Why do this problem?

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.

Possible approach

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 in pairs or small groups to look at the results in more detail and revise their rankings. Each member of the group/pair should explain their initial decision, and then each group/pair should agree on a preferred ranking. Explain that they will be expected to justify their ranking to the rest of the class, so they will need to think about the arguments others will make, and how to counteract them.

After the groups 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:

• whether the true value lies within the range suggested
• how far the true value is from the middle of the range suggested
• how wide the suggested range is

Finally, challenge groups 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 rankings if they believe their proposed scoring system is fairer.

You could bring these scoring systems together in a class discussion, drawing out common themes and inviting groups to share their ideas. Can the entire group agree on a fair scoring system?

Perhaps students could also use their system to score a real "Guess the Weight" competition.

Possible support

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?"

Possible extension

Retiring to Paradise provides a different context for considering the importance of spread as well as average when working with data.