# Statistical shorts

## Problem

*Statistical Shorts printable sheet*

**Below are some statistical statements. Can you decide whether each is always, sometimes, or never true?**

If they are sometimes true, give examples or conditions under which they are true and under which they are false.

If they are always true or never true, give convincing reasons why that is the case.

*Be sure to be clear about your statistical assumptions in each case.*

- It is just as likely to rain tomorrow as it is to rain the day after tomorrow.
- In a school, there will be two people who share a birthday.
- In a school, it will be someone's birthday every day.
- A randomly selected person from London will live to a greater age than a randomly selected person from Calcutta.
- If everyone in the class rolled two dice until someone threw a double six, there would be one winner.
- If I roll a die 100 times, I will get about the same number of 1s as 6s.
- If I flip a fair coin 20 times, I will get 10 heads.
- The mean, median and mode of a set of numbers can't all be the same.
- The mean cannot be less than both the median and the mode.
- Half of the students taking a test score less than the average mark.
- Nobody scores higher than the average mark in a test.
- In a game where you can only score an even number of points (0, 2, 10 or 50), the average score over a series of games must be an even number.

Could you adapt any of the statements that are sometimes true to make them always or never true?

*This resource is part of the collection Statistics - Maths of Real Life*

## Getting Started

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

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

## Student Solutions

Well done to Jamie, Oliver, George, Jamie, Sujay, Linden, James, Alexander, Lewis, James, Alex and Andre from Wilson's School, Ran Gu from Nanjing International School, Siddhartha from Beijing City International School, Florence, Erin, Connor, Barnett, Billie, Tom and Jack from Boulcott School in New Zealand, and Alistair and George from Crosscrake Primary School. A selection of your answers appear below.

- It is just as likely to rain tomorrow as it is to rain the day after tomorrow.
*This might be true because the weather conditions are similar from one day to the next and cloud cover might remain stable. It might depend on the location and the time of year.*

- In a school, there will be two people who share a birthday.
*It depends on the number of students.**There are usually over 366 people in a school; if there are more than 366 students it is certain that two will share a birthday. This may not always be true as some schools have only a few students.*

- In a school, it will be someone's birthday every day.
*This is sometimes true. There are more people than days in the year in the average secondary school. In a school with over 1500 pupils, there will be an average of 5 people celebrating their birthday per day. In a school with fewer pupils than there are days in the year, it is impossible.*

- A randomly selected person from London will live to a greater age than a randomly selected person from Calcutta.
*People in London have a greater life expectancy then those in Calcutta. However, there is always the possibility that the person randomly selected gets ill or gets involved in a fatal accident. There are some rich people in Calcutta who could afford better healthcare than some poorer people in London, but statistically it is likely that the person selected from London will live to a greater age.*

- If everyone in the class rolled two dice until someone threw a double six, there would be one winner.
*If they took turns then as soon as someone got 12 the game would end, allowing only one person the opportunity to win. If more than one person rolled at the same time then there could be more than one winner.*

- If I roll a die 100 times, I will get about the same number of 1s as 6s.
*Almost always true as there is a $\frac{1}{6}$ theoretical probability for each number.**The probability suggests that there will be a roughly equal amount of each number in 100 throws, but there is always a possibility for the amount of 1s or 6s to be different because there have not been enough trials.*

- If I flip a fair coin 20 times, I will get 10 heads.
*Sometimes true: I expect to get a head approximately half the time. With this low number of trials however, I might get 9 heads or 11 heads.*

- The mean, median and mode of a set of numbers can't all be the same.
*Not true. For example, if the numbers were 1, 1, 2, 2, 2, 3,3 then the mean would be two the median would be two and the mode would be two. If every number in the set was the same, all three averages would be the same.*

- The mean cannot be less than both the median and the mode.
*Not true: 1, 2, 3, 4, 5, 6, 6, 6, 7, 8, 9, 10*

mode = 6

median = 6

mean = 5.583333...

- Half of the students taking a test score less than the average mark.
*This could sometimes be true depending on which average you use.**For the mean, someone could get an extreme low score and everyone else gets just above average. The person with the extremely low score cancels out the others so that all but one pupil score above average.**It is true for the median, but the mode and mean do not necessarily correspond with the middle person's mark.**If we use the mode and most people score highly the modal average will be high.*

- Nobody scores higher than the average mark in a test.
*This is very unlikely but is still possible: if everyone gets exactly the same mark, this would result in no-one being above or below the average mark.**If the mode is the highest mark and we take "average" to mean mode, then this is true.*

- In a game where you can only score an even number of points (0, 2, 10 or 50), the average score over a series of games must be an even number.
*It is possible to get an odd number, for example 2 points out of 2 rounds results in an average of 1*.*If there are 3 games played and someone scored 4, 12 and*

6 you would not get an even number as the result is not divisible by 3, in fact you wouldn't even get a whole number.

## Teachers' Resources

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

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 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?"

### Possible extension

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