Moses, Newton and Lincoln are sitting at a bar. Jack Ladenson walks in, but the bartender has him thrown out. "TMSF," says the bartender, "— too many significant figures!"
As a topic of conversation, significant figures will likely bring to mind either personages such as Lincoln, or shapes such as Hedy Lamarr, depending on who you have cornered for your engaging conversation. But, most science types will eventually recall the "TMSF" (too many significant figures) written in red on their freshman year chemistry problem sets. A Google search on "significant figures" in fact homes in on this context quite precisely, referring one to numerous maths, chemistry and physics course notes that are devoted to the topic. The definition of significant figures varies, well, significantly among these notes, with some muddling of the concepts of accuracy and precision. I'll use the definition here that significant figures is the number of leading meaningful digits in a numerical result, the last of which conveys the scale on which there is uncertainty (1). The discussion here is to ask the question among readers: should chemistry lab results ever carry more than 3 SF?
I did a survey of SF reporting at my home institution some years ago (2) and discovered that we were regularly reporting as many as 6 SF for some assays (e.g., for high [TSH] using a sensitive TSH assay, with results such as 123.456 mIU/L), particularly for automated assays. Granted, I would not contend that there is any clinical care issue here – other than that clinicians can truthfully regard us as purveyors of sometimes-ridiculous numbers. We might do better.
Part of the problem lies in simple informatics associated with laboratory automation. Automated analyzers, lab middleware and LISs allow or require setting of numbers of decimal points in a result, but I am unaware of any that deal directly with SF. These systems can usually handle calculations associated with specifying SF, but such intervention usually must be implemented rather laboriously for each and every analyte. It seems to me that the industry of chemistry analyzers and software should approach this problem more directly.
Question: should there ever be more than 3 SF in chemistry reports? I have no assays in my laboratory with CVs less than 1%. It follows that uncertainty is usually at the scale of the second or third digit. For instance, at 5% CV, 4 SF is not appropriate for [PSA] = 141.1 ng/mL, yet we report that sort of number every day.
A related issue is whether we should simply report confidence intervals for some assay results. This might be unwelcome additional information for many clinicians, but it would be more complete information. In my experience, discussion with clinicians regarding imprecision of certain lab results often draws genuine surprise. Whereas they do not misinterpret patently ridiculous numbers such as the high TSH, they take asterisks (designations of abnormal results) very seriously, and they are surprised to learn that the PSA result of 4.1 ng/mL is in fact not distinguishable from 4.0 ng/mL -- after all, from their perspective, they have a report in front of them that says it is not 4.0 ng/mL.
I have only ever been asked directly about precision with respect to tumor markers -- e.g., whether the 387 units from 3 months ago is distinct from the 407 units reported yesterday. It's a good question. Note that there is a huge psychological component at play here related to units and scale -- human reactions to numbers and to differences between numbers is likely to be strongly dependent on scale. Would the clinician ask the question if units were such that results were instead either 0.387 units and 0.407 units, or 387,000 units and 407,000 units? In both cases, probably not, but for opposite reasons – the difference seems small in one case, and large in the other. A clinician once remarked to me that he thought the numbers corresponding to Troponin I elevations (in conventional units of ng/mL) “seem awfully small to be abnormal." Hah! Ultimately, assay imprecision information will certainly be required data for automated expert systems -- and a new era in which such systems are widely utilized to manage lab test ordering, interpretation, and response is probably not that far off.
Back to 3 SF: are there circumstances in chemistry wherein 4 SF is appropriate? How about in areas of quantitative microbiology and quantitative molecular diagnostics? I welcome any comments about whether requirements or rationale in any of these areas would indicate a need to go beyond 3 SF.
See references (3-5) for additional literature on the topic of SF.
A historical note: Hedy herself was a science type – see references (6,7)
- Levy G. Significant figures or significant nonsense? Clin Pharmacol Ther 1996;59(3):363 (PMID:8654000)
- Stickle DF, Shearon EC, Pirruccello SJ (2002). TMSF: Too Many Significant Figures in automated reporting of concentrations of some clinical chemistry analytes. Clin Chem 48(S6):A4 (#A-9).
- Hawkins RC, Johnson RN. The significance of significant figures. Clin Chem 1990;36(5):824. (PMID:2338006)
- Hawkins RC, Badrick T, Hickman PE. Over-reporting significant figures--a significant problem? Clin Chim Acta 2007;375(1-2):158-61. (PMID:16839534)
- Badrick T, Hickman PE. Significant figures. Clin Biochem Rev 2008;29 Suppl 1:S89-91. (PMID:18852865)
- Crease RP. Technology: inventing beauty. Nature 2011;479:474-5
- Hughes DR, Hendricks D. Spread-spectrum radio. Scientific American 1998(April);278:94-6.