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I am trying to find out whether two variables X,Y are distributed as bivariate normal. Here is my procedure:

  1. check whether X is normal

  2. check whether Y is normal

  3. check whether scatter plot is follow a ellipse

My tutor told me that I should remove the zeros of my data. I want to know, what is the point of doing that?

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    $\begingroup$ If you want to know why your tutor said to do something, why not ask your tutor? But you should give more details. What do the data consist of? How do zeros occur? Why are you checking bivariate normality? How are you assessing normality? How are you assessing elliptical contours? It might give us some better idea. It's hard to say much with so little to go on. $\endgroup$ – Glen_b Oct 9 '16 at 10:15
  • $\begingroup$ Surely zero is a legal value for a normal which has its support on the whole real line? So I agree with @Glen_b you need to politely ask your tutor what his/her understanding of the normal distribution is. $\endgroup$ – mdewey Oct 9 '16 at 12:40
  • $\begingroup$ @mdewey with continuous data the probability of an exact 0 is 0 (and if one actually occurred, would not need to be removed anyway); I believe the problem here is that the data are not truly continuous and that multiple zeros are in the data -- which is presumably why the tutor is saying to remove them, even though they're the biggest clue that the data are not drawn from a bivariate normal. $\endgroup$ – Glen_b Oct 9 '16 at 21:23
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If you have exact zeros in your data, the distribution they were drawn from will not be normal. So rather than remove them, they're grounds for immediately rejecting the null of bivariate normality.

Elliptical contours imply an elliptical distribution. A bivariate normal is elliptical, but so (for example) is a bivariate $t$. However, elliptical contours with normal margins would imply bivariate normality. So that part would be fine, if you knew that information about the distributions from which the data were drawn.

However sample margins that aren't obviously non-normal and sample contours that are not obviously non-elliptical doesn't mean that the distribution is bivariate normal. Similarly failure to reject in some hypothesis test of such things doesn't tell us that the distribution the data were drawn from is bivariate normal. Sometimes you can see that you don't have bivariate normality and sometimes you can't tell that you don't.

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