# Variance of sampling distribution of regression coefficients depends on the specific sample values?

this is my first post!

I know the sampling distribution of the sample mean is $N(\mu,\frac{\sigma^2}{n})$.

I also know that the sampling distribution of $\hat \beta_1$ is $N(\beta_1, \frac{\sigma^2}{\sum(x-\bar x)^2})$

However, this last variance depends on the specific values for a given sample! How can that be? I have run a simulation in R just to check.

Here's a demonstration. I am creating 1000 data points with $\beta_1=2$ and $\sigma^2=1$. I then take 1000 samples of size 50, estimate $\hat \beta_1$, and then take the standard deviation of the $\hat \beta_1$s. When I do, I find that it is

x<-seq(-5,4.99,.01)
y<-1+2*x+rnorm(1000,mean=0,sd=1)
dat<-data.frame(x,y)

betas<-vector()
sigma2<-vector()

for(i in 1:1000){
sum1<-summary(lm(data=dat[sample(nrow(dat),size = 50,replace = F),],y~x))
sum1
betas<-rbind(betas,coef(sum1)[1:2,1])