I have residuals of a two different linear regressions, the residuals are called
r.oc in the grpahics. I need to assess the normality of the residuals. I am considering the 4 approaches below:
- Shapiro-Wilk test
- Checking the percentage of values that are within each standard deviation
- A QQplot with normal quantiles as a standardization
- A histogram with overlaid normal approximation curve
The summary of my findings are as below. Can you comment as to whether the findings/interpretations are correct?
For both samples, the Shapiro-Wilk test shows a p-value far below $p=0.05$, which means I would reject the hypothesis (at the 0.05 level) that the residuals are normally distributed.
The percentage of values that are within 1, 2, and 3 standard deviations approximately follow the empirical/normal rule.
Both QQ plots show some normal distribution, but the deviating tails would suggest otherwise.
The first histogram appears approximately normally distributed while the second one shows a marked departure from normality.
In summary: the significance test hints at no normal distribution, the empirical rule supports a normal distribution, the QQ plots discredit the underlying normal rule, and the histograms show a departure from normality. I would like someone to correct me if i'm wrong, but I would personally lean towards saying the residuals show no or little normal distribution.
#shapiro test Shapiro-Wilk normality test data: r.popc W = 0.93131, p-value < 2.2e-16 Shapiro-Wilk normality test data: r.oc W = 0.97708, p-value < 2.2e-16 #percentage of values that are within 1,2 and 3 standard deviations. r.popc  0.6916542 0.9300350 1.0000000 r.oc  0.6951524 0.9440280 1.0000000
hist(r.popc,freq=FALSE) curve(dnorm(x,mean=mean(r.popc), sd=sd(r.popc)), add=TRUE)
hist(r.oc,freq=FALSE) curve(dnorm(x,mean=mean(r.oc), sd=sd(r.oc)), add=TRUE)