Why do this problem?

This task encourages students to engage with statistics without them needing to carry out detailed calculation. Discussing these statements will lead to a better understanding of statistical ideas (such as averages, expectation and sampling), as well as helping students to see the importance of stating ideas clearly when working with statistics.

This question gives students the opportunity to encounter the power of counter-examples in a mathematical analysis: for example, constructing a single example in which 'Half of the students taking a test DON'T 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.

### Possible approach

*This printable worksheet may be useful: Statistical Shorts.*
The statements (available as a worksheet

here) in this problem are designed to be short but thought-provoking, so could be used at the start of some statistics teaching. Here are some ways the statements could be used:

- Display one statement at the start of a lesson, give students some time to decide on their response, and then discuss as a class different students' ideas.
- Give all twelve statements out and invite students to discuss them in pairs before bringing the class together to share their answers and debate any disagreements.
- Give out different statements from the twelve to different pairs and then invite each pair to present their reasons for choosing "always", "sometimes" or "never", with the rest of the class acting as critical friends insisting on clear reasoning.

Key questions

Can you think of a situation when this statement **isn't** true?

Can you think of a situation when this statement **is** true?

How can you convince me that this will **never** happen?

How can you convince me that this will **always** happen?

### Possible extension

For similar statements using statistical ideas at a more advanced level, see Stats Statements.

### Possible support

Pose these two questions to students who are finding it difficult to decide which category each statement falls into:

"Can you think of a situation when this statement **isn't** true?"

"Can you think of a situation when this statement **is** true?"