I have found various and seemingly related explanations:

  1. Multiplying by 10,000 [patient days or patients] standardizes the rate so it can be compared to other hospitals / populations that may have fewer or greater number of patient days or patients.
  2. Multiplying by 10n converts decimal fractions to a standard population size which is a more understandable description of the prevalence within a population.
  3. Multiplying by 10n is performed because the frequency of the numerator compared to the denominator is usually rather low.

The third explanation makes the most sense to me, since a relatively small numerator and large denominator can result in very small fractions with lots of leading zeros (e.g., 0.000045) and multiplying by 10,000 would give an easier to read/interpret number (e.g., 4.5). Contrastingly, explanation #1 above is not satisfying since it seems legitimate to compare hospital A with a rate of 0.000045 to hospital B with a rate of 0.000067, for example, without the need to compare these same rates only after "standardizing" / multiplying by 10,000 (e.g., 4.5 vs 6.7).

What is the underlying theory for this type of standardization and why is it so common in health and epidemiological measures (e.g., Clostridium diff. infections, SSERs, etc.)?


Just for convenience, like you point out. The same thing is done in nearly every science, often by changing the SI units scale. We talk about weights of humans in kilograms, but measure drug doses in milligrams and micrograms. For relatively rare cases in a population, the equivalent to talk in terms of N per 10^k.

It's difficult to tell at a glance that 0.00010 is larger than 0.000094, and gets harder and harder the more decimal points one has, whereas 1.0 versus 0.94 is less ambiguous. They also take up less space in a table, and are easier for humans to comprehend.

I don't think it makes sense to try to decide on one reason. There is no formal reason at all, but rather a collection of motivations.

  • Thank you, Bryan. This confirmed my intuition. As you stated, there are many motivations for multiplying by 10^n, but could it really be considered a type of mathematical or statistical "standardization" like reasons #1 and #2 suggest? To me, it is not but rather functions to move the decimal point to make the number easier to read. – Lucas Jan 27 '20 at 16:22
  • @Lucas There are two completely different procedures here. One is moving the decimal point to express the outcomes as ratios rather than probabilities (i.e., 1 per 1,000 rather than a probability of 0.001). The other is using a probability or ratio rather than a simple N (like saying something occurred in 120 people) without reference to the sample size. There are cases where reporting a straight N makes sense (the current news around 2019-nCov is an example) and cases where reporting the ratio makes sense. – Bryan Krause Jan 27 '20 at 16:35
  • Makes sense. But to clarify, can either of these two procedures be considered types of "standardization" that statically allows a practitioner to compare one population to another and without which comparisons should not be performed? – Lucas Jan 27 '20 at 17:12
  • @Lucas I would not consider it standardization, no; it's just a different way to display the same number. In the situation of considering cases versus probability or ratios, I would not really think of that as standardization, either, it's just a completely different scale of measurement. Total cases is useful if you are, say, allocating funding. Probability is useful if you want to compare rates/odds. – Bryan Krause Jan 27 '20 at 17:17
  • @Lucas I would add that, when doing statistical comparisons, you would not want to treat a number like 1/10,000 as .0001 or normalize it to "1 per 10,000" and treat it as if it's the number "1". You would want to do your statistics as a logistic regression where you explicitly express both the number of cases and the size of the total population. – Bryan Krause Jan 27 '20 at 17:25

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