How to Read Uncertainty From a Ruler, Burette, or Digital Meter

Rule Statement

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.