What is the difference between standard errors and residuals in OLS? I'm trying to get a deeper understanding of how OLS works. One thing that I thought I understood is the difference between standard errors and residuals.
Here are two definitions

*

*Standard errors: The average distance that the observed values fall from the regression line.

*Residuals: The difference between the actual value and the value predicted by the model ($y_i - \hat y_i$) for any given point.

Where I always assumed that number 2 was unobservable (Actually in this post they claim it's the other way around: https://stats.stackexchange.com/a/232588/334202).
But if I run a simple regression in R like this I get both! So how can I think about this?
library(tidyverse)
library(broom)

mtsmall <- mtcars |>
  rownames_to_column(var="carnames") |> 
  as_tibble() |> 
  select(mpg,hp,wt)

model1 <- lm(mpg ~ hp, mtsmall)

mtsmall_predicted <- augment_columns(model1, mtsmall) |> 
  rename(.mpg_hat = .fitted) 

mtsmall_predicted |> head(5)

Output:
    mpg    hp    wt .mpg_…¹ .se.fit .resid   .hat .sigma .cooksd
  <dbl> <dbl> <dbl>   <dbl>   <dbl>  <dbl>  <dbl>  <dbl>   <dbl>
1  21     110  2.62    22.6   0.777 -1.59  0.0405   3.92 3.74e-3
2  21     110  2.88    22.6   0.777 -1.59  0.0405   3.92 3.74e-3
3  22.8    93  2.32    23.8   0.873 -0.954 0.0510   3.92 1.73e-3
4  21.4   110  3.22    22.6   0.777 -1.19  0.0405   3.92 2.10e-3
5  18.7   175  3.44    18.2   0.741  0.541 0.0368   3.93 3.89e-4


You have errors and residuals, which are different and probably what you understood.
You seem to be speaking about 'standard error'. That is an indirect quantification of the 'error'. The standard error quantifies it in terms of an estimate for the standard deviation of the (estimated) sample distribution of that value.
