I'm assuming you are using multiple regression or a similar technique. There are no normality assumptions about the distribution of predictors (Xs) in the regression or the outcome (Y). It is assumed that the errors from a normality distribution and thus the residuals should be approximately normal. Transforming the predictors or outcome will alter the distribution of residuals and may be sensible. Residuals may also change if you have different predictors (including interactions). However, you'd expect extreme some residuals in a large sample and what matters more is influence. I'd first look at something like Cook's distance to see if any of your extreme values have high influence (and are therefore materially changing your parameter estimates).

Transformations are best in my view when they have a meaningful interpretation - for example the log transform and reciprocal are often useful depending on context.

There can be 2 different situations:

1) a Gaussian distribution with some percentage of outliers, and you want to remove outliers (detected assuming the normality)

2) the distribution is not Gaussian

For situation 1 you don't need to transform your variables, you can use leave-one-out diagnostics, such as DFFITS.

For situation 2 you do need to transform. You need to apply a transformation that gives the best results in terms of the behaviour of residuals. If needed, using a maximum likelihood method you can transform both variables to joint normality, see:

Velilla, S. (1993). A note on the multivariate Box-Cox transformation to normality. Statistics and Probability Letters, 17, 259-263.

But usually you won't need the multivariate Box-Cox, you can use these simple guidelines:

https://xdplot.blogspot.com/p/data-transformations-guidelines.html

I'm assuming you are using multiple regression or a similar technique. There are no normality assumptions about the distribution of predictors (Xs) in the regression or the outcome (Y). It is assumed that the errors from a normality distribution and thus the residuals should be approximately normal. Transforming the predictors or outcome will alter the distribution of residuals and may be sensible. Residuals may also change if you have different predictors (including interactions). However, you'd expect extreme some residuals in a large sample and what matters more is influence. I'd first look at something like Cook's distance to see if any of your extreme values have high influence (and are therefore materially changing your parameter estimates).

Transformations are best in my view when they have a meaningful interpretation - for example the log transform and reciprocal are often useful depending on context.

Adding to what Thom Baguley wrote--and I heartily second his suggestion about looking at measures of influence--I suggest that normality of the errors is a sufficient condition, but not a necessary condition. The necessary condition is (approximate) normality of the sampling distributions of the parameter estimates (i.e., the constant and other coefficients). As n increases, those sampling distributions converge on the normal distribution, even if the error distribution is not normal. Jeff Wooldridge addresses this when he reviews the assumptions of OLS linear regression in his well known econometrics textbook. See my summary here:

• https://www.researchgate.net/project/Silly-or-pointless-things-people-do-when-analyzing-data/update/5a78ce78b53d2f0bba501534

HTH.

PS- I said "(approximate) normality" above, because as George Box noted, normal distributions (and straight lines) do not exist in the real world. See Section 2.5 of his famous article, Science and Statistics.

• http://mkweb.bcgsc.ca/pointsofsignificance/img/Boxonmaths.pdf

Please find attached a problem where such a procedure was necessary. It may be problem(data) dependent. Best wishes, D. Booth