Shapiro -Wilk assumptions I am doing statistical analysis in a df and I am trying to check the normality of the data. I did the Shapiro Wilk test. I think I have to check if the data meets some assumptions, this is the case in the t-test. Basically I am trying through research to identify which are the assumptions that the data has to satisfy. I am new in statistics so any help will be appreciated ..thank you
 A: You don't say how many observations you have. For small $n$ the
Shapiro-Wilk test can fail to reject data sampled from non-normal populations. [Using R.] Neither test below has a P-value below $0.05 = 5\%.$
set.seed(808)
shapiro.test(rbeta(10, 2, 2))$p.val
[1] 0.8723119
shapiro.test(rgamma(15, 4, .2))$p.val
[1] 0.4677453

For small $n,$ it is probably best to assume that markedly
asymmetrical samples, especially ones with extreme outliers
in one tail, are not normal.
For very large $n,$ the S-W test may reject because of unimportant
quirks in a few observations. And of course, the test will reject
at the 5% level in 5% of samples that randomly sampled from
normal populations. After several tries, I got the following
result for data from a t distribution with 100 degrees of freedom,
which is normal for most practical cases. [The implementation of
the S-W test in R will not accept samples of size greater than 5000.]
shapiro.test(rt(5000, 100))$p.value
[1] 0.01595207

Nevertheless, I think it is worthwhile to look at normal probability plots (for distinctively nonlinear patterns), boxplots (for skewness or extreme outliers), and S-W tests (for P-values near $0)$ before performing a procedure that may give inaccurate results with non-normal data. The exponential sample of size $n = 35$ shown below would not be mistaken for a
normal sample by any of these methods.
set.seed(1234)
x = rexp(35, .2)
shapiro.test(x)$p.val
[1] 0.001157203


par(mfrow = c(1,2))
 boxplot(x, col="skyblue2", pch=20)
 plot(qqnorm(x)); qqline(x, col="blue")
par(mfrow = c(1,1))


Notes: (1) A sample of size $n=35$ from an exponential population with $\mu = 5$ will almost always be rejected by the S-W test:
pv = replicate(10^5, shapiro.test(rexp(36,.2))$p.val)
mean(pv <= .05)
[1] 0.9892

Moreover, if one uses a t test for $H_0: \mu=5$ against $H_a: \mu \ne 5,$ then the rejection rate for a test intended to be at the 5% level will be closer to 7% than to 5%.
pv = replicate(10^5, t.test(rexp(36,.2), mu=5)$p.val)
mean(pv <= .05)
[1] 0.06892

(2) By contrast, the correct distribution theory with
$\bar X/\mu \sim \mathsf{Gamma}(35,35)$ leads to a test
that rejects with exactly the intended significance level.
An exact 95% confidence level for the sample x above
is $(3.27, 6.36):$
mean(x)/qgamma(c(.975,.025), 35,35)
[1] 3.265594 6.364287

(3) While none of the links under "Related" in the margin of this page is an exact duplicate of your question, you might get useful insights by looking at them.
