Probability and uncertainty: the interval travels with the number
Introduced| English | Meaning |
|---|---|
| confidence interval/ˈkɒnfɪdəns ˈɪntəvl/ | the interval that would capture the true mean in a stated share of repeats |
| uncertainty/ʌnˈsɜːtənti/ | the quantified doubt attached to a measurement |
| propagation of uncertainty | carrying input uncertainties into a computed result, root-sum-square for independent inputs |
| expanded uncertainty | k times the combined uncertainty; k = 2 for about 95% coverage |
A rumour with decimals
- Ten micrometer readings of the pivot shaft: mean 25.000 mm, sd 0.0031 mm, standard error 0.00097 mm - and the report line is not any of these.
- 95% confidence interval 置信区间: $25.000 \pm 0.0022$ (t = 2.262, 9 df).
The budget, top-down
- Type A (repeat readings) 0.00097; resolution $0.01/\sqrt{12} = 0.00289$; calibration $0.005/\sqrt{3} = 0.00289$.
- Combined (root-sum-square): $u_c = 0.0042$ mm; expanded $U = 2u_c = 0.008$ mm. The report: 25.000 plus or minus 0.008 mm.
*Attack the largest term: ninety more readings change the fourth decimal
Which term is this?
Route each contribution to its place in the budget.
sd 0.0031 over ten readings: the standard error (five decimals)?
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0.0031 over root 10.
Resolution 0.01 mm as an uncertainty (five decimals)?
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0.01 over root 12.
Match each number to its name.
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Four numbers, four different jobs.
Propagation carries it forward
- $A = \pi d^2/4$: $u_A = \frac{\pi d}{2}u_d = 0.165$ mm2 - report $490.9 \pm 0.3$ mm2 (k = 2).
- The mount stiffness: $\delta = 2.50$ mm carries 0.32%, dominating the mass’s 0.1% - the certification margin must exceed the band or the test decides nothing.
RSS of 0.00097, 0.00289 and 0.00289 (four decimals)?
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sqrt of the sum of squares - the two 0.00289s dominate.
d = 25.000 mm: the area (one decimal)?
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pi x 625 / 4 = 490.87.
The budget is dominated by resolution and calibration. The efficient fix:
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The instrument is the ceiling - the Type A term is already an order smaller.
The report line for the shaft is:
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U = 2 u_c - the expanded uncertainty, not the scatter or the se.
Three uncertainty slips
- Reporting the mean’s precision as the measurement’s. sd, se, U - three different numbers.
- Adding uncertainties. Root-sum-square for independent terms.
- Shrinking the easy term. The instrument is the ceiling.
Independent uncertainties combine by adding them.
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Root-sum-square - addition overstates the band.
A pass/fail margin narrower than the uncertainty band certifies nothing.
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The test must out-resolve its own doubt.
Recap
- Mean 25.000, sd 0.0031, se 0.00097; budget dominated by resolution and calibration: $u_c = 0.0042$, U = 0.008.
- Propagation of uncertainty 不确定度传播 by RSS: $490.9 \pm 0.3$ mm2.
- The expanded uncertainty 扩展不确定度 travels with the number, forever.