How would you score it?
Problem
A game was played at a fair in which children guessed the weight of Mr Jones, the rugby teacher.
Five children made the following guesses:
1. Between 70 kg and 90 kg
2. Between 75 kg and 100 kg
3. Between 88 kg and 140 kg
4. Between 90 kg and 95 kg
5. 85 kg exactly
It turns out that Mr Jones weighs 88 kg.
Each child argues that they should win the prize.
How might each child argue that their guess was best?
If you had to award the prize, who would you choose to be the winner, and how would you justify your choice?
Devise a scoring system that penalises those whose guess has a very wide range, and rewards guesses where the middle of the range is very close to the actual weight.
Does your scoring system reward the child you thought should be the winner?
Getting Started
Imagine trying to hit a target in archery.
Is it easier to hit a large or a small target?
Do you score more for hitting the target at the edge or in the middle?
Now imagine that the children's guesses are like shots at a target...
Student Solutions
Victor from St. Pauls School in Brazil, and Alice from the British School in Manila suggested similar arguments for each child:
Child number 1 would argue that they have the smallest correct range.
Child number 2 would argue that 88 is almost exactly in the middle of their range.
Child number 3 would argue that 88 was the first number in their range.
Child number 4 might say that the correct weight is just 2 away from the first number in their range.
Child number 5 could argue that he was the only one to enter a weight, not just guess a range of weights (so in an unfair way, he had the smallest chances of getting it right).
They would choose child 5 as he had by far the smallest chances and even so he almost got it right.
Huw from Cowbridge Comprehensive School in Wales devised this scoring system that penalised students who had chosen a wide range of values and rewarded students whose median value was close to Mr Jones' actual weight.
Ben and Joe, from the same school, and Daniel from King's School in New Zealand, suggested the same scoring system. Here is how Ben and Joe worked out who had done best:
Matthew from Moonee Ponds Central School in Australia suggested this scoring system:
Find the difference between the actual weight and both ends of the range.
Pick the larger number and subtract it from 100 to get your score.
The highest score wins.
Thank you to Sophie from The King's Junior School in Chester and to the many students from Mount Pleasant Primary School who also offered their opinions.
Teachers' Resources
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.