Uncertainty, Precision and Significant Figures in a Measurement
Report measurements with uncertainty based on the instrument and method. Distinguish accuracy from precision, random scatter from systematic offsets, and individual-reading spread from uncertainty in an average. Averaging suitable independent repeats can reduce the random uncertainty of the mean, but does not remove a shared zero offset.
What a learner can do afterwards
- Quotes a measurement with an absolute and a percentage uncertainty and rounds the value to match
- Separates a random scatter from a systematic offset in a set of repeated readings and says what fixes each
- Chooses an instrument and a method that make the dominant uncertainty smaller, and says which one it was
1 · Read
Accuracy means closeness to a true or accepted reference value. Precision means close agreement among repeated readings under the same conditions. A tightly grouped set away from the reference is precise but inaccurate. A precise reading is not exact. If a single-position reading is rounded to the nearest 1 mm mark and other effects are negligible, its rounding bound is ±0.5 mm. State those assumptions.
Report a value with its absolute uncertainty in the same unit. Round the value to the uncertainty's decimal place. To round to tenths, check the hundredths digit: 5 or more rounds the tenths up; less than 5 leaves it unchanged. If an uncertainty is 0.3 cm, round the value to tenths: 18.274 cm becomes 18.3 ± 0.3 cm.
Random effects cause readings to vary. Taking more readings does not shrink the spread of individual readings. Averaging independent, unbiased repeats under stable conditions can reduce the random uncertainty of the mean. A systematic offset shared by all readings remains in that mean.
Percentage uncertainty is absolute uncertainty divided by the magnitude of the measured value, multiplied by 100%. Percent means per hundred: a fraction of 0.025 is 2.5%. A length of 12.0 ± 0.3 cm has percentage uncertainty (0.3 ÷ 12.0) × 100% = 2.5%. This percentage compares the uncertainty with the value's size.
Check a suspected zero offset against a known reference. If an empty balance repeatedly reads 0.4 g, reset its zero or apply a documented correction before measuring. Repeats with the uncorrected balance will not remove that offset. Other systematic effects may need calibration or a changed method.
For basic products or quotients of measured values, round to the fewest significant figures among those inputs. Exact counts and defined conversion factors do not limit the count. Nonzero digits and enclosed zeros count; leading zeros do not. Integer trailing zeros need context. Scientific notation states significance: 10³ = 10 × 10 × 10 = 1000, so 1.3 × 10³ and 1.30 × 10³ have two and three significant figures.
The largest uncertainty contribution depends on the calculation. For a sum, compare absolute uncertainties in the same unit, not percentages alone. Convert each percentage to an absolute amount by multiplying the value's magnitude by the percentage divided by 100. Improve the largest contribution first.
Choose improvements that address the main uncertainty source. If human reaction time dominates a stopwatch measurement, extra display digits will not help much. Electronic start and stop detection can reduce that timing uncertainty. If scale resolution dominates a length reading, finer suitable equipment may help.
State uncertainty and its basis. Distinguish scatter from offsets, and choose an improvement that addresses the main contribution.
Original lesson, questions and flashcards written for this dataset.
Readings written for this dataset. Additional flashcards written for this dataset.
2 · Watch
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Where it sits
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Where this leads
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.