Why do this problem?
will get students into statistics without them necessarily needing to engage in detailed calculation (although some students, depending on their level, might wish to calculate various statistical quantities). It will lead to a better intuitive grasp of statistics, which will inevitably lead to a better grasp of
various statistical techniques. The questions are approximately listed in order of difficulty.
You may wish to use the problem Statistical Shorts alongside or before this one.
This problem is very well suited to discussion. As there are no numbers, it is very easy for all students to get into this problem at a level which suits them. One approach might be to ask for an immediate, instinctive response to the questions before asking them to assess them in more detail. Were their gut-feelings right or wrong? Are there any surprises?
This problem offers a good chance to practise explaining complicated ideas in statistics. Students could try to explain their thoughts verbally to each other. To give a good explanation would require a sound analysis of the statistics. Does the audience think that the explanation is sound or convincing?
To give a sound analysis will require some quantification of the concepts of the key words: 'sometimes', 'always', or 'never'.
This question gives an opportunity to explore the power of counter-examples in a mathematical analysis: for example, constructing a single example in which 'Half of the students taking a test DONT score less than the average mark' shows that the statement 'Half of the students taking a test score less than the average mark' cannot ALWAYS be true.
Assessing the meanings of 'sometimes' and 'nearly always' will be more open to discussion. This could easily lead to discussions of normal distributions, statistical testing and confidence limits.
It would be good to use this problem at an early (intuitive) stage in the study of statistics and then revisit it towards the end of a course of statistics (once computation skills are developed). Comparison of answers at these two stages would be an interesting exercise.
It is important to note that this problem is likely to raise many questions (such as the meaning of the word 'average'). All questions are valid and exploration of the issues raised will lead to a stronger intuitive understanding of statistics, which can only impact positively the subsequent learning of more formal statistical techniques.
This problem may feel very 'open' to certain students. Questions should be chosen to encourage students to think their way into each statement and to consider possible, concrete scenarios in which each one might or might not be true.
The last two parts of this question are challenging in themselves. For further work on these statistical ideas, you might refer students to Random Inequalities
, in which bounds on statistical quantities are explored.
Students could focus on the first four parts of the question and invent scores for the people in their class and then see how the averages work out. You might also refer to the problem Misunderstanding Randomness