Is it problematic to use percentages to describe a sample with less than 100 people?

Question

This is a very basic, dumb question, but I couldn't find an answer, and on the top of that I'm generally suspicious of common sense/intuition.

I feel like it is "cheating" when saying "80% of people in the sample liked the banana flavor" when there are less than 100 people in the whole sample. But maybe I'm wrong, hence my question. I see at least two problems:

1. it could be somehow misleading if I omit to mention the sample size

2. there are percentages impossible to attain, for instance it's impossible to have any percentage between 0 and 10% if there are just 10 people in the sample

Am I overcautious here, and is it actually OK to use percentages like that? Or is it really a problem? Does it have other problems I did not identify?

Someone commented that it depends on the context, but I don't have a specific example in mind so my question is "it depends on what?". When is it fine to report percentages with a sample size less than 100, and when is it not fine?

TL;DR: If I understand correctly the current answers, it is always fine to report percentages as long as I report the sample size too. Not reporting the sample size along percentages is always problematic. Is that statement correct, or are there additional things to consider? Thank you.

Answer 1

Percentage is just a simple way to express a number

Percent, %, is just a number expressed as a fraction (or ratio) with the denominator being one hundred. It derives from Latin 'per hundred'. We also have permille, ‰, which means 'per thousand'. And you can go further with permyriad, per cent mille (also pcm), parts per million (ppm), or parts per billion (ppb).

All these numbers are used to make (dimensionless) numbers easier to express. With especially the large ones this is clear, ppm and ppb, can help to write small fractions or ratios like 0.00000123 much easier.

For percent it helps because people have a concept of it. We know what 8% tax means and it is easier than a 0.08 fraction tax.

Continuing with the tax example. A percentage doesn't need to indicate that the denominator in the case that it is applied to is equal to a hundred. We can speak about 8% of half a dollar, which is 4 cents.

In $9 \cdot 10^1\%$ of the cases with percentages, typically no significant digits are implied.

Also percents are often used without considering scientific notation and indicating the accuracy or imply that the last digit is an indication of the significance. 12.5% of a pizza means that it is 1 out of 8 slices, it can be way more inaccuracy than ±0.05%, A 33% often means one third. When a poll about elections results in 49%, but this figure has a ±5% error, then it is not reported as 5x10¹%. You never see such expressions.

For your sample the high accuracy, if you insist on interpreting it that way, is also technically not incorrect. Say, if your sample has 4 out of 5 people that like banana, then it is very accurately 0.8 which you can safely express as 80% without indicating that this number is too precise. The statistics about a sample can be very accurate (if you ignore errors in observing your sampled specimens, e.g. they might lie about bananas or you could have made a counting error).

A different issue

It are the estimates about the population that you infer from it which are inaccurate. For that reason, it is good to also report sample sizes. And if your sentence implies in it's context to refer to an estimate, then present a confidence interval, standard error or other way to express the accuracy.

This is a different issue than the usage of percentages to express numbers.

Alternative ways to write your sentence

Asside from the question is it good or not, you can wonder whether something else is better.

If your sample has size 10, then writing

• 8 of the 10 sampled people like banana flavor.

is easy to understand and more informative than

• 80% of the sampled people like banana flavor

However if your sample is size 321, then the following might not directly ring a bell with everyone

• 257 out of the 321 sampled people like banana flavor.

and computing it to 80% helps your readers to directly see what $\frac{257}{321}$ is.

Answer 2

It depends on the situation and how you communicate it.

The short answer is: Yes, it is problematic, and you should not do this.

I am coming from an applied perspective; most of my career has been using techniques, ranging from simple to complex, but then explaining the results concisely and at the level of the audience's understanding. There are still times where stakeholders or clients will want to see percentages—even after I warn them that I'm suppressing numbers for a sample size that is too small. They insist they know the limitations and still want to see them. In this situation, I try to overstate that this is descriptive (only related to the given data I have) and not meant to be inferential (meant to draw conclusions about relationships generally).

Science is a social process, so communication is vital, and part of the skill of being a scientist is explaining these concepts in a way that is interpretable to your audience. Rhetorically, I find the idea that, "One person represents more than one percent" being problematic to be convincing to most audiences, albeit arbitrary.

Yes, it is problematic, but you will find yourself in situations you will have to do it, if you are working in applied science of any kind. The key there is communication.

Studying the communication numbers is just as important and fundamental to the practice of statistics as learning methodologies, software, etc. If you are interested in learning more about statistics and communication, I highly recommend Statistics As Principled Argument by Robert Abelson. Required reading for anyone whose job it is to communicate data to a wide variety of people, in my opinion.

Answer 3

The percentage (or proportion) is the maximum likelihood estimator for the true percentage (or proportion) for a binomial distribution*. That does not really change with the sample size. However, on its own it is not a sufficient statistic (i.e. only reporting the percentage throws away important additional information), while percentage (assuming no rounding) and total number are, similarly the number of positive cases and the total number, or the number of positive and negative cases, or percentage with confidence intervals etc. all are sufficient statistics.

To give an example 100% based on 1 trial tells you much less about the underlying parameter than 100% based on 100,000 trials. Not making it clear which it is is not good practice.

* Footnote on the binomial distribution bit: This of course assumes that there's no difference between all the trials that we know about (e.g. if we look at the proportion of deaths from any cause in a year across the population of a country, one would be silly to ignore additional information and think that this is an appropriate estimate for the proportion for all future years, even if e.g. the age distribution of the population changes, or medical treatment and/or sanitation options improve

Answer 4

If you want to be really technical about it, I suppose you can consider the "percent" operator to be a correspondence (not a function) that maps the decimal value to all possible pairs of non-negative integers or perhaps real numbers giving that decimal value. Let $\%$ be the "percent" correspondence and $x$ be the value (e.g., $x = 10\% = 0.10$).

$$ \%(x) \in \left\{ \left(a,b\right)\in\mathbb Z_{\ge 0}\times \mathbb Z_{\ge 0} \mid \frac{a}{b} = x \right\} $$

In this formulation, there is no concern about how many observations there were: $80\%$ corresponds to both $80/100$ and $4/5$.

(Or we can just use the word the way people have used it for years while causing minimal problems in communication, despite the fact that almost no one has heard of the formal mathematical term correspondence.)