This problem offers a fairly informal introduction to the importance of considering spread as well as average when working with data. Students will intuitively make arguments for particular winners, and these intuitive ideas can be honed into more formal statistical statements about why some guesses should be valued more highly than others.
This printable worksheet may be useful: How Would You Score It?
Display the question with the five guesses and give students a short amount of time to rank the five guesses in order.
Collect a few of the students' rankings on the board.
Now give students time to look at the results in more detail and to make a case for their preferred ranking. Explain that they will be expected to justify their ranking to the rest of the group, so they will need to think about the arguments others will make, and how to counteract them.
After students have had time to consider their arguments, give those with differing views a chance to convince the class of the merits of their ranking. (If there is consensus within the class, challenge them to convince you.)
Key ideas that should emerge are:
Finally, challenge students to use these key ideas to produce a fair scoring system that could be published in advance of future "Guess the Weight" competitions. They could test their scoring system to check it agrees with their suggested rankings, modifying it if needs be, or they might reject their initial ranking if they believe their proposed scoring system is fairer.
Perhaps students could also use their system to score a real "Guess the Weight" competition.
Use the analogy of hitting a target in archery to help students to think about how to rank the guesses. "Is it easier to hit a large or a small target?" "Do you score more when you hit the target at the edge, or in the middle?"
Retiring to Paradise provides a different context for considering the importance of spread as well as average when working with data.