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I'm not sure how to explain it. I'm measuring time passed (not normal) against my dependent variable (a grade, with distribution close to normal). What I can see from the scatter plot is almost triangular. That is, I have both low and high grades when less time has passed and it becomes flatter as time passes.

I'm not sure if it´s linear because it seems both positive and negative. I'm also not sure is monotonic. I've recently heard about "distance correlation". Is that the case here?

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    $\begingroup$ It's impossible to interpret this plot reliably due to the overplotting of points. To a certain extent, the expanded ranges at low grades can be explained by their numerousness: the larger the sample, the more extreme its range tends to be. It's difficult to determine why you are concerned about this, though: what is the objective of your analysis of these data? $\endgroup$
    – whuber
    May 6, 2021 at 15:07
  • $\begingroup$ That´s a good point. This time variable is "time passed since formal education". My boss thinks that as time passes, grades tend to be lower. I'm trying to see if it's true. $\endgroup$
    – ana.la
    May 6, 2021 at 15:13
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    $\begingroup$ Superimpose a robust smooth on this plot: it will show you, at any level of detail you want, how the typical grade varies with time passed. $\endgroup$
    – whuber
    May 6, 2021 at 15:42
  • $\begingroup$ @ana.la that hypothesis doesn't quite make sense: one's grades remain quite constant after graduation. I think you are conditioning on being employed at a particular job, say civil engineering. In this case, the hypothesis is that average GPA tends to be lower among employees having a greater duration since formal education. $\endgroup$
    – AdamO
    May 6, 2021 at 16:05
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    $\begingroup$ @AdamO the grade isn't GPA. It's actually a score on a test that was applied recently and people of different ages (thus, different times since graduation) took it. $\endgroup$
    – ana.la
    May 6, 2021 at 23:33

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The phenomenon you think you are seeing is called heteroscedasticity. That, for the more extreme values of time on the $X$-axis, the apparent variability in GPA ($Y$-axis) seems to go down. The issue in this case isn't how to measure correlation: the usual measures can be used. Rather, it's how you estimate the uncertainty. Putting that aside, I actually don't see much evidence of heteroscedasticity in this plot.

The problem with the default settings on a plot like this is that it doesn't show point density.

Between time 30 and 40 I can easily count the sample size. However there are too many points to even count at time 0. Our eye tends to impute the sample size somewhat conservatively in these cases.

Some better plotting choices can help you to see the sample size more apparently:

  • Use transparent points so that slightly overlapping points can be seen as distinct values rather than one blob
  • Do not use scatterplots if there are more than, say, 300 points. Rather, use hexagonal binning, or using a partition of the $X$ domain, create box-and-whisker plots or violin plots
  • Mark quantiles or deciles or other percentiles as appropriate on the X-axis or include the cumulative $N$ at each $X$-axis tick mark
  • Include a smoothed estimate of mean trend as a function of the $X$ such as lowess or smoothing spline
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  • $\begingroup$ Thank you, that made perfect sense. I've followed your tips and made a few changes on my plots and it's easier to see it now. $\endgroup$
    – ana.la
    May 6, 2021 at 23:34

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