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?

What is this problem about?
This problem is about looking at data and considering what it might mean. This is an important skill that is gradually developed in students as they go up through the school. There are increasingly sophisticated statistical ideas that are to be found in the curriculum that all point in the direction of understanding and interpreting data. These reach their peak in the tests that exist for use by actuaries in large insurance companies. Here the interpretation of data can mean profits or losses for these companies.

In this particular problem we are only concerned that students think about the data and what it might mean. There is no correct answer to this problem. The important thing is that the students think about how the data might be have been collected and how it might be interpreted.

This problem solving lesson is different to others that we have on the site in that it does not require that the children use one of the usual problem solving strategies. However it is as close to a problem-solving lesson that we can get for the data-collection Achievement Objectives in the Statistics Strand.

Achievement Objectives
Statistics (Level 3)
- use their own language to talk about the distinctive features, such as outliers and clusters, in their own and others’ data displays.
Mathematical Processes
- devise and use problem solving strategies (draw a picture, use equipment, think).

Resources
- Picture of sprinter to introduce the problem
- Copy master of the problem (English)

Specific learning outcomes
The children will be able to:
- use their own language to talk about features of a set of data.

1. Introduce the problem by talking about the recent world record for the 100m set by Maurice Greene (USA) at 9.79s (16 June 99).  Discuss whether Maurice always runs that time.
   What might you expect the range of his times to be? Why?
2. Pose the problem to the class. Check that the children understand that there are two parts to the problem: assessing Nellie's time and replying to the coach and the principal.
3. As the children 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 the Principal said that Nellie isn't trying at training? Was that a reasonable comment to make? Why?
4. Ask the children to be prepared to share their statements about the data. This could be a written or verbal report.
5. Share interpretations and responses to the coach and principal.

Other Contexts
This kind of problem can be posed in a variety of sporting contexts where times or distances are important. So athletics, swimming and cycle (or any other kind of) racing are just three possible contexts.

Solution
As we have said above, there are various ways to interpret this data. For a start though, it might be worth looking at the outliers and considering how they came into being. Consequently, we think that the 19.2-second time should be deleted from consideration. Her Coach noted 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.

We think 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.