Significant Figures, Rounding and How a Result Is Reported
Two certificates report the same lot. One says 98.7%, the other says 98.65%. The material did not change and neither laboratory made an error — the difference is in how each chose to report a number, and those choices are governed by conventions that are rarely stated on the document itself.
What a significant figure claims
Every digit written down is an assertion that the digit is meaningful. Reporting 98.7% claims the tenths place is real; reporting 98.70% claims the hundredths place is too. The second is a stronger claim about the method, not a more careful description of the material.
The number of figures a result can honestly carry comes from the method’s precision, not from the instrument’s display. Software will happily print 98.6741%, and four decimal places on a chromatographic purity is a display artefact rather than a measurement — the spread in measurement uncertainty usually swamps everything past the first decimal.
Rounding happens once, at the end
The rule that causes most of the avoidable discrepancies: intermediate values carry extra digits, and rounding is applied once, to the final reported result.
Rounding at each step accumulates error in a direction that depends on the arithmetic, and the effect is not negligible when three or four operations are chained — a net content calculation combining chromatographic purity, water content and counter-ion content, of the kind set out in net peptide content, can move by several tenths of a percent depending purely on where the rounding was applied.
Half-up, half-even, and why laboratories differ
When the digit to be dropped is exactly five with nothing after it, two conventions are in use:
- Round half away from zero. 98.65 becomes 98.7. Taught in schools, and it introduces a small upward bias across many values.
- Round half to even. 98.65 becomes 98.6, while 98.75 becomes 98.8. The bias cancels over a large set of values, which is why it is the default in most statistical software and in several standards.
Neither is wrong. What matters is that the convention is fixed in advance and applied consistently, because a laboratory that switches conventions between lots has introduced a difference that looks like a change in the material.
Rounding against a specification limit
This is where the arithmetic stops being cosmetic. A result of 97.96% against a limit of “not less than 98.0%” passes if the result is rounded to the precision of the limit first, and fails if it is compared as measured.
The ordinary convention is that the result is rounded to the same number of decimal places as the limit and then compared — so 97.96 becomes 98.0 and complies. That convention is defensible, and it is also the reason a limit should be written with the precision actually intended: “not less than 98.0%” and “not less than 98%” are different specifications, and a lot at 97.6% passes the second and fails the first. The relationship between limit and method capability is covered in how a specification limit gets set.
Where extra digits are actively misleading
- Chromatographic area percentage. Integration decisions — where a baseline is drawn, where a shoulder is split — move the result more than the third decimal place ever could, as described in peak integration.
- Mass measurements. A calculated mass quoted to four decimals is meaningful; an observed mass quoted to four decimals is only meaningful if the instrument’s accuracy supports it, which is the subject of mass accuracy in parts per million.
- Anything below the quantitation threshold. A value reported as 0.03% by a method that quantifies down to 0.05% is a number the method cannot support, per limit of detection and limit of quantitation.
- Converted units. Converting 5 mg to 5000 µg is exact; converting a measured 5.0 mg to 5000.0 µg invents a digit.
The trailing zero, which is not decoration
98% and 98.0% differ. The trailing zero says the tenths place was determined and found to be zero, which is a claim about resolution. The same applies to a stated quantity: a vial labelled 5 mg and one labelled 5.00 mg are making different statements about how well the fill was controlled.
This is the one convention that is routinely violated in both directions — zeros dropped where they were earned, and added where they were not.
What to do with two certificates that disagree in the last digit
Check the precision of each result before concluding anything. A difference confined to a digit that neither method can resolve is not a disagreement about the material; it is two laboratories making different display choices.
A difference in a digit both methods can resolve is a real finding, and the reasons it happens — different columns, different gradients, different integration — are the subject of why certificates disagree on purity. Telling the two situations apart takes one glance at the number of figures, and it is the difference between a genuine discrepancy and an argument about rounding.
