how about using proportions or percentages? Even plots would be more revealing with less cluttered points.

cancelled_prop = (cancelled_num / flights_num),

OR even better:

cancelled_prop = (cancelled_num / flights_num) * 100,

ggplot (cancelled_per_day, aes (x = cancelled_prop, y = avg_dep_delay)) + geom_point (color = 'blue') + geom_smooth (se = FALSE)

ggplot (cancelled_per_day, aes (x = cancelled_prop, y = avg_arr_delay)) + geom_point (color = 'red') + geom_smooth (se = FALSE, color = 'red')