An issue of sampling bias: doctors think people are sicker than they actually are, because most of the patients they see are sick. This is also known as the “clinician’s illusion”. Say you’re a doctor, and you have six “well” patients that you see once a year and one “sick” patient that you see once a month. Then out of your eighteen annual appointments, two-thirds will be with “sick” patients even though only one-seventh of your patients are sick! Patricia and Jacob Cohen wrote about this in 1984; here’s a recent explanation from a web site about addiction.
This is also a hazard of teaching: and perhaps related to the “80-20” rule: you spend 80 percent of your time dealing with 20 percent of the students. Certainly grading feels this way to me, since I teach classes where the questions generally have “right” answers, and I give partial credit – my impression of the average student is probably less favorable than the actual average student. The ones who get things right are easy to grade, so they take up less time than the ones who get things wrong, since I have to read their wrong answers carefully to decide where they went wrong. Perhaps grading feels less like this on more open-ended assignments. I’d be interested to hear.
Another educational example is that by simply making all classes at an institution the same size, one can reduce the average class size experienced by students without actually having to hire more faculty. Say your institution has one class of thirty students and one of sixty. Then if you pick a student uniformly at random, one-third will say “there are thirty students in my class” and two-thirds will say “there are sixty students in my class”, for an average of (1/3)(30)+(2/3)(60)=50. If you rebalance the classes to have forty-five students in each class, then the average class size experienced by students is 45. (The average class size experienced by students, by the way, is always greater than or equal to the average class size experienced by instructors, with equality if and only if all classes are the same size.)
Finally: how many children, including you, did your mother have? The average, averaging over women, is somewhere around 2 in the US. But say that the proportion of women having i children is . Then if there are N total mothers, there are people in families of i children. The total number of children is . So the average family size, averaging over children, is
Let be the mean number of children per woman, and let be the variance; then this is , or . Again, the mean as experienced by the children is substantially larger than the mean as experienced by the parents.