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). 

For instance, for the percentages given above, we could have had 8 out of 11 women (72.7…%) saying “Yes! My hair no longer looks unhealthy”, or 7 out of 8 cats (87.5%) purring approvingly through a mouthful of processed meat. That is, for each percentage %, we want to find the least integer , such that there is an integer that satisfies:

To get 1%, then is the smallest integer such that there is a so that .

Using a simple computer program, I’ve made the following table. Surprisingly, perhaps, 86 of these 100 percentages require fewer than 20 respondents (I could have written that unhelpfully as “86% of these percentages”). The ‘outliers’ requiring the most are those sitting closest to simple fractions: 1-4% (near to 0), 34% (near to 1/3), 48%, 49% (which are near 1/2), and the seven others are given by symmetry: 100% minus the ones already mentioned here.

Percentage	Minimum no. of respondents	Percentage	Minimum no. of respondents
1%	67	51%	35
2%	41	52%	21
3%	29	53%	15
4%	23	54%	13
5%	19	55%	11
6%	16	56%	9
7%	14	57%	7
8%	12	58%	12
9%	11	59%	17
10%	10	60%	5
11%	9	61%	18
12%	17	62%	13
13%	8	63%	8
14%	7	64%	11
15%	13	65%	17
16%	19	66%	29
17%	6	67%	3
18%	11	68%	19
19%	16	69%	13
20%	5	70%	10
21%	14	71%	7
22%	9	72%	18
23%	13	73%	11
24%	17	74%	19
25%	4	75%	4
26%	19	76%	17
27%	11	77%	13
28%	18	78%	9
29%	7	79%	14
30%	10	80%	5
31%	13	81%	16
32%	19	82%	11
33%	3	83%	6
34%	29	84%	19
35%	17	85%	13
36%	11	86%	7
37%	19	87%	15
38%	8	88%	8
39%	18	89%	9
40%	5	90%	10
41%	17	91%	11
42%	12	92%	12
43%	7	93%	14
44%	9	94%	16
45%	11	95%	19
46%	13	96%	23
47%	15	97%	29
48%	21	98%	40
49%	35	99%	67
50%	2	100%	1

(Sorry, if you scrolled all the way down here looking for more content: that was the end of the post).

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