# Tag Archives: percentages

## Percentages for sceptics: part III

I wanted to do some self-criticism of my previous two posts in this series:

1. You can calculate the minimum of responses from a single percentage by hand (no need for computer programmes or look-up tables).
2. I’ve made a very rough model to estimate how many people the program typically returns when fed six percentages (as I did several times here).
In between, I’ve collected some links to demonstrate how great continued fractions can be.

Handy calculations

There are many ways of writing real numbers (fractions and irrationals) apart from in decimal notation. You can represent them in binary, for instance $\pi = 11.001001000011\ldots$, or in other bases. These have their uses: there is a formula to calculate the $n$th binary digit of $\pi$ without calculating all the preceding digits.

For our purposes we will use continued fractions. People write $\pi$ as $[3; 7, 15, 1, 292, 1, 1, 1, 2, 1, ...]$: this notation means that 3 is a good first approximation to $\pi$, the well-known $3+\frac{1}{7}=\frac{22}{7}$ is the closest you can be with any fraction $\frac{p}{q}$ with $q \leq 7$. Then $3+\frac{1}{ 7 + \frac{1}{15}}=\frac{333}{106}$ is the best with $q\leq 106$, and the fourth term

$3+\frac{1}{7+\frac{1}{15+\frac{1}{1}}}=\frac{335}{113}$

is a very good approximation, as the next number in the square brackets, 292, is very large (I’ll motivate this observation at the end of the section). The golden ratio is sometimes called the ‘most irrational’ number because it has a continued fraction expansion with all ones, so the sequence converges slowly.

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## Percentages for Sceptics: Part II

In the first percentages for sceptics post, I showed that, if you are given a percentage, you can work out the minimum number of people to whom you would have to pose a yes-or-no question to be able to get that percentage. Ideally, I hope to add to your scepticism of percentages that are unaccompanied by the number of respondents. It’s easy to be suspicious of nice, round percentages like 10%, 20%, 50% etc., but in fact all but 14 of the whole number percentages can come from polls with 20 or fewer people.

The aim of this post is to take this approach to the next level. After a quick quiz, I’ll go through two examples, the second where I reverse-engineer a pie chart is the cleaner of the two. Don’t get hung up on any of the particulars of the numbers, especially in the dating example, what they are isn’t important, it’s more the fact that we can get them: most of the post functions as a demonstration of the principle.

Warm-up puzzle: A special case

In some survey 22% of people answered “yes”, 79% answered “no” (both to zero decimal places). Each person interviewed chose exactly one of the two options. What is the least number of people that could have been interviewed to get this result? Answer at the end of this post. It’s an on-topic mathematical question, not involving any silly tricks.

Dating data

Let’s take a horrible press release reported as news by the Daily Mail, (commented on by the Neurobonkers blog) under the succinct headline: The dating rule book is being rewritten with one in four single girls dating three men at a time and a third happy to propose.

Given the trivial nature of the survey, alarm bells should be ringing; and the fact that it is ”according to the study by restaurant chain T.G.I. Friday’s” means, like their food, this ‘research’ might best be taken with a pinch of salt.

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## Percentages for sceptics

Let’s suppose you see a newspaper article or an advert on television that claims “73% of women reported healthier looking hair” or “88% of cats prefer Meonards to other leading feline food brands”, but they don’t give the number of respondents to their survey (perhaps you also have to suppose it happened in the past—the Advertising Standards Agency in Britain seems to have clamped down on this behaviour). What is the minimum number of people they could have asked?

Clearly, we can get any integer percentage by asking 100 people (or pets with strong brand preferences), and to get, say, 75% we only need to have asked four (with three responding positively). But if we assume that they rounded to zero decimal places in the standard manner, the number of respondents may be far fewer than expected (or at least fewer than I expected).  Continue reading

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