You have a key error. Take the highest serial number, add that number divided by the sample size, subtract one. You can then intuitively observe that as sampl

You have a key error. Take the highest serial number, add that number divided by the sample size, subtract one. You can then intuitively observe that as sample size increases, the "m/n" component shrinks- our highest serial number is probably going to be a lot closer to the maximum. As sample size decreases all the way to 1, the "m/n" component grows- our largest sampled serial number is probably not very close to the maximum.