From Ernie Cohen comes a scintillating stumper of a survey:
You’re reviewing some of the survey data that was randomly collected from the residents of Riddler City. As you’ll recall, the city is quite large.
Ten randomly selected residents were asked how many people (including them) lived in their household. As it so happened, their answers were 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10.
It’s your job to use this (admittedly limited) data to estimate the average household size in Riddler City. Your co-worker suggests averaging the 10 numbers, which would give you an answer of about 5.5 people. But you’re not so sure.
Would your best estimate be exactly 5.5, less than 5.5 or greater than 5.5?
Your co-worker would be correct if the question were, 'What is the average number of people in each resident's house?'. This is from the residents' perspective.
However, household size is from the houses' perspective.
To achieve the random selection as described in the problem and a whole number of residents for each household size, assume the number of residents for each household size is:
$$LCM(1,2,3,...,10) = 2^3\cdot3^2\cdot5\cdot7 = 2520$$
The table below summarizes the household demographics of Riddler City.
| Number of Residents per Household | Number of Residents | Number of Households |
|---|---|---|
| $1$ | $2520$ | $2520$ |
| $2$ | $2520$ | $1260$ |
| $3$ | $2520$ | $840$ |
| $4$ | $2520$ | $630$ |
| $5$ | $2520$ | $504$ |
| $6$ | $2520$ | $420$ |
| $7$ | $2520$ | $360$ |
| $8$ | $2520$ | $315$ |
| $9$ | $2520$ | $280$ |
| $10$ | $2520$ | $252$ |
The average household size is thus:
$$2520 \cdot \left(\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10} \right)$$
$$ \approx 3.41 \textrm{ residents per household}$$
The appropriate guess would be less than 5.5.
This follows the form of the friendship paradox.