# Avoiding fractional digits when referring to an average number of people

### Initial examples

Having studied students in a sufficiently large number of classes, you might say something like:

The average number of professional athletes per class is 0.15.

On average, there are 0.15 professional athletes per class.

This seems awkward, because there are no fractions of people in a class. Of course the phrase is technically (and I think even semantically) correct, because the average is in essence a fraction.

One way could be to define the quantity first, but this doesn't seem to be possible for all cases (my first example for instance):

Define household size as the number of people living in a household. The average household size in the US is 2.54. 1

### Solution for expressing percentages

With percentages (not exactly the same, but a neat example) you could use the following phrases to convey the same meaning (numbers are made up):

60% of people like eating cheese.

3 out of 5 people like eating cheese.

### Question statement

Is there a comparable way to avoid the awkward fraction in the example about averages?

### 'The examples are perfectly fine, why look for another phrase?'

I agree that the examples are fine, concise and correct. I am asking the question here, on a forum of English language enthusiasts, because (for the sake of this question) I am interested in how to communicate to the general public, including younger people, those who are new to the English language and those who only engage with numbers and statistics sporadically.

An interesting read on communicating science to the general public can be found in this article on Scientific American. This partial quote from the article that sums it up nicely:

”The broader audience science can reach, the bigger the benefit in terms of the new ideas you are transmitting as a scientist.”

The quote is from M. Du Sautoy, who according to the article is the Professor for Public Understanding of Science and a Professor of Mathematics at the University of Oxford.

• I don't really see the problem. You could of course say, 'Therefore an average class would have no professional athletes', as an aside.
– JDF
Commented Mar 20, 2018 at 15:05
• It's not awkward, just statistically suspect unless the sample size is mentioned. Commented Mar 20, 2018 at 15:06
• @PhilSweet I'd argue the opposite, it may sound awkward, but as a statistic it doesn't have to be suspect. Take for example rare diseases, i.e. diseases which affect fewer than one in 200.000 (US definition).
– JJJ
Commented Mar 20, 2018 at 15:10
• Depends on your audience. For those who are numerate and understand how averages work, non-integers are perfectly acceptable and meaningful. For those who aren't you can always fudge: 'the average household has between 2 and 3 people.'. Also, forcing integers may be just as confusing: converting '62.5% or people like cheese' to '125 people out of 200 like cheese' is confusing. Just say 'roughly 60%' Commented Mar 20, 2018 at 15:17
• @JJJ It doesn't have to be suspect when you have a feel for the sample size and have reason to accept the number with confidence. That is there in your medical example, but missing in the athlete example because we may reasonably wonder if there isn't just one athlete and six classes. Commented Mar 20, 2018 at 15:31

You and the other answerer are correct in that the following are all reasonable ways of conveying the same information:

1a. The average number of professional athletes per class is 0.15.

1b. On average, there are 0.15 professional athletes per class.

2a. On average 15% of classes have a professional athlete.

2b. About 1 in 7 classes have a professional athlete.

Although you and @MarkPerryman are right to point out that:

• 1a and 1b are equivalent to each other, and
• 2a and 2b are equivalent to each other (roughly, with some rounding), and yet
• 1a and 2a are not exactly equivalent because 2b ignores instances of more than 1 athlete in a class

So, I advocate for phrasing this information in terms of natural frequencies. This is a concept I learned from Gerd Gigerenzer's book, Calculated Risks: How to Know When Numbers Deceive You. In that book, he shares results from studies that show how even doctors themselves have great difficulty understanding probabilities associated with false positives and incidence rates when testing for a disease. When those same probabilities are phrased, not as percentages, but as natural frequences (proportions with large denominators that are powers of 10, like 10,000), then understanding is greatly improved. For this reason, I think such a method will be most useful for your context, because you are looking to reach a broad, non-technical audience.

So, for example, I would state the information as follows:

Among 100 classes, there will be 15 athletes, on average.

This avoids "ugly" decimals, fractions, and percentages entirely. It also skirts the small issue in the other answer by avoiding the phrase "per class" and considering how many athletes there are among a large number of classes, as a whole.

• Thanks for your answer, as Mark and myself discussed in the comments of his answer, your 1 and 2 are equivalent and your 3 and 4 are equivalent. You also point out that 3 and 4 are different from the first two, I think you should also start with that in your first sentence (perhaps rename them to 1a, 1b, and 2a, 2b). With that is stated clearly and considering your added references to (popular) literature, I think I can accept your answer.
– JJJ
Commented Mar 22, 2018 at 20:04
• @JJJ: Done. I hope it makes more sense now. Commented Mar 22, 2018 at 20:08

If you were unconcerned about being too precise (and willing to ignore classes with two or more athletes), you could say

On average 15% of classes have a professional athlete.

or

About 1 in 7 classes have a professional athlete.

But, for mathematically literate people, I don't think that the initial phrasing is awkward at all.

• Thanks, these ways of phrasing it is exactly what I'm looking for, instead of focusing on the people, focus on the classes. One downside, however, is that this assumes a class has at most 1 professional athlete. Suppose you have 100 classes of which 1 has 15 professional athletes and the others have none, then there is only 1/100=1% of classes with a (one or more) professional athlete(s). The average number of professional athletes per class, however, is still 15/100=0.15.
– JJJ
Commented Mar 20, 2018 at 15:16
• Agreed, if I wished to be mathematically precise then stick with the 0.15. But if you are fairly loosely dealing with averages I think you can get away with ignoring multiple athletes in a class. Commented Mar 20, 2018 at 15:20
• Your answer would be correct if there could be only one occurrence. For example, the fraction of people who suffer from a certain disease; you either suffer from it or not, you don't count people having the disease twice. If you indicate that in your answer, I think it is a good answer to the question (for certain cases). I will not mark this as the best answer (yet), because I'm wondering if there are solutions that also work for the class-athlete example.
– JJJ
Commented Mar 20, 2018 at 15:31