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I am not quite familiar with the concept of correlation. The Pearson's correlation coefficient is defined as:

$\rho_{X,Y}=\mathrm{corr}(X,Y)={\mathrm{cov}(X,Y) \over \sigma_X \sigma_Y} ={E[(X-\mu_X)(Y-\mu_Y)] \over \sigma_X\sigma_Y}$

Which makes use of Mean and Standard deviation. But, is it strict to the normal distributed data ? Since Gaussian distribution is configured by mean and variance.

I currently have some which is apparently not following normal distribution. When assessing the correlation between them, is correlation appropriate here ?

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    $\begingroup$ The correlation coefficient exists whenever the relevant expectations exist. Very much not confined to the normal. $\endgroup$ – André Nicolas Jun 4 '14 at 20:14
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Correlation makes sense in any case where the two standard deviations are finite and not zero

As a non-normal distribution example, if you have two independent Poisson distributed random variables, $X$ with mean $\lambda$ and $Z$ with mean $\mu$, and you let $Y=X+Z$ then you can make the following useful statements:

  • $Y$ is a Poisson distributed random variable with mean $\lambda + \mu$
  • $X$ has standard deviation $\sigma_X^{\,}=\sqrt{\lambda}$
  • $Y$ has standard deviation $\sigma_Y^{\,}=\sqrt{\mu}$
  • the covariance of $X$ and $Y$ is $\mathrm{cov}(X,Y)=\lambda$
  • the correlation of $X$ and $Y$ is $\rho_{X,Y}^{\,}=\mathrm{corr}(X,Y)=\dfrac{1}{\sqrt{1+\frac{\mu}{\lambda}}}$
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The Pearson's correlation can be used only for normal distributions. If you have some non-normal distribution, you can use the Spearman's correlation, which does not require the normality assumption.

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