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I searched online and looked video tutorials but I'm still not sure. Would you consider the below data normally distributed? I know the ideal fit in theory would be that most of the points are on the line. However data in the real world can be different. So would like to hear your opinion from a practical point of view. Would it be safe to perform a regression analysis on this dataset?

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Skewness .291 Excess Kurtosis 2.489

Both Shapiro and Kolmogorov show significance at .000 level (therefore not normal)

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    $\begingroup$ Sorry I should have been more clear, this is the output for the residuals (Y-axis = Zresiduals and X-axis = Zpredictors. I followed this tutorial to check the assumptions on the model youtu.be/liiDHEeEH_I $\endgroup$ – JohnKimble Aug 20 '18 at 21:51
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    $\begingroup$ @John is that kurtosis figure you gave actual kurtosis (average 4th standardized moment) or is it excess kurtosis? $\endgroup$ – Glen_b Aug 21 '18 at 9:07
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    $\begingroup$ This is SPSS you're using? That would generally use excess, I believe. $\endgroup$ – Glen_b Aug 21 '18 at 9:17
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    $\begingroup$ @Glen_b and Nick I correct my answer, I believe its the excess kurtosis reported by SPSS, since its equal to Excel's KURT function. If I enter the values "2, 3, 4, 5 and 6'' in SPSS and run the descriptive analysis, it shows a skewness of 0 and kurtosis of -1,2 $\endgroup$ – JohnKimble Aug 21 '18 at 9:50
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    $\begingroup$ Excel really isn't a standard for statistics calculations but you've confirmed informed guesses from @Glen_b and me that you're showing results for excess kurtosis. A uniform distribution has kurtosis 1.8 and excess kurtosis $-$1.2. Kurtosis must be $\ge$ 1. $\endgroup$ – Nick Cox Aug 21 '18 at 10:50
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You should calculate and report the sample skewness and kurtosis of your residual distribution. Even without this, it appears from your histogram that it is probably leptokurtic; it has a higher peak, lower shoulders and fatter tail than the normal distribution. From the histogram it looks quite close to a Pearson Type VI distribution with positive excess kurtosis and possibly some slight positive skew. Fitting the distribution to this family would probably give a reasonable fit.

Deviation from normality of errors is not fatal for a regression model, since many of the results are robust to deviations from this distributional assuption. This deviation from normality means that your underlying error distribution is probably slightly leptokurtic. Your coefficient estimates should still be fine, but you will want to take the excess kurtosis into account if you construct prediction intervals for individual values. The excess kurtosis means that there is a higher probability of high errors in either direction than would be predicted by the normal regression model.

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  • $\begingroup$ I might mention in the paper that the distribution looks quite close to a Pearson Type VI distribution with some positive excess kurtosis as you mentioned, followed by a statement like "...accordingly this distribution would probably give a reasonable fit...'' Is there a academic paper I can reference as additional support for this conclusion? (sorry I'm quite new to this). I did found this paper, academicjournals.org/article/article1379928288_Lahcene.pdf but I found it hard to link this, without diving too much into it $\endgroup$ – JohnKimble Aug 21 '18 at 16:07
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    $\begingroup$ If you want to make this claim in a paper, you need to actually fit the data to that distribution and establish that it is a better fit than the normal (e.g., through goodness-of-fit statistics). You could reference a paper on the Pearson distribution, but obviously there will be no paper that asserts that your data has a reasonable fit to this distribution. (If there were, it would be the most prescient paper ever!) If you just want general references to the distribution, I would start by looking at the references here. $\endgroup$ – Ben Aug 21 '18 at 22:47
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To answer my own question based on discussion with others: The data looks quite close to being normally distributed. No distribution with real data is exactly normal, there will be always small deviations. In this case its quite close to normal and can be therefore treated as normal distribution.

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    $\begingroup$ This is distinctly non-normal: this distribution has noticeably higher probability in the center (negative excess kurtosis). Whether it could be treated as Normal depends on what you will be doing with the data. "Regression analysis" is too vague to permit further comment. $\endgroup$ – whuber Aug 21 '18 at 0:16
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    $\begingroup$ I'd expect all of those to be present in CAR. I'd also suggest leaning toward using a Q-Q plot rather than a P-P plot (your title says Q-Q but you actually have a P-P). P-P plots "squash" differences in the tail, which you will probably want to avoid making hard to identify $\endgroup$ – Glen_b Aug 21 '18 at 0:41
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    $\begingroup$ @whuber: I think this would actually be excess positive kurtosis, not negative. You have a high centre, low shoulders and fat tails (as indicated by standardised residuals of approximately five standard deviations). It would be best for the OP to calculate and report the higher moments so that we can see, but I would expect it to have positive excess kurtosis. $\endgroup$ – Ben Aug 21 '18 at 3:40
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    $\begingroup$ Ben, you're correct; heavier tail (and to some extent higher peak) tend to be associated with positive excess kurtosis. I think that's probably a slip of the fingers on whuber's part. $\endgroup$ – Glen_b Aug 21 '18 at 4:28
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    $\begingroup$ @Ben Yes, I agree--I misread the plot. The QQ plot that has since been plotted makes the interpretation clear. $\endgroup$ – whuber Aug 21 '18 at 13:27

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