Questions tagged [statistics]

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Qubit ensemble measurement error from single-shot measurements with fidelity

I feel like I should have been able to find the answer to the question, but I have been unsuccessful so far. This question relates to measuring a two level system, and determining an ensemble or ...
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Particle concentration dependency by particle volume for a given mass

I have a mixture with a lot of nano-particles (suspended in water) with diameter ranging from 15nm to 40nm with a skewed distribution. I know the count of each particle by size. I know the mass of the ...
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Spherical harmonics with 2 cosmological probes : considering $a_{\ell m, photo}$ as a constant and $\hat{a}_{\ell m, spectro}$ as an estimator

I am in cosmological context where the survey on which I am working has 2 probes : a photometric galaxy clustering ($GC_{ph}$) probe and a spectroscopic galaxy clustering ($GC_{sp}$ probe). We use an ...
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In the Statistical Mechanics Mark E. Tuckerman 4.4.7

When deriving the energy fluctuations in the canonical ensemble, a step is made to approximate Cv≈N, why is this?
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1 answer
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Error calculation for a relative difference with only one standard deviation (STD)

I have a quantity that is calculated with two measurements, but I only have a tabulated standard deviation for one of the measurements. How would I calculate the error on the calculated quantity? The ...
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What does $C^{XY}_{\ell}$ mean when we weasure $a_{\ell m}$ in the sky?

In cosmology context, we have the general formula for the angular power spectrum $C_{\ell}$ : $$C_{\ell}=\left\langle a_{l m}^{2}\right\rangle=\frac{1}{2 \ell+1} \sum_{m=-\ell}^{\ell} a_{\ell m}^{2}=\...
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Equivalent Formulation of Least Squares Method in Terms of Entropy

I have learned that people in the field of machine learning and statistics often use the least squares method. Is there any alternative formulation of the least squares method using entropy? (It might ...
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What is the probability of pulses from 3 detectors (with known rates) coinciding? [closed]

I have three detectors giving output (on/off signals) with the following properties: Detector 1: pulse width = 40 ns, pulse rate = 1 kHz Detector 2: pulse width = 40 ns, pulse rate = 1 MHz Detector 3:...
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2 answers
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Ambiguous propagation of uncertainty

Suppose I have a voltage versus current graph with various "peaks" in current at given voltages. For example, this is what is seen in a Franck Hertz experiment. I can measure the value of ...
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Covariance of transformed random variable

So i have the covariance $cov (x_0 , x_1)$ and i know $m_b = -2.5 log_{10} (x_0) + constant$, then how do i calculate $cov (m_b , x_1)$ i found this but i guess it isnt useful here since it is ...
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On the statistical occupancy of energy levels for a Boltzmann distribution at high temperatures

Suppose we have a two-level thermodynamical system: In what follows, we will adopt the following convention: $E_a=0$, $E_b=\cal E$. For a system of $N$ classical (non-interacting) particles, the mean ...
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Where does the error from Origins fitting come from?

Im using Originpro to do some linear fitting of my data points. Origin calculates the gradient and gives an error. Where does this error come from? Ive assumed its how much the data points are ...
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1 answer
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Why kinetic energy is omitted from the Boltzmann factor?

I am studying an example of application of Boltzmann's distribution Concepts in Thermal Physics (specifically the example 4.4 p.43). It is stated that given an isothermal atmosphere, the probability ...
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1 answer
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How to quantify the uncertainty of the time series average?

Usually, we measure a fixed physical quantity, then we simply average MEAN on the measurement results and calculate the standard deviation STD, then the measurement result can be written as MEAN±STD. ...
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3 answers
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Total standard deviation for a set of measurements

I have a doubt about the calculation of the standard deviation for a set of measurement. To explain better what I mean, I will illustrate what I am doing in my experiment. I am collecting the spectra ...
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1 answer
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What is the difference of two Chi-square formula, and how can I use them properly?

I found two formula for chi-square: $$\chi^2 = \sum_i ^N \frac{(\text{measured}_i - \text{expected}_i)^2}{\text{variance}}$$ and $$\chi^2 = \sum_i ^N \frac{(\text{measured}_i - \text{expected}_i)^2}{\...
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Test for correlation in photon detection

I have data from an experiment as follows. $x^A_t = [0,0,1,0,2,0,3,...]$ and $x^B_t = [0,1,1,0,2,2,0,...]$. These numbers are photon counts measures in two different photon-counting detectors. I have ...
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Converting reach reported at $1\sigma$ to $95\%$ C.L

I'm comparing these two papers, Refs. (1) and (2), and they both constraint the ratio of doubly-charged scalar mass ($M_{H^{\pm\pm}}$) and its Yukawa coupling $(f_{ee})$ to electrons. In Ref. (1), the ...
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1 vote
1 answer
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Example for a physical distribution without a well-defined standard deviation

Is there a physical example of a distribution that has a diverging standard deviation (like the Cauchy distribution) and is there an intuitive reason for the standard deviation diverging? Is there a ...
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How do I find the uncertainty range of experimental data given this constraint on the standard deviation?

We have an inequality relating the standard deviation of position, $σ_x$, and the standard deviation of momentum, $σ_p$: $$σ_x × σ_p ≥ \,\frac{h}{4π}$$ Where $\frac{h}{4π} = 5.27285909 × 10^{-35} \...
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How to compute errors from a fit?

I have a set of data with errors, how do I compute the error for the fit $f(x)=a$? I remember there are formulas for the errors in parameters for fit $f(x)=ax+b$, i.e. $\delta a$ and $\delta b$. Where ...
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1 answer
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Error propagation for the mean of time-series

From what I have read, when measure repeatedly the same quantity X N times and the measurements follow a normal distribution the uncertainty of the mean is $σ_{mean} = \frac{σ}{\sqrt{N}}$ where σ is ...
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How the finite size scaling is derived?

Let us define that $A$ is a geometrical or physical observable quantity of a system, and $L$ is the linear size of the system. One can write $$\frac{A_L}{A_\infty} = f\left(\frac{L}{\xi}\right), \tag{...
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Physical interpretation of FFT frequencies

I need to calculate the PSD of a discrete signal and want to compare it to other processes. By Nyquist theorem, I only can account half of the frequencies. Assume I have a signal of length $N=100$, ...
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1 answer
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How to plot the 95% CLs exclusions when working with MG5/MA5?

So I am a complete noob to phenomenology and I need to plot CL contours for a model I am working on. My model looks kinda like this $$\mathcal{L} = g_{1}\{Fields\} + g_2\{Fields\}$$ and I need to plot ...
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4 answers
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In a least squares fit, is the resulting error in the fit parameters considered statistical/random error or systematic error?

In a least squares fit, if there is no error estimate on the input data points, is the resulting error in the fit parameters $\sigma_y$ (which is calculated from residuals) considered statistical/...
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2 answers
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What type of probability distribution, Gaussian, Poisson, etc, does time independent wavefunction or $|Ψ|²$ usually take, or is it completely random?

What shape the probability distribution for finding a particular particle in 3D space usually takes at any given time, for free particles not subject to any external influence? and does this shape ...
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Calculating error on manually chosen estimated fit

I understand that this should be basic knowledge, but despite searching all over the internet, I could not pinpoint the answer to this. Suppose I have data, and it can be linearised to a basic y=mx+c ...
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(Detailed) Fluctuation Theorems/Relations and their implied symmetry

I'm currently reading up on non-equilibrium statistical mechanics, in particular so-called fluctuation theorems or fluctuation relations. In Section 3.1.2 of arXiv:1205.4176, the author introduces the ...
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Comparing the variance of the position of particles of a sample of matter with the sample's temperature

As the title suggests i have attempted comparing the variance of the position of particles of a sample of matter with the sample's temperature. Now firstly we need to clarify the basics: method for ...
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30 votes
7 answers
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Least squares fit versus averaging

Suppose I want to find the spring constant of a spring from measured $F$ and $x$ data points. There are two basic ways to do this. I could calculate the spring constant for each data point and ...
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1 answer
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What is a proper resolution, and how many times of measurement needed for a credible experiments? [closed]

This question aims to understand a basic intuitive understanding and proper doing in practical situations for designing experiments. Resolution Suppose there is an object large as a hand. What is ...
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1 answer
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How to calculate the "observed" line in Brazil Band plots?

Look, for example, fig 5a in the following paper. My understanding was that, to calculate the 'expected' line, we would calculate the predicted signal+background cross section $\sigma_{s+b}$ (since ...
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How to scale Poissonian light?

In quantum optics, coherent light with constant frequency, phase, and amplitude shows poissonian photon number statistics: $$P(n) = \frac{\bar{n}^{n}}{n!}e^{-\bar{n}}.$$ A well-known result for ...
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Data science for physics recommendations

As a mathematical statistics student, I was wondering about learning data science application for physics, in general probabilistic modelling for physics. I don't mean that much engineering ...
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Validity of power law fit (and analytical fit choice)

Context I am carrying out molecular dynamics simulations of trapped ions in a radio-frequency field of amplitude $U_{RF}$ + DC (see details https://arxiv.org/abs/2102.04098). I am studying radio-...
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1 vote
1 answer
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Is the integral of the probability density of finding a paricle symmetric about the mean?

Is the probability density integral of finding a particle symmetric about the expectation value? Also is the probability integral symmetric with respect to standard deviations from the mean? Like if I ...
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1 answer
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What is the total signal strength of the Higgs?

Dealing with the signal strength modifier of the Higgs, a few things escape my grasp : 1/ In the PDG booklet the combined result is presented with an uncertainty, although some of the measured signals ...
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Reconstruction of Normal Distribution Function (NDF) using Wheeler's algorithm

I have a polydisperse bubble in liquid simulation which calculates statistical moments, and my next target is to reconstruct the Normal Distribution Function (NDF) that represent bubble size ...
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1 answer
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How do I calculate the uncertainty of a fraction raised to a power? [closed]

What is the uncertainty of: $y=(\frac{a}{b})^{n}$ Would it be: $\Delta y = n (\frac{\Delta a}{a}+\frac{\Delta b}{b}) y $
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3 votes
4 answers
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How large does $N$ need to be for statistical mechanics to be a good approximation?

About how many components ($N$) does a system need for statistical mechanics to apply to that system? I took stat mech and biophysics from the same professor in undergrad and I distinctly remember him ...
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Poisson error for detected photons

Assume that there are photons arriving at a detector according to a Poission distribution. Let's say we detect 100 photons over 10 seconds. Now I see two ways to calculate the statistical error $s$, ...
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In the uncertainty principle, are $\Delta x$ and $\Delta p$ uncertainties? [duplicate]

I was thinking about a very old exam where the UP was expressed as $$\Delta x\Delta p\geq\frac{\hbar}{2}.$$ From what I understood in my measurement course, the uncertainty of a measurement model $Y$, ...
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Cross correlation for the fluctuating part of two timeseries. Can it attain values gretaer than unity

I have written a code to estimate the lagged cross correlation between the fluctuating parts of two quantities $(X, Y)$. To estimate the fluctuating parts I am using the following equations: $X^{'}(t) ...
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Fitting of experimental data affected by different kinds of errors

It is quite easy to evaluate the best-fit curve for a set of n data points when the dependent variable is affected by a statistical error (namely when you have n triplets $(x_i,y_i,\sigma_{y_i})$. I ...
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Inclusion of a systematic uncertainty in the upper limit on cross-section

Let's imagine the following simplistic experiment: we search for a certain particle of a known mass at the LHC, and set an upper limit on its production rate. We use the CLs method and claim, for ...
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How do you weight a chi square for uncertainties in $x$ and $y$ (different units)?

So I have experimental data for energy vs angle with uncertainties in both the $x$ and $y$ direction for every point. I am comparing this to a known curve/data points. Is it possible to calculate a ...
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1 answer
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Particle interference

According the experiments of interference of the particles one by one*, we see that it is the distribution of the impacts of all the particles on the screen (detector) which draws (constructs) the ...
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How to calculate the uncertainty and mean of multiple measurements with different errors?

In common situation,each measurement with the same error,so we can simple average the measurement value to get the result and uncertainty.But some times the error of each measurement is not same,that ...
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What does "a least-squares approximation to a sequence of data is simply their averaged value (mathematical expectation)" mean?

I am reading the paper "Multiscale coarse graining of liquid-state systems" (https://doi.org/10.1063/1.2038787). In the paper, right after equation (1), they say "a least-squares ...
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