Significant digits and reported precision

Reported precision should reflect the information a measurement supports. Significant digits, calculation rules and rounding conventions help prevent extra digits from implying unwarranted certainty or too few digits from hiding a meaningful difference.

Measurement and uncertainty

Significant digits, or significant figures, comprise the certain digits in a measured value plus one uncertain digit. ¹ For example, measurements of the same object on a kitchen scale, triple beam balance and analytical balance may be reported as 4 g, 3.6 g and 3.582 g, with one, two and four significant digits respectively. ^{1, 2} Reporting too many or too few significant digits can misrepresent scientific data and affect decisions. ^{1, 2}

Accuracy describes agreement with the true value, whereas precision describes agreement among repeated measurements under the same conditions. ³ A precise measurement can be inaccurate, and an accurate measurement can be imprecise. ³ Both inaccuracy and imprecision contribute to uncertainty in measurements and in quantities calculated from them. ³ Uncertainty can be expressed through limited significant digits, in measurement units or as a percentage of the value. ³ For example, 25.2 mL ± 0.05 mL corresponds to approximately 25.2 mL ± 0.2%. ³

Reporting 3.50 implies finer precision than reporting 3.5. ³

Counts can also require approximation: New York City's 2018 population was reported as 8.40 million. ³ A value whose precision supports 5.4 should not be reported as 5.38761. ³ Software's ability to display many decimal places does not establish that they are meaningful. ¹

Counting significant digits

Five rules determine the significant digits in a measured quantity. ^{4, 5}

1. All nonzero digits are significant: 4.63 mg has three significant digits and 9.8 °C has two.
2. Zeros between nonzero digits are significant: 208 g has three significant digits and 5.004 µg has four.
3. Leading zeros are placeholders, not significant digits: 0.04 mL has one significant digit and 0.053 nm has two.
4. Trailing zeros to the right of the decimal point are significant: 1.0630 dL has five significant digits and 0.70 mg has two.
5. Trailing zeros in whole numbers are not necessarily significant: 140 cm may have two or three significant digits and 52,400 km may have three, four or five.

^{4, 5}

Scientific notation removes ambiguity about trailing zeros: 52,400 km can be written as 5.24 × 10⁴ km, 5.240 × 10⁴ km or 5.2400 × 10⁴ km to indicate three, four or five significant digits. ⁵

Exact numbers do not limit the significant digits in a calculation because they are treated as having infinitely many significant digits. ⁵ Examples include the conversion 1 in = 2.54 cm, the constants π and e, and a count of 27 students in a classroom. ⁵

Calculations

Calculations using measured inputs cannot acquire greater precision merely through arithmetic. ⁶ For addition and subtraction, the rounding rule uses the number of decimal places in the least precise input: 135 + 53.487 = 188.487, rounded to 188. ⁶ For multiplication and division, it uses the smallest number of significant digits among the inputs: 6.2 × 24.63 = 152.706, rounded to 150 with two significant digits. ⁷

Logarithms and antilogarithms

For base-10 logarithms, the integer part is the characteristic and the fractional part is the mantissa. ⁸ The characteristic represents the power of ten, so the number of digits retained after the decimal point corresponds to the significant digits in the original value. ⁸ Thus log(3.00 × 10⁴) = 4.47712125472 is rounded to 4.477, retaining three decimal places. ⁸

For an antilogarithm, the result has as many significant digits as there were digits in the logarithm's mantissa. ⁸ A pH of 6.47 corresponds to a hydrogen ion concentration of 3.4 × 10⁻⁷, and a pKa of 4.578 corresponds to Ka = 2.64 × 10⁻⁵. ⁸ Similarly, 10^(2.53) = 338.8442 is rounded to 340, or 3.4 × 10², with two significant digits. ⁸

Rounding conventions

The following rounding rules limit reported precision, rounding bias and intermediate calculation error. ⁹

1. If the digit to be dropped is greater than 5, add 1 to the last retained digit: 84.7 becomes 85.
2. If it is less than 5, leave the last retained digit unchanged: 26.4 becomes 26.
3. If it is 5 followed by any nonzero digit, add 1 to the last retained digit: 28.51 becomes 29.
4. If it is 5 followed only by zeros, increase an odd last retained digit by 1 and leave an even one unchanged: both 47.5 and 48.5 become 48.
5. In multistep calculations, retain as many intermediate digits as practical, at least one more than the final result, to avoid round-off error.

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Rounding ties to an even retained digit is intended to avoid bias by averaging rounding errors across calculations. ⁹

Planning reported precision

The statistical analysis plan should specify the appropriate precision for each output variable so that tables, figures and listings report meaningful values. ¹⁰ This is particularly relevant to clinical study reports, which combine data from many sources. ¹⁰

A worked reporting example used a bioanalytical method validated over a calibration range of 7.754–3969 ng/mL and recommended four significant digits for plasma concentrations throughout that range: 7.754–9.999, 10.00–99.99, 100.0–999.9 and 1000–3969 ng/mL. ¹⁰ A reported value of 3938.748 ng/mL has seven significant digits, exceeding the precision supported by the method in that example. ¹⁰

Medical writers, statisticians, clinical reviewers and others involved in research reporting share responsibility for appropriate reported precision. ¹¹ Meaningful results support decisions by clinicians, regulators, policymakers and other stakeholders. ¹²

Reasons to round and reasons to retain digits

Statistical reproducibility can limit useful digits: simulations of correlation coefficients across different coefficients and sample sizes found two decimal digits statistically meaningful only with samples above 500 and three only with samples above 100,000. ¹³

Limiting numbers to at most two significant digits can aid comprehension. ¹³ If greater precision is unnecessary, rounding a procedure cost of $32,833 to $33,000 should improve comprehension and recall. ¹³

Practical meaning also matters: a mean age of 32.81 years includes about 296 days beyond 32 years, and changing 0.81 to 0.8 years changes it by about four days. ¹⁴ Rounding to the nearest year is pragmatic advice for this example, with similar reasoning applicable to other demographic and clinical characteristics. ¹⁴

Terminal digit bias is a tendency to round recorded values to zero, an even digit or five. ¹⁵ It has been reported for manually measured blood pressure, emergency department discharge times, birth weights, self-reported age among cancer patients in developing countries and self-reported menopause onset year. ¹⁵

Among almost 29,000 women, blood pressure measurements ended in zero in 78% of cases, another even digit in 15%, five in 5%, and another odd digit in 2%. ¹⁵ In that study, changing the hypertension threshold from at least 140 mm Hg to greater than 140 mm Hg reduced estimated prevalence from 26% to 13% overall, from 15% to 6% in the low-risk subgroup and from 43% to 25% in the high-risk subgroup. ¹⁵ In another study, caregivers recorded diastolic pressures of 88–89 rather than 90–99 mm Hg for patients they thought did not need treatment; this difference was associated with a relative risk for excess cardiovascular death of 2.6 (95% confidence interval 1.4–4.6). ¹⁵

Consequences of inappropriate precision

Three examples illustrate the potential effects of inappropriate precision. ¹¹

1. A misleading claim of precision in an epidemiological study of phenylpropanolamine and stroke was suggested to have influenced policymakers and contributed to withdrawal of the product from the US market.
2. A potassium concentration rising from 4.8 to 5.2 mmol/L with assay imprecision of 0.1 mmol/L was considered a likely patient change rather than assay imprecision alone; reporting both values as 5 mmol/L would hide the 0.4 mmol/L increase. Laboratory results reflecting measurement imprecision give clinicians greater confidence in judging whether changes reflect normal variability or possible disease.
3. A scientific advisory panel to the US Environmental Protection Agency concluded that extra significant figures in cancer risk estimates could imply false precision and mislead risk managers.

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