When reading a measurement from an instrument, the recorded value must reflect both the magnitude and the uncertainty of the reading. The uncertainty is determined by the instrument’s resolution and the measurement context. For analog instruments (ruler, burette), the uncertainty is typically taken as half the smallest scale division, though some conventions use the full division. For digital instruments, the uncertainty is usually ±1 in the last displayed digit, unless the manufacturer specifies a different value. The number of significant figures in the recorded value must be consistent with this uncertainty: the last significant digit should be in the same decimal place as the uncertainty.

This rule aligns with the Guide to the Expression of Uncertainty in Measurement (GUM), which states that the uncertainty should be given to at most two significant figures and that the measurement result should be rounded to match the uncertainty (JCGM 100:2008, Clause 7.2.6). For practical work, we use the instrument’s resolution to estimate the uncertainty, following the principle of resolution-limited uncertainty.

Worked Examples

Example 1: Ruler with millimeter divisions

A standard metric ruler has divisions of 1 mm. The smallest division is 1 mm, so the uncertainty is ±0.5 mm (half the division). When reading a length, you should estimate to the nearest 0.1 mm (one decimal place beyond the smallest division) to get a meaningful reading. For instance, if the object’s edge falls between 23 mm and 24 mm, you might estimate it as 23.4 mm. The recorded value is 23.4 mm ± 0.5 mm. This has three significant figures, with the last digit (tenths place) matching the uncertainty’s decimal place.

Example 2: Burette with 0.1 mL divisions

A typical burette has graduations every 0.1 mL. The uncertainty is ±0.05 mL (half of 0.1 mL). When reading the meniscus, you estimate to the nearest 0.01 mL. For example, if the meniscus is at 12.35 mL, the reading is 12.35 mL ± 0.05 mL. The recorded value has four significant figures, with the hundredths place matching the uncertainty.

Example 3: Digital multimeter displaying 3.456 V

For a digital meter, the resolution is the smallest change in the displayed value. If the display shows 3.456 V, the resolution is 0.001 V. The uncertainty is typically ±0.001 V (or sometimes ±0.0005 V if the manufacturer specifies a more precise uncertainty). The recorded value is 3.456 V ± 0.001 V. Do not add extra digits beyond what is displayed; the display already includes the uncertainty in its last digit.

Counter-Examples

These common errors illustrate how NOT to read uncertainty.

• Overestimating precision on a ruler: Reading a ruler with 1 mm divisions to 0.01 mm (e.g., 23.45 mm) is incorrect because the uncertainty is ±0.5 mm, so the hundredths place is meaningless. The correct reading is 23.4 mm ± 0.5 mm (or 23.5 mm if you estimate that precisely).
• Ignoring the last digit on a digital meter: If a digital meter displays 3.456 V, writing 3.46 V (rounding to three decimal places) loses the resolution. The uncertainty is ±0.001 V, so the reading should be 3.456 V, not 3.46 V.
• Using the full division as uncertainty: Some textbooks say the uncertainty is the smallest division, not half. For a ruler with 1 mm divisions, this would give ±1 mm. While this is a conservative estimate, it is not the standard convention for a single reading. The GUM recommends half the division for analog scales when the operator can interpolate (JCGM 100:2008, Clause 4.6.1).

Convention Comparison Table

Note: Some disciplines (e.g., chemistry) may adopt a different convention, such as using the full division for analog instruments when the scale is coarse. Always follow your laboratory or industry standard.

Standards Citation

The following standards provide authoritative guidance on uncertainty and significant figures:

• JCGM 100:2008 (GUM): Guide to the Expression of Uncertainty in Measurement. Clause 7.2.6 recommends that the uncertainty be given to at most two significant figures and that the measurement result be rounded to the same decimal place as the uncertainty.
• ASTM E29-13: Standard Practice for Using Significant Digits in Test Data to Determine Conformance with Specifications. This standard defines how to round test results and how to determine the number of significant digits based on the measurement resolution.
• ISO 80000-1:2009: Quantities and units – Part 1: General. Annex C provides guidance on rounding and significant figures, emphasizing that the number of significant digits should be consistent with the uncertainty.
• NIST Technical Note 1297: Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results. This document offers practical advice on uncertainty estimation for physical measurements.

When in doubt, consult the specific standard applicable to your field. For general scientific work, the GUM is the primary reference.

Common Mistakes

1. Recording too many digits: For analog instruments, recording more than one decimal place beyond the smallest division is a common error. For example, reading a ruler with 1 mm divisions to 0.01 mm is false precision.
2. Recording too few digits: For digital instruments, dropping trailing zeros or rounding off the last digit loses information. If a digital meter displays 3.000 V, record all four digits, not 3 V.
3. Confusing resolution with accuracy: The uncertainty from resolution is only one component. The instrument’s accuracy (e.g., ±0.5% of reading) may be larger. Always check the manufacturer’s specifications.
4. Inconsistent rounding: When reporting a result, the uncertainty and the value must have the same decimal place. If the uncertainty is 0.05, the value must be reported to the hundredths place (e.g., 12.35 ± 0.05, not 12.3 ± 0.05).
5. Ignoring parallax and calibration: These are not directly related to reading uncertainty, but they affect the overall uncertainty. Always align your eye perpendicular to the scale to avoid parallax error.

Practice Problems

Test your understanding with these exercises. Answers are provided below.

1. A burette has graduations every 0.1 mL. You read the meniscus at 24.7 mL. What is the correct reading with uncertainty?
2. A digital thermometer displays 36.5 °C. What is the uncertainty and the correct recorded value?
3. A ruler with 0.5 cm divisions is used to measure a length. The edge falls between 4.5 cm and 5.0 cm. Estimate the reading and its uncertainty.

Answers:

1. 24.7 mL ± 0.05 mL (since the smallest division is 0.1 mL, half is 0.05 mL).
2. 36.5 °C ± 0.1 °C (assuming the display shows one decimal place, the resolution is 0.1 °C).
3. 4.7 cm ± 0.25 cm (half of 0.5 cm). Estimate to 0.1 cm, so 4.7 cm is appropriate.

Quick Reference Table

For more on significant figures and rounding, see our significant figures calculator and the rounding rules article.

Sources & Further Reading

• JCGM 100:2008, Evaluation of measurement data — Guide to the expression of uncertainty in measurement, BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP, OIML.
• ASTM E29-13, Standard Practice for Using Significant Digits in Test Data to Determine Conformance with Specifications, ASTM International.
• ISO 80000-1:2009, Quantities and units — Part 1: General, ISO.
• NIST Technical Note 1297, Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results, NIST.
• Taylor, J. R., An Introduction to Error Analysis, University Science Books, 1997.

For further reading on related topics, see our articles on Accuracy vs Precision and Error Propagation.

The post How to Read Uncertainty From a Ruler, Burette, or Digital Meter appeared first on SignificantFiguresCalculator.

Rule Statement

The formal definition of measurement uncertainty, as given in the Guide to the Expression of Uncertainty in Measurement (GUM, JCGM 100:2008, clause 2.2.3), is:

“Non-negative parameter characterizing the dispersion of the quantity values being attributed to a measurand, based on the information used.”

This definition emphasizes three key aspects:

• Dispersion – uncertainty is a spread, not a single point.
• Quantity values – it applies to the values of a measurand, not to the measurement process itself.
• Based on the information used – uncertainty depends on the available knowledge, including both statistical data and scientific judgment.

Uncertainty is not an error. An error is the difference between a measured value and the true value; it is a single, often unknown quantity. Uncertainty, on the other hand, is a parameter that describes the range within which the true value is expected to lie, with a given level of confidence. It is a measure of doubt, not a mistake.

Worked Examples

Example 1: Simple Length Measurement

Suppose you measure the length of a metal rod using a ruler with 1 mm divisions. You take five readings: 152 mm, 153 mm, 152 mm, 154 mm, 153 mm.

Step 1: Evaluate Type A uncertainty (statistical)

Calculate the mean: (152+153+152+154+153)/5 = 152.8 mm. The standard deviation of the mean (standard uncertainty) is the experimental standard deviation divided by √5. The sample standard deviation is approximately 0.84 mm, so the standard uncertainty is 0.84/√5 ≈ 0.38 mm.

Step 2: Evaluate Type B uncertainty (non-statistical)

The ruler’s resolution is 1 mm. Assuming a rectangular distribution, the standard uncertainty from resolution is 1/(2√3) ≈ 0.29 mm. If the ruler has a calibration certificate stating an expanded uncertainty of 0.2 mm (k=2), the standard uncertainty is 0.1 mm. Combine these Type B components: √(0.29² + 0.1²) ≈ 0.31 mm.

Step 3: Combine standard uncertainties

Combine Type A and Type B: √(0.38² + 0.31²) ≈ 0.49 mm. This is the combined standard uncertainty, u_c.

Step 4: Expanded uncertainty

For a 95% confidence level, use a coverage factor k=2 (assuming normal distribution). Expanded uncertainty U = 2 × 0.49 ≈ 0.98 mm. Report as 152.8 ± 1.0 mm (k=2).

Example 2: Calibration of a Thermometer

A thermometer is calibrated against a reference standard. The reference standard has a certified uncertainty of 0.05 °C (k=2). The thermometer is read 10 times, giving a standard deviation of 0.07 °C. The resolution of the thermometer is 0.1 °C.

Type A: u_A = 0.07/√10 ≈ 0.022 °C.

Type B: From reference standard: 0.05/2 = 0.025 °C. From resolution: 0.1/(2√3) ≈ 0.029 °C. Combined Type B: √(0.025² + 0.029²) ≈ 0.038 °C.

Combined standard uncertainty: √(0.022² + 0.038²) ≈ 0.044 °C. Expanded (k=2): 0.088 °C, report as 0.09 °C.

Counter-Examples

Treating Uncertainty as an Error

Wrong: “The measurement uncertainty is ±0.5 mm, so the true value could be 0.5 mm away.” This implies the uncertainty is an error bound, but uncertainty is a probabilistic dispersion, not a hard limit.

Using Tolerance Instead of Uncertainty

Wrong: “The part has a tolerance of ±0.1 mm, so the measurement uncertainty is ±0.1 mm.” Tolerance is a specification for the product, not a property of the measurement. Uncertainty must be evaluated from the measurement process itself.

Ignoring Type B Contributions

Wrong: “I only report the standard deviation of my readings as the uncertainty.” This neglects systematic effects like calibration, resolution, and environmental factors. The GUM requires a complete uncertainty budget.

Convention Comparison Table

Standards Citation

The following standards and guides provide the authoritative basis for evaluating and reporting measurement uncertainty:

• GUM (JCGM 100:2008) – Evaluation of measurement data — Guide to the expression of uncertainty in measurement. Clauses 2.2.3 (definition), 4.2 (Type A), 4.3 (Type B), 5.1 (combined standard uncertainty), 6.2 (expanded uncertainty).
• NIST TN 1297 – Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results. Provides practical implementation of GUM, with specific guidance on significant figures (Section 7).
• ISO 5725-1 – Accuracy (trueness and precision) of measurement methods and results. Defines repeatability and reproducibility, which are components of uncertainty.
• ASTM E29 – Standard Practice for Using Significant Digits in Test Data to Determine Conformance with Specifications. Governs rounding of test results, which is directly related to how uncertainty is reported.

Common Mistakes

• Omitting the coverage factor – Reporting “± 0.5” without specifying k or confidence level is meaningless.
• Rounding the result before the uncertainty – Always round the uncertainty first, then round the measured value to the same decimal place.
• Using the standard deviation of the sample instead of the standard deviation of the mean – The uncertainty of the mean is smaller than the spread of individual readings.
• Forgetting to include resolution effects – Every digital or analog instrument contributes a Type B uncertainty from its resolution.
• Double-counting components – For example, including both a calibration correction and its uncertainty when the correction is already applied.

Practice Problems

Problem 1: A balance has a calibration certificate stating an expanded uncertainty of 0.02 g (k=2). The balance’s resolution is 0.01 g. You weigh a sample 4 times: 10.23, 10.25, 10.22, 10.24 g. Compute the combined standard uncertainty and report the result with expanded uncertainty (k=2).

Solution: Type A: mean = 10.235 g, sample std = 0.0129 g, u_A = 0.0129/√4 = 0.00645 g. Type B: calibration: 0.02/2 = 0.01 g; resolution: 0.01/(2√3) ≈ 0.00289 g. Combined Type B: √(0.01² + 0.00289²) ≈ 0.0104 g. Combined standard: √(0.00645² + 0.0104²) ≈ 0.0122 g. Expanded: 0.0244 g, round to 0.024 g. Result: 10.235 ± 0.024 g (k=2).

Problem 2: A pH meter has a stated accuracy of ±0.02 pH units. You take 3 readings: 7.01, 7.02, 7.01. Estimate the standard uncertainty (assume the accuracy is a rectangular distribution).

Solution: Type A: mean = 7.0133, sample std = 0.00577, u_A = 0.00577/√3 ≈ 0.00333. Type B: accuracy as rectangular: 0.02/√3 ≈ 0.01155. Combined: √(0.00333² + 0.01155²) ≈ 0.0120 pH.

Quick Reference Table

Sources & Further Reading

• JCGM 100:2008, Evaluation of measurement data — Guide to the expression of uncertainty in measurement (GUM).
• NIST TN 1297, Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results.
• ISO 5725-1:1994, Accuracy (trueness and precision) of measurement methods and results.
• ASTM E29-13, Standard Practice for Using Significant Digits in Test Data to Determine Conformance with Specifications.

For more on rounding and significant figures, see our Significant Figures Rules and Rounding Methods articles.

The post What Measurement Uncertainty Actually Is (and Why It Matters) appeared first on SignificantFiguresCalculator.

The universally accepted rule is that an uncertainty (also called measurement uncertainty, standard deviation, or expanded uncertainty) should be reported with one significant figure if the leading digit is 1 or 2, and with two significant figures if the leading digit is 3 or greater. This rule is a compromise between precision and practicality: one sig fig is often too coarse for small leading digits (e.g., 0.14 vs. 0.1 loses 40% of the information), while two sig figs for large leading digits (e.g., 9.3 vs. 9) adds little value and may imply false precision.

This rule is endorsed by several metrology authorities, including the GUM (Guide to the Expression of Uncertainty in Measurement), NIST Technical Note 1297, and ISO 80000-1. It is also a common requirement in scientific journals and engineering standards.

When the uncertainty is reported with one significant figure, the measurement result must be rounded to the same decimal place. When two sig figs are used, the result is rounded to the same place as the last digit of the uncertainty. This ensures that the uncertainty and the result are consistent in precision.

Worked Examples

Example 1: Leading digit 1 or 2 → one sig fig

Suppose a length measurement yields a value of 12.345 cm with a calculated standard uncertainty of 0.0142 cm. The leading digit of the uncertainty is 1, so we round it to one significant figure: 0.01 cm. The measurement result must then be rounded to the same decimal place (hundredths): 12.35 cm. The final reported value is 12.35 ± 0.01 cm.

Example 2: Leading digit 3 or higher → two sig figs

Consider a mass measurement of 50.123 g with an uncertainty of 0.0467 g. The leading digit is 4, so we keep two significant figures: 0.047 g. The result is rounded to the same decimal place (thousandths): 50.123 g. The reported value is 50.123 ± 0.047 g.

Example 3: Edge case – leading digit exactly 2

If the uncertainty is 0.00298, the leading digit is 2, so one sig fig is used: 0.003. The result is rounded to the same place (thousandths). This avoids the awkwardness of reporting 0.0030 which would imply two sig figs.

Example 4: Expanded uncertainty (k=2)

For an expanded uncertainty (e.g., 95% confidence) of 1.34, the leading digit is 1, so we report 1. The result is rounded to the ones place. If the result is 25.6, it becomes 26 ± 1.

Counter-Examples

Common errors include reporting too many or too few significant figures in the uncertainty. Here are typical mistakes:

• Reporting three or more sig figs: Writing 0.0142 as the uncertainty when the leading digit is 1 is overkill. It implies a precision that the uncertainty itself does not justify.
• Reporting one sig fig when the leading digit is 9: If the uncertainty is 9.8, rounding to 10 (one sig fig) loses the distinction between 9.8 and 10.2. The rule says two sig figs for leading digit ≥3, so report 9.8.
• Inconsistent rounding of the result: If the uncertainty is rounded to 0.01, but the result is reported as 12.345, the last digit of the result is meaningless. Always round the result to the same decimal place as the uncertainty.
• Using the number of sig figs of the uncertainty to determine the sig figs of the result: The result’s rounding is determined by the decimal place, not by the count of sig figs in the uncertainty.

Convention Comparison Table

Standards Citation

The rule is not arbitrary; it is codified in international standards. Key clauses include:

• GUM (JCGM 100:2008), Section 7.2.6: “The uncertainty should be given to two significant figures if the leading digit is 1 or 2, and to one significant figure otherwise.”
• NIST Technical Note 1297 (1994), Section 7.5: “The numerical value of the uncertainty should not be given with more than two significant figures.”
• ISO 80000-1:2009, Annex C (C.3.4): “The uncertainty of a measurement should be expressed with one or two significant digits.”
• ASTM E29-13, Section 6: Discusses rounding of test data, recommending that the number of significant figures be consistent with the uncertainty.

These standards are widely adopted in physics, chemistry, engineering, and metrology. Adherence ensures that reported uncertainties are neither overly optimistic nor unnecessarily conservative.

Common Mistakes

1. Always using one sig fig: Some textbooks simplify the rule to “always one sig fig,” but this fails for uncertainties like 0.14, where 0.1 is a 40% change. The leading-digit rule is more accurate.
2. Always using two sig figs: This overstates precision for uncertainties like 9.3, where 9.3 vs. 9 is a negligible difference.
3. Rounding the uncertainty before rounding the result: Always round the uncertainty first, then round the result to match the decimal place.
4. Using the same number of decimal places as the original data: If the original measurement has more decimal places than the uncertainty, the extra digits are meaningless.
5. Ignoring the leading digit rule when the uncertainty is exactly 1.0 or 2.0: The rule applies to the first non-zero digit, so 1.0 has leading digit 1 and should be one sig fig (1), not two.
6. Reporting an uncertainty with a leading zero after the decimal: For example, 0.04 has one sig fig; 0.040 has two. The rule applies to the significant digits, not the decimal places.

Practice Problems

Test your understanding with these exercises. Answers are provided at the end.

1. Uncertainty = 0.000345, result = 1.23456. How many sig figs for the uncertainty? Round the result.
2. Uncertainty = 1.89, result = 45.6. What is the reported value?
3. Uncertainty = 0.020, result = 0.100. Should the uncertainty be 0.02 or 0.020? What is the final result?
4. Uncertainty = 0.00012, result = 0.000345. Report the value.

Answers:

1. Leading digit 3 → two sig figs: 0.00035. Result rounded to same place (5th decimal): 1.2346. Reported: 1.2346 ± 0.00035.
2. Leading digit 1 → one sig fig: 2. Result rounded to ones: 46. Reported: 46 ± 2.
3. Leading digit 2 → one sig fig: 0.02. Result rounded to hundredths: 0.10. Reported: 0.10 ± 0.02.
4. Leading digit 1 → one sig fig: 0.0001. Result rounded to 4th decimal: 0.0003. Reported: 0.0003 ± 0.0001.

Software Behavior Note

When using scientific software, be aware of how it handles uncertainty rounding. For example:

• Excel: There is no built-in function for rounding to significant figures. You must use custom formulas or the ROUND function with a calculated number of digits. Excel’s ROUND uses half-away-from-zero, which may differ from standard rounding.
• Python (NumPy/SciPy): The round() function uses banker’s rounding (half-to-even). When reporting uncertainties, you should implement the leading-digit rule manually.
• MATLAB: The round function rounds half away from zero. Use sigfig from the File Exchange or write a custom function.
• R: The signif() function rounds to a specified number of significant digits, but it does not apply the leading-digit rule automatically. You must decide whether to use 1 or 2 sig figs.
• LaTeX (siunitx): The num{...} command can round to a given number of significant figures, but you must specify the number manually.

Always verify that the software’s rounding method matches your intended rule. For critical work, implement the rule explicitly.

Quick Reference Table

Sources & Further Reading

• JCGM 100:2008, Evaluation of measurement data — Guide to the expression of uncertainty in measurement (GUM).
• NIST Technical Note 1297, Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results (1994).
• ISO 80000-1:2009, Quantities and units — Part 1: General.
• ASTM E29-13, Standard Practice for Using Significant Digits in Test Data to Determine Conformance with Specifications.
• IUPAC, Quantities, Units and Symbols in Physical Chemistry (Green Book), 3rd ed.

For a deeper dive, see our related articles on Sig Figs in Measurement Uncertainty and Rounding Rules.

The post How Many Sig Figs Should an Uncertainty Have? (Usually One or Two) appeared first on SignificantFiguresCalculator.