Tools and Inquiry — IB Physics skills
T.3.7 Processing Uncertainties — IB Physics HL Revision Notes & Topical Questions
5 min read 9 parts Lesson 12 of 22 1 quiz
Curriculum Equation Key قوانين الرقم الهيدروجيني والاتزان الأيوني - pH & Acid-Base Equilibrium FormulasFast Exam Solution
pH = -log[H⁺] | pOH = -log[OH⁻] | pH + pOH = 14 | [H⁺][OH⁻] = 10⁻¹⁴
Word Equation & Conditions: الرقم الهيدروجيني = -لوغاريتم تركيز أيون الهيدروجين | pH + pOH = 14
Location & Specifications: الاتزان الأيوني، كيمياء 3 ثانوي، التحصيلي، و AP Chemistry.
Lesson summary
T.3.7 Processing Uncertainties — IB PHYSICS SL notes: Express uncertainties, propagate them through calculations and quote results to an appropriate number of significant figures.Uncertainty = a quantitative indication of the quality of a result.It is a range around a measurement…
- Uncertainties are not errors
- Mean of repeated readings
- Absolute uncertainty
- Fractional uncertainty
- Percentage uncertainty
- Absolute uncertainties have the units of the quantity
Part 1 of 7What uncertainty is
What uncertainty is
Uncertainty
Uncertainty a quantitative indication of the quality of a result.
- It is a range around a measurement within which the true value is expected to lie, and is an estimate.
Uncertainties are not errors
- Uncertainties are not errors errors come from equipment or techniques that make a reading differ from the true value.
Uncertainties are quoted as a range (±) to an appropriate precision.
Part 2 of 7Estimating uncertainty
| Measurement | Uncertainty |
|---|---|
| A reading | ± half the smallest division |
| A measurement | at least ± one smallest division |
| Repeated data | ± half the range, ½(largest − smallest) |
| Digital readings | ± the last significant digit, unless quoted |
- Mean of repeated readings
- the uncertainty is half the range of the data.
Part 3 of 7Absolute, fractional and percentage uncertainty
Absolute uncertainty
Absolute uncertainty the actual amount by which a measurement is uncertain — it has the units of the quantity.
- L = 5.0 ± 0.1 cm has ΔL = ±0.1 cm.
Fractional uncertainty
ΔL / L — 0.1 / 5.0 = 0.02, with no units.
Percentage uncertainty
Percentage uncertainty absolute uncertainty / measured value × 100% — 2% for the same measurement.
Uncertainty in constants such as π is taken to be zero.
Part 4 of 7Propagating uncertainties
| Operation | Rule | ||
|---|---|---|---|
| Addition or subtraction y = a ± b | Δy = Δa + Δb (add absolute uncertainties) | ||
| Multiplication or division | Δy/y = Δa/a + Δb/b (add fractional or percentage uncertainties) | ||
| Power y = an | Δy/y = | n | Δa/a (multiply the fractional uncertainty by the power) |
1
Uncertainties in trigonometric and logarithmic functions are not tested.
2Absolute uncertainties have the units of the quantity
- Absolute uncertainties have the units of the quantity fractional and percentage uncertainties have none.
Part 5 of 7Significant figures with uncertainties
1Quote an uncertainty to one significant figure
- 7.4 ± 0.2 has percentage uncertainty 2.703%, quoted as 3%.
- Exceptionif the first digit is 1, two significant figures may be used (1.8).
2The quantity must match the decimal places of its absolute uncertainty
- 36.4652 ± 5% is quoted as 36.5 ± 1.8.
3
Raw data match the resolution of the instrument.
- Metre rule15 ± 1 mm.
- Vernier callipers15.4 ± 0.1 mm.
- Micrometer15.42 ± 0.01 mm.
4
Processed data have no more significant figures than the least precise measurement.
- The mean of 5.11, 5.14, 5.12 and 5.10 is 5.1175, quoted as 5.12.
Part 6 of 7Comparing uncertainty and error
Random uncertainty is reduced by repeats and averaging
- Random uncertainty is reduced by repeats and averaging it can be estimated from the spread.
Systematic errors shift all results the same way
- Systematic errors shift all results the same way they are not reduced by repeating.
- (e.g. zero error)
Largest percentage uncertainty dominates the final result
- Largest percentage uncertainty dominates the final result improve that measurement first.
A measured value agrees with an accepted value if the ranges overlap.
Part 7 of 7More from the specification
More from the specification
Combined uncertainties in force and speed
- Combined uncertainties in force and speed propagating fractional uncertainties in applied force and measured speed determines absolute uncertainty in power.
Significant figures
- Significant figures quote a result to the same number of significant figures as the least precise data used, with uncertainty to one or two.
Significant figures
- Significant figures quote a result to the same number of significant figures as the least precise data used, with uncertainty to one or two.
Derived unit
- A unit formed by multiplying or dividing SI base units.
Percentage difference
- Difference divided by accepted value × 100
Scale drawing
- Accurate drawing with a chosen scale.
Parallelogram method
- Vector addition using a parallelogram of vectors.
Vector
- Quantity with magnitude and direction.
Gradient
- Slope of a graph line.
Distance
- Total length of path travelled, a scalar.
Variable
- A factor that can change in an investigation.
Length
- Resistance is directly proportional to the length of a wire.
Displacement
- Change in position.
Proportional
- Changing by the same factor as another quantity.
Scalar
- Quantity with magnitude only.
a
− ω2x
What the symbols mean: a = acceleration (m s−2) · ω = angular frequency (rad s−1) · x = displacement from equilibrium (m); the minus sign means a always points back to the centre
Significant figures
- Significant figures quote a result to the same number of significant figures as the least precise data used, with uncertainty to one or two.
Key terms
| Term | Meaning |
|---|---|
| Uncertainty | Range around a measurement within which the true value is expected to lie. |
| Absolute uncertainty | Actual amount of uncertainty, with units. |
| Fractional uncertainty | Absolute uncertainty divided by the measured value. |
| Percentage uncertainty | Fractional uncertainty × 100%. |
| Propagation | Combining uncertainties through a calculation. |
| Systematic error | Error that shifts all readings in the same direction. |
Avoid these mistakes
1
Confusing percentage error with percentage uncertainty.
2
Adding absolute uncertainties when multiplying — add percentage or fractional uncertainties.
3
Forgetting to multiply the fractional uncertainty by the power for y = a^n.
4
Quoting an uncertainty to several significant figures.
5
Leaving the quantity with more decimal places than its uncertainty.
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