Combining Uncertainties and Reading a Result off a Graph
Combine stated uncertainty bounds through calculations. Add absolute bounds for sums and differences, and use first-order percentage bounds for products and quotients. Calculate gradients of acceptable extreme lines, then report their midpoint with half their difference as a bound. Compare reported intervals using an explicit compatibility criterion.
What a learner can do afterwards
- Combines uncertainties through a multi-step calculation and names which measurement dominates the result
- Draws stated error bars, calculates gradients of acceptable extreme lines, and reports their midpoint with half their difference as an uncertainty bound
- Decides whether two published values agree, using their stated uncertainties rather than their central values
1 · Read
Uncertainty describes limited knowledge of a measurement. Error is the difference between a measured value and the true value; it is often unknown. In this lesson, stated ± amounts are uncertainty bounds, not standard deviations. Accuracy compares with a reference; precision describes repeat agreement.
For a sum or difference, add the absolute uncertainty bounds in common units. Subtraction does not subtract uncertainties. For example, adding lengths of 7.0 ± 0.2 cm and 2.0 ± 0.4 cm gives 9.0 ± 0.6 cm.
For this laboratory's measured results, report a value, unit and uncertainty bound, and state how the bound was estimated. Without that information, a reader cannot judge the result's stated measurement limits. Exact counts and definitions are a different case.
For a product or quotient of positive values with small relative bounds, this lesson uses a first-order upper-bound estimate: add percentage uncertainties. Higher-order terms are neglected. For first-power factors, compare their percentage contributions. Powers or weights change the contributions; sums require absolute amounts.
Plot (x, y) with x horizontal and y vertical. For (2 s, 5.0 ± 0.3 m), draw a vertical bar from 4.7 to 5.3 m at x = 2 s. Here x has negligible uncertainty. Gradient is change in y divided by change in x. Find the shallowest and steepest straight lines crossing every bar. Report their gradient midpoint with half their difference as a bound; it need not equal a separately fitted best-line gradient.
Carry the matching bound through each calculation stage. When adding lengths, add their absolute bounds. Divide the summed bound by the magnitude of the summed length to get fractional uncertainty. Multiply that fraction by 100 to express it as a percentage. If the sum is multiplied by another positive measurement, add their percentages for the first-order product estimate.
Our interval comparison calls two results compatible when their stated closed intervals overlap. Find each interval by subtracting and adding its bound. Overlap means a common value is allowed by both intervals; it does not prove equality. This is an interval rule, not a statistical significance test.
For basic multiplication or division of measured inputs without explicit propagated uncertainty, round to the fewest significant figures among the inputs. For addition or subtraction, use the coarsest decimal place. Exact counts and defined factors do not limit either rule. Keep guard digits during working.
If acceptable line gradients run from 1.4 to 1.8 m/s, their midpoint is (1.4 + 1.8) ÷ 2 = 1.6 m/s. Half their difference is 0.2 m/s. Under our convention, report 1.6 ± 0.2 m/s.
State the uncertainty model, propagate each contribution through the calculation, and distinguish interval compatibility from equality.
Original lesson, questions and flashcards written for this dataset.
Rewritten readings were written for this dataset. Additional flashcards written for this dataset. Reporting measurements video by Khan Academy.
2 · Watch
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Where it sits
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.