What is Correlation: Definition and 366 Discussions

In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. In the broadest sense correlation is any statistical association, though it commonly refers to the degree to which a pair of variables are linearly related.
Familiar examples of dependent phenomena include the correlation between the height of parents and their offspring, and the correlation between the price of a good and the quantity the consumers are willing to purchase, as it is depicted in the so-called demand curve.
Correlations are useful because they can indicate a predictive relationship that can be exploited in practice. For example, an electrical utility may produce less power on a mild day based on the correlation between electricity demand and weather. In this example, there is a causal relationship, because extreme weather causes people to use more electricity for heating or cooling. However, in general, the presence of a correlation is not sufficient to infer the presence of a causal relationship (i.e., correlation does not imply causation).
Formally, random variables are dependent if they do not satisfy a mathematical property of probabilistic independence. In informal parlance, correlation is synonymous with dependence. However, when used in a technical sense, correlation refers to any of several specific types of mathematical operations between the tested variables and their respective expected values. Essentially, correlation is the measure of how two or more variables are related to one another. There are several correlation coefficients, often denoted


{\displaystyle \rho }


{\displaystyle r}
, measuring the degree of correlation. The most common of these is the Pearson correlation coefficient, which is sensitive only to a linear relationship between two variables (which may be present even when one variable is a nonlinear function of the other). Other correlation coefficients – such as Spearman's rank correlation – have been developed to be more robust than Pearson's, that is, more sensitive to nonlinear relationships. Mutual information can also be applied to measure dependence between two variables.

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  1. M

    A Vacuum Energy from Correlation Functions

    In QFT the objects of interest are the n point Correlation functions which contain all the information about the theory and can be used to compute any expectation value in principle. However I cant figure out how to compute the vacuum energy from the correlation functions alone and cant find any...
  2. F

    I Looking for the most suitable distance for binary clustering

    Hello everyone. I have a pandas dataset in python which has n+1 columns and t rows. The first column is a timestamp that goes second by second during a time interval, and the other columns are the names of the people who log in the server. The t rows of the other columns indicate if the person...
  3. T

    I Correlation Matrix of Quadratic Hamiltonian

    I am struggling to rederive equations (61) and (62) from the following paper, namely I just want to understand how they evaluated terms like ##\alpha\epsilon\alpha^{T}## using (58). It seems like they don't explicitly solve for ##\alpha## right?
  4. S

    I Question on multiple rebrightening gamma ray bursts and gravity waves

    Hi Guys. I am interested to find out if anyone at the IPTA or other relevant organizations have correlated gravity waves with multiple rebrightening gamma ray bursts where there is a constant time (t) between 3 or more rebrightening's? If so, did the detection of the gravity wave occur between...
  5. A

    A The standardized and unstandardized canonical correlation coefficients

    The output of SPSS 27 Canonical Correlation gives the standardized and unstandardized canonical correlation coefficients. What exactly are the standardized and unstandardized canonical correlation coefficients and what is the difference between them?
  6. Ranku

    I Correlation between size and mass of particles

    Is there a correlation between the size of a matter particle (defined as its matter wavelength) and the mass of the particle? With the photon, its wavelength and its energy/mass are inversely correlated. Is it also true of matter particles?
  7. chwala

    Calculate the correlation coefficient in the given problem

    Unless there is another alternative method, i would appreciate...ms did not indicate working...thought i should share my working though... Let Waistline= ##X## and Percentage body fat =##Y## and we know that ##n=11## ##\sum X=992, \sum XY=13,772## and ## \sum Y=150## Then it follows that...
  8. N

    Delay estimation using cross correlation

    Hi Suppose there are two continuous signals of same frequency say 4 KHz. The time corresponding to its one cycle is around 250 us. If we delay one signal by 4010 us (i.e >> one cycle delay), can we use cross correlation techniques to estimate this delay accurately? Thanks
  9. N

    Calculating delay b/w signals using correlation

    Hi all I am generating two signals of same frequency, i introduce a fixed delay in one of the signal and then try to find out the simulated delay using MATLAB 'gccphat' and 'finddelay' functions, but not able to measure correct delay. MATLAB Code script is given below: clc clear close...
  10. codebpr

    A Does the Maximum Lyapunov exponent depend on the eigenvalues?

    I am currently reading this paper where on page 8, the authors say that: This correlates with Figure 8 on page 12. Does it mean that there is a real correlation between eigenvalues and Lyapunov exponents?
  11. S

    Correlation between the level of education and the number of friends

    I have heard, that a study was performed, maybe in Brazil, and it showed that a correlation was found – more educated people have less friends. I was unable to google this work. Maybe somebody here knows it? I found only the following...
  12. F

    A Coefficient correlation between 2 cosmological probes

    Hello, I have the demonstration below. A population represents the spectroscopic proble and B the photometric probe. I would like to know if, from the equation (13), the correlation coeffcient is closed to 0 or to 1 since I don't know if ##\mathcal{N}_{\ell}^{A}## Poisson noise of spectroscopic...
  13. A

    A The expectation of the sampling distribution of Pearson's correlation

    The shape of the sampling distribution of the Pearson product moment correlation coefficient depends on the size of the sample. Is the expectation of the sampling distribution of the Pearson product moment correlation coefficient always equal to the population correlation coefficient, regardless...
  14. chwala

    Calculate the rank correlation coefficient of the given problem

    Find the problem and solution here; I am refreshing on this topic of Correlation. The steps are pretty much clear..my question is on the given formula ##\textbf{R}##. Is it a generally and widely accepted formula or is it some form of improvised formula approach for repeated entries/data? How...
  15. gremory

    A Computing Correlation functions

    Hello, recently I'm learning about correlation functions in the context of QFT. Correct me with I'm wrong but what i understand is that tha n-point correlation functions kinda of describe particles that are transitioning from a point in space-time to another by excitations on the field. So, what...
  16. T

    A Brownian Motion (Langevin equation) correlation function

    So the Langevin equation of Brownian motion is a stochastic differential equation defined as $$m {d \textbf{v} \over{dt} } = - \lambda \textbf{v} + \eta(t)$$ where the noise function eta has correlation function $$\langle \eta_i(t) \eta_j(t') \rangle=2 \lambda k_B T \delta_{ij} \delta(t -...
  17. W

    Coding of Spin Spin correlation function

    I tried to code spinoperators who act like $S_x^iS_x^j$ (y and z too) and to apply them to the states, which works fine. I am not sure about how to code the expectation value in the product Space. Has anyone pseudo Code to demonstrate that?
  18. U

    I Noise Whiteness hypothesis in Kalman filtering

    In Kalman filter mathematical treatment I have always read that a foundamental hypothesis is represented by the whiteness of the process noise. I have tried to do again the mathematical steps in the Kalman filter derivation but I can't see where such hypothesis is crucial. Could you help me...
  19. G

    Comparing Near-Infrared Spectra: What Stat Method?

    I'd like to compare 2 or more near-infrared spectra. The data consists of measured light intensity in different wavelengths (range 600 nm to 1100 nm). I'm wondering which statistical method would be appropriate? I noticed when searching online that pearson correlation might be inaccurate as...
  20. L

    B Why isn’t quantum entanglement just correlation?

    They say spin up and spin down is correlated at at any distance and that it can’t be explained by basic logic. say I rip a photo in two, shuffle them and put them in two boxes and send them light years away. No matter which box I open, I can’t know which half I have but when I open it the photo...
  21. M

    Bound correlation coefficient for three random variables

    Hi, I just found this problem and was wondering how I might go about approaching the solution. Question: Given three random variables ## X##, ##Y##, and ## Z ## such that ##\text{corr}(X, Y) = \text{corr}(Y, Z) = \text{corr}(Z, X) = r ##, provide an upper and lower bound on ##r## Attempt: I...
  22. chwala

    Determine the type of correlation for the two variables given in the data

    Kindly see the attached problem below (i find the topic to be easy and straightforward). My concern is only on the highlighted part: In my understanding, to define the type of correlation i have always approached a straightforward approach. For value ##1## perfect positive correlation and...
  23. entropy1

    I Correlation vs. Causation: Knowing the Difference

    If we observe that: IF A happens THEN B happens, we speak of correlation and not of causation, right? So when do we speak of causation?
  24. JD_PM

    Loop Feynman diagram contributions to correlation functions

    My understanding of the n-correlation function is \begin{equation*} \langle \phi(x_1) \phi(x_2) ... \phi(x_n)\rangle = i \Delta_F (x_1-x_2-...-x_n) \end{equation*} Where ##\Delta_F## is known as the Feynman propagator (in Mathematics is better known as Green's function). Let us analyze...
  25. A

    A Correlation between parameters in a likelihood fit

    Hello community! I am facing a conceptual problem with the correlation matrix between maximum likelihood estimators. I estimate two parameters (their names are SigmaBin0 and qqzz_norm_0) from a multidimensional likelihood function, actually the number of parameters are larger than the two I am...
  26. P

    Explaining Quantum Correlation by Comparing It To Road Traffic

    I know quantum correlation means that the particles are entangled and so the state of each cannot be determined independently of the other. However I'm not sure how it applies to this particular scenario - If there are more cars on the M25 I suppose we could say technically there are less on...
  27. M

    MHB Distribution, expected value, variance, covariance and correlation

    Hey! :giggle: Let $X$, $Y$ and $Z$ be independent random variables. Let $X$ be Bernoulli distributed on $\{0,1\}$ with success parameter $p_0$ and let $Y$ be Poisson distributed with parameter $\lambda$ and let $Z$ be Poisson distributed with parameter $\mu$. (a) Calculate the distribution...
  28. F

    I Understanding 2 equivalent formulations of both data set measures

    I have two independant experiments have measured ##\tau_{1},\sigma_{1}## and ##\tau_{2},\sigma_{2}## with ##\sigma_{i}## representing errors on measures. From these two measures, assuming errors are gaussian, we want to get the estimation of Ï and its error (i.e with a combination of two...
  29. W

    A Good Examples of Causation does not Imply Correlation

    Ok, so if the causality relation between A,B is not linear, then it will go unnoticed by correlation, i.e., we may have A causing B but Corr(A, B)=0. I am trying to find good examples to illustrate this but not coming up with much. I can think of Hooke's law, where data pairs (x, kx^2) would...
  30. Rzbs

    I Electron correlation vs electron exchange

    What is the difference between electron correlation and electron exchange? Which of them is due to the spin of electrons and which is due to charge of electrons?
  31. Danny Boy

    A Quantum Ising model correlation function query

    In this paper, on quantum Ising model dynamics, they consider the Hamiltonian $$\mathcal{H} = \sum_{j < k} J_{jk} \hat{\sigma}_{j}^{z}\hat{\sigma}_{k}^{z}$$ and the correlation function $$\mathcal{G} = \langle \mathcal{T}_C(\hat{\sigma}^{a_n}_{j_n}(t_n^*)\cdot\cdot\cdot...
  32. entropy1

    I Ontology of measurement and correlation

    If a non-commuting measurement is made on a quantum property (like spin), this can be seen as the wavefunction being prepared. So you can't tell if the outcome represents the property, or that the property is prepared. However, if the property is prepared, we can predict the correlation with a...
  33. J

    I Correlation between Symmetry number & Total wavefunction

    Some rotational quantum states are not allowed for a rotating particle. At quantum level, these "forbidden" quantum states is based on the requirement of the total wavefunction being either symmetrical or anti-symmetrical, depending on whether the particle is a fermion or boson. The particle's...
  34. Addez123

    Correlation coefficient of a jumping particle

    What I need help with is how I would start.. I can say p(X, Y) = (1,0) = 1/4, and same for the other 3 coordinates. P = 0 for all other coordinates. This doesn't give me anything to work with though. C(X, Y) = E(XY) - E(X)E(Y) What is XY? I don't even know what X is.
  35. G

    MHB Linear Correlation: Is Evidence Sufficient?

    The accompanying table lists the numbers of Internet users per 100 people and numbers of scientific award winners per 10 million people for different countries. Construct a​ scatterplot, find the value of the linear correlation coefficient​ r, and find the​ P-value of r. Determine whether there...
  36. Danny Boy

    A Correlation functions of quantum Ising model

    I have a single technical question regarding a statement on page 7 of the paper "Dynamical quantum correlations of Ising models on an arbitrary lattice and their resilience to decoherence". The paper up until page 7 defines a general correlation function ##\mathcal{G}## of a basic quantum Ising...
  37. entropy1

    I (In)dependence in entanglement experiments

    If an entanglement experiment, whereby an entangled pair of particles is measured at both ends, is independent of the next entanglement experiment with another pair of entangled particles, how can there be a correlation? It seems that each independent run does not influence the next run, but...
  38. J

    Studying Correlation between contest math training and grasping abilities?

    I have a notion that students involved in contest math/physics since high school 'develop' a better ability to pick up concepts (not in the context of contest math/physics) quicker and solve relatively more 'complex' problems. In high school I heard about contest math but never really immersed...
  39. O

    I Higher-Order Time Correlation Functions of White Noise?

    Suppose I have Gaussian white noise, with the usual dirac-delta autocorrelation function, <F1(t1)F2(t2)> = s2*d(t1-t2)*D12 Where s is the standard deviation of the Gaussian, little d is the delta function, and big D is the kronecker delta. For concreteness and to keep track of units, say F...
  40. G

    A Correlation function of a Klein-Gordon field

    First, let me introduce the notation; given a Hamiltonian ##H## and a momentum operator ##\vec{P}##, and writing ##P=(H,\vec{P})##. Let ##|\Omega\rangle## be the ground state of ##H##, ##|\lambda_\vec{0}\rangle## an eigenstate of ##H## with momentum 0, i.e. ##\vec{P}|\lambda_\vec{0}\rangle=0##...
  41. W

    A Joint Used to Show lack of Correlation?

    Hi All, I think I have some idea of how to interpret covariance and correlation. But some doubts remain: 1)What joint distribution do we assume? An example of uncorrelated variables is that of points on a circle, i.e., the variables ##X ##and ##\sqrt{ 1- x^2} ##are uncorrelated -- have...
  42. bagasme

    Correlation between Tire Pressure and Acceleration of a Motorbike

    I have asked this question on Stack Exchange: SE question. I often encountered this sticker on most motorbikes (especially matic ones) [credit: cintamobil.com]: There, when the tire pressure was measured from cold condition, the tire pressure are same regardless of loadout (29 psi and 33 psi...
  43. jk22

    A Does there exist a super-quantum correlation in mixed-state formalism

    Suppose a two point covariance : ##C(a,b)=\langle A\otimes B\rangle## with the eigenvalues of A and B in {-1,1}. Does there exist a mixed state such that ##CHSH=C(a,b)-C(a,b')+C(a',b)+C(a',b')>2\sqrt{2}## ?
  44. J

    B Correlation between shifting graph of a function and shifting the axes

    1.To shift the graph of a function : Vertical Shifts : ## y=f(x) +h## where the graph shifts ##k## units up if ##k## is positive and downwards when ##k## is negative. Horizontal Shifts : ##y=f(x+h)## where the graph shifts to the left by ##h## units when positive and to the right when ##h## is...
  45. P

    Mean, variance and correlation function of Langevin equation

    Homework Statement I have simulated Langevin equation (numerically in Matlab) for some specific conditions, so I have obtained the solution ##X(t)##. But now, with the solution I have obtained, I have to calculate ## <X(t)|x_0>, <X^2(t)|x_0>-(<X(t)|x_o>)^2 ## and the conditional correlation...
  46. A

    I Why is there this extra term for this correlation function?

    Let's say we have a Dirac field ##\Psi## and a scalar field ##\varphi## and we want to compute this correlation function $$<0|T \Psi _\alpha (x) \Psi _\beta (y) \varphi (z_1) \varphi (z_2)|0>$$ $$= \frac {1}{i} \frac{\delta}{\delta \overline{\eta}_\alpha(x)} i \frac{\delta}{\delta \eta_\beta(y)}...
  47. haruspex

    I What is the expected value of Cov(x,y)2 in an independent X and Y scenario?

    all the references I can find on the net to justifying a correlation treat it as a matter of judgment, and, quite correctly, that it depends on the application. But it seems to me that one could compare the fit to the data of a horizontal line (i.e. average y) with that of the linear regression...
  48. J

    CO2 and the correlation with rising atmospheric temperatures

    I just read the policy for this forum "Earth" so i hope my next question will be approved, it is with the best intentions. My friends and i where having a discussion about CO2 and the correlation with the raising temperature, we know that CO2 is a greenhouse gas. About that, CO2 is a greenhouse...
  49. entropy1

    B Can this be called 'a coincidence'?

    Suppose we have two variables A and B. A has a truly random distribution over {0,1} with P(0)=P(1)=0.5 . B has the same distribution. Now suppose that A and B always show both a 1 or both a 0. This would be a strong correlation between A and B. Now could this be called 'a coincidence'? And if...
  50. F

    I Infer the distance of a closest neighbor galaxy from correlation

    I am trying to estimate the distance of closest galaxy neighbor knowing the expression of number of neighbors into a volume ##\text{d}V##, the mean density ##n_\text{gal}## and the correlation function, i.e with this expression : ##\text{d}N=n_{\text{gal}}\,\text{d}V\,(1+\xi(r))## with...