proper states, i.e., elements of $L^2(\mathbb R)$ and not, for instance, eigenfunctions of the momentum operator. Actually, the ...


11

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 ...


11

Answers here have generally addressed the different question of whether empirical variables should be Gaussian, but 21joanna12 asked about experimental errors, which admit a completely different analysis. The best resource on that question I can recommend is Chapter 7 of Probability Theory: The Logic of Science by E T Jaynes. In short, there are good reasons ...


11

The thermal motion has to be taken into account in any understanding of the example. It’s just as important as gravity. if there was no gravity no gravitational force, zero gravitational potential) the gases would fully mix if the temperature was zero (no thermal energy, no thermal motion) the gases would fully separate out. When both effects are present,...


10

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 ...


10

On the deepest level, particles are indistinguishable if and only if they have the same quantum numbers (mass, spin, and charges). However, in statistical mechanics one often studies effective theories where there are additional means of distinguishing particles. Two important examples: In modeling molecular fluids, two atoms on the same molecule are ...


10

This is something that particle physicists are perfectly well aware of. For any given observed effect, there is always a nonzero probability that the observation will be a false positive that was caused by a random fluctuation. The name of the game is taking enough data that this probability is small enough. In general, the more data you take, the less ...


9

If there are enough data and the prior is not completely unreasonable, the frequentist and the Bayesian approach give essentially the same answer. This is related to the central limit theorem. If data are fairly scarce, the two approaches may differ a lot. In this case the Bayesian approach is far preferable but only if the prior reflects true prior ...


9

A chi-squared test is used to compare binned data (e.g. a histogram) with another set of binned data or the predictions of a model binned in the same way. A K-S test is applied to unbinned data to compare the cumulative frequency of two distributions or compare a cumulative frequency against a model prediction of a cumulative frequency. Both chi-squared ...


9

Half life is, by definition, the amount of time until half of an infinitely large sample would decay. That's precisely equivalent (according to the frequentist interpretation of probability, if that matters to you) to the time until an individual particle's probability of decay reaches one half. The half life is a theoretical quantity that doesn't depend on ...


9

Model two consecutive measurements as the real values plus some noise. Call the first measured temperature $T_1$ and the second $T_2$. Call the measured noises $\gamma_1$ and $\gamma_2$, and suppose that they are drawn from a distribution $\Gamma(\gamma)$ and are uncorrelated. The (approximation to the) derivative is $$\text{Derivative} \approx \frac{(T_2 +...


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