How do I write orders of magnitude in scientific notation when margin of error is involved?

Question

My TA has me confused about how to use scientific notation when margin of error and orders of magnitude are involved.

The number I wrote: 3.78 ± 0.02 * 10^-3

My TA told me that this means: 3.78 ± 0.00002

I thought that this meant: 0.00378 ± 0.00002

Which one is correct? I've been looking online to find something that explains the proper notation when writing out margin of error with orders of magnitude but I haven't been able to find anything (I'm at a loss of what terms to search for). If what I wrote isn't the correct way to say 0.00378 ± 0.00002, then what is? Writing it as 3.78 * 10^-3 ± 0.02 or 3.78 * 10^-3 ± 0.02 * 10^-3 seems kind of foolish.

Answer 1

Sometimes you need to go the extra mile to be entirely unambiguous. Your original formulation (3.78 ± 0.02 * 10^-3) could be interpreted in two ways, the way you intended and the way your TA suggested. The majority of people, I expect, would use context to interpret it correct as you intended, but some (as your TA demonstrates) will not.

There is no clear cut widely agreed standard that will resolve this. But you can apply standard mathematical notation (which will look superfluous to some) to make what you write entirely unambiguous. As Ivan suggested in a comment the simple addition of brackets giving (3.78 ± 0.02) * 10^-3 makes your statement unambiguous in its interpretation. This is more compact than 3.78 * 10^-3 ± 0.02 * 10^-3 but just as unambiguous.

If you were using properly typeset formulae, you might well be able to avoid some of the apparent redundancy of the last formulation, but that would take more effort. Better to accept that redundancy is sometimes required to make things clear.

Answer 2

I think your TA is wrong, based on the following argument.

When you write a number derived from calculation or experiment, and following standard protocols for the propagation of uncertainties, you regard the last reported integer in the number as uncertain. Therefore, it simply does not make sense to report an uncertainty of ±0.00002 for 3.78, since the uncertainty in 3.78 must be at least ±0.01. I would say that although there may be cases in which explicit notation should be followed, it is generally safe to write

$3.78 ± 0.02 \times 10^{-3}$

as meaning

$0.00378 ± 0.00002$

A safe alternative (consult a standard analytical chemistry textbook such as Skoog, West and Holler's) is to write

$3.78 (±0.02) \times 10^{-3}$

The use of parentheses emphasizes that the uncertainty defines a range of possible values.

Postscript: Another similar argument can be made for the placement of units, for instance should one write

$3.78 ± 0.02 ~kcal/mol$

or

$3.78 ~kcal/mol ± 0.02 ~kcal/mol$

Clearly the second case borders on the absurd. This is a recurrent problem during drafting of scientific text. Another similar situation is the following: should one write

A and B were determined to be 1.0 and 2.0 kcal/mol, respectively.

or

A and B were determined to be 1.0 kcal/mol and 2.0 kcal/mol, respectively.

or

We determined values of A=1.0 kcal/mol and B=2.0 kcal/mol.

Clearly the first case is acceptable, even if it could be construed as ambiguous, and is not necessarily the most concise. Scientific writing is subject to different rules than standard prose.

Clarity is very important, conciseness too, and there are, in general, recommended conventions. See this answer for instance, which suggests that using the "±" symbol is not recommended unless you mean to use the uncertainty to estimate the distribution of errors.