When a bunch of data generates a nicely-fit line with a nearly zero slope, R^2 approaches zero. Why? The line clearly fits well. We think the answer is because R^2 assesses the suitability of a slope-dependent term in the linear model. And a nearly-horizontal line doesn't need a slope-dependent term in its model.

r-squared-intuition.R

library(ggplot2)
library(magrittr)

set.seed(1001)

time_spent_lecturing <- c(
    0.09
  , 0.14
  , 0.21
  , 0.76
  , 0.82
)

enrollment <- c(
    183
  , 111
  , 149
  , 146
  , 146
)

jt_data <- data.frame(x, y)

p <- ggplot(
  aes(
    x = time_spent_lecturing, 
    y = enrollment)
  , data = jt_data
)

p <- p + geom_point()
p <- p + geom_smooth(method = "lm")

print(p)

fake_y <- rnorm(n = 1000, mean = 1000, sd = 0.1)
fake_x <- 1:1000

p <- ggplot(aes(x = fake_x, y = fake_y), data = data.frame(fake_x, fake_y)) + geom_point() + geom_smooth(method = "lm")
print(p)

lm(fake_y ~ fake_x) %>% summary()