What are the best methods to assess the normality of residuals?

- I agree, but to explain the intuition I would make the following reason be the first one: "First, it indicates that the model is adequately capturing the 'signal' in the data and leaving the 'noise' in the residuals."

Ideally, Regression models should produce residuals which are normally distributed. Residuals should carry zero mean and they should have a constant variance. Residuals should not have auto-correlation. They also should not have correlation with other variables in data. Residuals should be independent. The above assumptions in regression should be true to ensure that the regression models' prediction is unbiased. The distribution of residual is different from the distribution of dependent variables in data. However, if dependent variables in data are not normally distributed, then it is hard to achieve normal distribution for residuals.

A simple, if unlikely, though experiment may help. Assume you're explaining height with weight. There is a clear but imperfect correlation. After fitting the model, you see that for height values 5.0-5.5 feet there is variance around the fitted weight values (known as the residuals). For those height values over 5.5 feet, you calculate twice as much residual variance for est. weight values. In each case the correlation is the same. If you do two regressions, you should get the same Beta coefficient, but a higher certainty in Beta for 5.0-5.5. Intuitively and mathematically the residual variation is a term in the variance of Beta. But if you combine these two sub-datasets, you can't calculate the level of Beta uncertainty accurately.

I typically use visual methods like histogram plots and quantile-quantile (Q-Q) plots to assess the normality of residuals. Additionally, statistical tests such as the Shapiro-Wilk test or the Anderson-Darling test can provide quantitative measures of normality, but I rely more on visual inspection in practice.

Violations of the normality assumption can lead to biased estimates of regression coefficients and inaccurate predictions. If the residuals are not normally distributed, the model may not capture the true relationship between the independent and dependent variables effectively

- Agree with most remedies, but would caution against removing outliers or influential observations too liberally or for the wrong reason. - These cases should certainly be investigated. - The goal should be to have a better model (not just to make residuals normal.) - Outliers caused by errors in data capture should be deleted or corrected, but outliers could also represent infrequent real-world phenomena that should not be ignored. To be clear, we mean model outliers (big residuals.) A predictor outlier isn't (per se) a problem for regression. - Influential observations may be OK to remove if the intent is to better represent the larger population during inference, even at the expense of representing one or few influential entities.

The definition of residuals is accurate but I would add color. After this sentence "Residuals are the differences between the observed and predicted values of the response variable, and they reflect how well the model fits the data" I would add: "If the regression model is a good representation of the real world, then the residuals (or the error of the model) should be relatively small, not exhibit any noticeable pattern, have zero mean, constant variance, and be normally distributed. The rest of this article focuses on the last item." Even if this article is dedicated to residual normality, it would be good to mention all the other attributes that residuals must meet. Maybe even point to a higher-level page that lists them all.