Units, significant figures, accuracy, precision, and uncertainty
Report measurements with meaningful units and honest digits, while separating closeness to a reference from repeatability.
Field diagramAccuracy and precision
Estimated reading
13 min
Estimated practice
30 min
Equipment
Calculator or spreadsheet; ruler or supplied measurement dataset; no hazardous materials.
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Objective
By the end, you can:
Use quantity and unit together and convert without invented precision.
Distinguish accuracy, precision, repeatability, and uncertainty.
Round and report results so the digits match the measurement process.
A number needs a unit
A measurement is a quantity expressed with a unit and a method. “12” is incomplete: 12 mL, 12 g, and 12 s mean different things. Use a consistent unit system for calculations and label axes, tables, and notebook values. NIST's SI guidance standardizes symbols and advises against carrying more digits through a conversion than the source measurement justifies.
Accuracy describes closeness to a reference or accepted value; precision describes how closely repeated results agree under stated conditions. A set can be precise but biased, or variable but centered near a reference. Uncertainty is a quantified description of doubt in a result based on relevant components such as repeatability, calibration information, resolution, and assumptions. It is not a synonym for error or a promise that the true value lies in a particular interval unless the reporting convention says so.
Repeatability: precision under closely repeated conditions.
Reproducibility: agreement under changed conditions such as operators or laboratories, when defined by the field.
Resolution: the smallest display or scale increment, not automatically the measurement uncertainty.
Significant figures: digits supported by the method and data, not a reward for extra decimal places.
Report with discipline
Keep guard digits during intermediate arithmetic, then round the final result using a stated rule. In addition and subtraction, decimal-place precision matters; in multiplication and division, relative precision or significant figures matters. Exact counts and defined conversion factors can be treated differently from measured inputs. Do not report a calculated volume as 10.0000 mL because a formula produced many digits when the original readings support only hundredths.
When comparing measurements, state the reference, method, and uncertainty basis. If no suitable reference exists, describe repeatability or observed spread and say that accuracy was not established. Avoid calling a result “accurate” merely because it looks plausible.
Put units in every raw column and carry them through equations.
Record the instrument or scale resolution and relevant specification.
State whether a spread is range, standard deviation, standard uncertainty, or another defined quantity.
Round once at the end unless a method requires a different rule.
Worked example
A qualified statement
The arithmetic mean is 10.03 mL. The readings cluster within 0.02 mL of one another, while the mean is 0.03 mL above the stated reference.
01Mean = (10.02 + 10.04 + 10.03) / 3 = 10.03 mL.
02Observed range = 0.02 mL; this is a descriptive spread, not automatically an uncertainty.
03Relative to the supplied reference, the mean differs by +0.03 mL; reference suitability and method bias remain assumptions.
04Report: “Three readings gave a mean of 10.03 mL and range 0.02 mL under the stated method.”
Materials
Set out what you need.
Calculator
Unit-conversion and rounding worksheet
Supplied readings with a stated reference for comparison
Safety & stop conditions
Pause if the work no longer fits the plan.
No hazardous procedure is required. Measurement and calculation practice should not be used to justify safety-critical or regulated decisions without qualified method review.
Method
Work through the steps.
01
Annotate the input
Write quantity, unit, method, resolution, and reference or calibration information for each value.
02
Convert coherently
Use dimensional cancellation in the equation and retain only digits supported by the source values.
03
Compare and qualify
Compute the chosen summary, identify uncertainty components or limitations, and state whether the claim concerns accuracy or precision.
Checkpoints
Observe, record, investigate.
Every number in the result has a unit or is explicitly dimensionless.
The reported digits match the least precise relevant input.
Accuracy and precision are not used as synonyms.
Common mistakes
Correct the process, preserve the record.
Adding trailing zeros to make a result look more precise.
Report only digits supported by the method and state the resolution or uncertainty basis.
Calling repeated agreement accuracy.
Repeated agreement supports precision; accuracy requires comparison to a suitable reference.
Using “error” to mean uncertainty without defining it.
Name the quantity: observed difference, standard deviation, standard uncertainty, or another defined measure.
Practice
Measure and report
Use supplied readings 10.02, 10.04, and 10.03 mL with a reference of 10.00 mL whose basis is stated as suitable for this exercise. Calculate a mean and describe spread; do not claim more than the data support.
What units and digits belong in the mean?
Does the set show accuracy, precision, or both relative to the reference?
Which uncertainty components are missing from this simple exercise?
