Training
This problem solving activity has a statistics focus.
Nellie is preparing for the 100 metres for her school athletics team.
The times below are in seconds and were recorded during training over a three-week period.
We list them here in increasing order rather than sequential order.
19.2; 20.9; 21.2; 21.3; 21.3; 21.3; 21.4; 21.4; 21.4; 21.5; 21.5; 21.5; 21.5; 21.5; 21.5; 21.6; 21.6; 22.0; 22.5; 23.8.
Her coach and the school principal both looked at these figures and made the following comments.
Coach: "The Inter-School 100 metre record for Nellie’s age group is 20.5 seconds. There’s no doubt in my mind that when the Inter-School’s meet takes place in two weeks time, Nellie will come close to breaking that record."
Principal: "I am very impressed with Nellie. She has done well in the 100 metres since she started at this school three years ago. Her time of 19.2 seconds says that when she is really trying she can be world class. The fact that most of her times are clustered around 21.6 seconds tells me that she doesn’t try hard enough at training."
How would you assess Nellie’s times?
What would you say to the Coach and the Principal?
- Use their own language to talk about features of a set of data.
- Devise and use problem solving strategies (draw a picture, use equipment, think).
This problem asks students to look closely at data and consider what it might mean. Meaningful interpretation of data is a very important skill to develop.
This problem does not require that the students use one, specific problem solving strategies as there is no correct answer to this problem. It is posed to generate worthwhile discussions with your students. They should be encouraged to think about how the data might be have been collected and how it might be interpreted.
Discussions of central tendency will arise, particularly of the mode (the number that occurs most often in a set of numbers), and of the average, or mean. Range and outliers should also be defined and considered.
The solution suggests considerations that you may wish to raise in your class discussion.
The Problem
Nellie is preparing for the 100 metres for her school athletics team. The times below are in seconds and were recorded during training over a three-week period. We list them here in increasing order rather than sequential order.
19.2; 20.9; 21.2; 21.3; 21.3; 21.3; 21.4; 21.4; 21.4; 21.5; 21.5; 21.5; 21.5; 21.5; 21.5; 21.6; 21.6; 22.0; 22.5; 23.8.
Her coach and the school principal both looked at these figures and made the following comments.
Coach: "The Inter-School 100 metre record for Nellie’s age group is 20.5 seconds. There’s no doubt in my mind that when the Inter-School’s meet takes place in two weeks time, Nellie will come close to breaking that record."
Principal: "I am very impressed with Nellie. She has done well in the 100 metres since she started at this school three years ago. Her time of 19.2 seconds says that when she is really trying she can be world class. The fact that most of her times are clustered around 21.6 seconds tells me that she doesn’t try hard enough at training."
How would you assess Nellie’s times?
What would you say to the Coach and the Principal?
Teaching Sequence
- Introduce the problem by talking about the world records for the 100m (e.g. 9.74s set by Asafa Powell). Discuss whether Asafa always runs that time.
What might you expect the range of his times to be? Why? - Pose the problem to the class. Check that the students understand that there are two parts to the problem: assessing Nellie's time and replying to the coach and the principal.
- As the students work ask questions that focus on the interpretation of data:
What time do you think Nellie will run in the championships? Why?
What do you think about what the Principal said that Nellie isn't trying at training? Was that a reasonable comment to make? Why? - Ask the students to be prepared to share their statements about the data. This could be a written or verbal report.
- Share interpretations and responses to the coach and principal.
Solution
There are various ways to interpret this data. Firstly, consider the outliers and how they came into being. The 19.2-second time could be deleted from consideration. Her coach notes that the record for Nellie’s age group is only 21.1 seconds. So it is unlikely that she could have taken nearly two seconds off that record in training. It is more likely that whoever was timing her started the stopwatch too late or stopped it too early.
Similarly her time of 23.8 seconds can probably be ignored too. This was either another timing problem (maybe the timer forgot to stop the stopwatch as Nellie crossed the line or started it too early); or a result of Nellie trying some new system (perhaps a new starting position); or maybe Nellie was just tired when she ran that time. Certainly Nellie is unlikely to run that time in a race.
The most reliable times appear to range between 21.2 seconds and 21.6 seconds. If we forget the two outliers, Nellie seems to average about 21.5 seconds per run.
It appears that the coach is being a little hard on Nellie. It would seem that she is unlikely to reduce her times by one second in the space of two weeks, especially as she has only gone under that time once in the last three weeks and even that time is suspicious.
It also appears that the principal too is being a little hard. First of all Nellie’s times cluster around 21.5 and not 21.6 seconds. Secondly, Nellie seems to run consistently in the 21.3 to 21.6 second range. This is probably her true ability level and has nothing to do with the fact that she doesn’t put all her effort into her training.