My answer to a question on Quora: What are some unusual ways you’ve applied the math you learned in high school to your life?

I once was asked to act as a reviewer of a paper submitted for publication in an academic journal on mathematics education. It was a double blind review: the draft paper sent to me contained no names of authors or their affiliation.

The paper described how the authors set up a website and run online questionnaire among staff at mathematics departments of two British universities on the following issue: what kind of examinations,* closed book*, or *open book*, better discriminates between different levels of students’ attainment, and what kind is preferred by the respondents? Three pieces of data were given by the authors:

- Closed book examinations were selected as the most discriminating or second most discriminating of the assessment methods by 79% of the participants.
- Closed book examination was selected by 86% of the respondents as their most preferred of the assessment methods.
- The response rate of the questionnaire was 15%,

What surprised me is that the total number of responses to the on-line questionnaire has not been given in the paper, although omitting the size of the sample from statistical data was unacceptable in published academic research.

However, I calculated the number of responses, and explained in my report to the editors how I did that essentially by mental arithmetic.

This is a cute arithmetic problem; one more general piece of information is needed for solution, but it is something commonsense. **Try to think for yourself, it is easy.** A solution is given below these warning signs:

**Spoiler below!**

**Spoiler below!**

**Spoiler below!**

**Spoiler below!**

**Spoiler below!**

**Spoiler below!**

**Spoiler below!**

**Spoiler below!**

**Spoiler below!**

**Spoiler below!**

**Spoiler below!**

**Spoiler below!**

**Spoiler below!**

**Spoiler below!**

**Spoiler below!**

**Spoiler below!**

**Solution. **Indeed, 79% and 86% rates of positive answers to particular questions suggest that 86% – 79% = 7% corresponded to an integer number of people (those who answered positively to one question but not to the other). If 7% consists of 1 person, the number of respondents is 14 or 15. If 7% consist of 2 persons, then the number of respondents is between 28 and 30, but in this case, since the response rate was 15%, the two departments have about 200 mathematics lecturers, which was unlikely in UK universities (here the common sense is used). Hence there were 14 or 15 respondents.

Very conveniently, 11/14 rounds up to 0.79 and 12/14 to 0.86 (here I used a calculator – previous steps had been done by mental arithmetic) 15 respondents would produce not so good rounding of percentages.

I recommended to reject the paper — in my opinion, the paper contained no representative data; a chat in a staff lounge during coffee break, or, even better, on in a pub after a seminar was likely to yield a more representative sample. However, the editors accepted the paper for publication, but asked the authors to reveal the number of respondents – indeed, it was 14.

## Leave a Reply