Lesson summary
1.3.2 Calculating Uncertainties — AS PHYSICS 9702 notes: Express absolute, fractional and percentage uncertainties and combine them for sums, products, powers and graph gradients.Uncertainty — the range of values around a measurement within which the true value is expected to lie…
- Uncertainties are not the same as errors
- A result is written x ± Δx
- Absolute uncertainty Δx
- Fractional uncertainty
- Quote the uncertainty to one significant figure
- Reading from an analogue scale (such as a voltmeter)
Section 1 of 10What uncertainty is
What uncertainty is
Uncertainty
- Uncertainty the range of values around a measurement within which the true value is expected to lie;
- it is an estimate.
Uncertainties are not the same as errors
- Uncertainties are not the same as errors errors are issues with equipment or method that make a reading differ from the true value.
A result is written x ± Δx
- A result is written x ± Δx for example 12.50 ± 0.05 cm from readings of 12.55, 12.45 and 12.51 cm.
Section 2 of 10Ways to state an uncertainty
Ways to state an uncertainty
Absolute uncertainty Δx
- Absolute uncertainty Δx a fixed quantity with the same unit as x.
Fractional uncertainty
- Δx ÷ x.
Percentage uncertainty = (absolute uncertainty ÷ measured value) × 100%.
For 25.0 ± 0.5 cm the fractional uncertainty is 0.02 and the percentage uncertainty is 2%.
Quote the uncertainty to one significant figure
- Quote the uncertainty to one significant figure and round the value to the same decimal place.
Section 3 of 10Uncertainty in a single measurement
Uncertainty in a single measurement
Reading from an analogue scale (such as a voltmeter)
- Reading from an analogue scale (such as a voltmeter) at least ± half the smallest division.
Measurement with a ruler
- Measurement with a ruler at least ± 1 smallest division.
Digital reading
- ± the last significant digit unless otherwise quoted.
Repeated data
- ± half the range, Δx = ÷ 2.
- (largest − smallest)
A length found from two readings
- A length found from two readings each reading carries an uncertainty and they add.
Section 4 of 10Combining uncertainties
1Adding or subtracting
- Adding or subtracting add the absolute uncertainties, so Δ(a ± b) = Δa + Δb.
2Subtraction does not reduce uncertainty
- Subtraction does not reduce uncertainty the difference of two nearly equal readings can have a large percentage uncertainty.
3Multiplying or dividing
- Multiplying or dividing add the percentage (or fractional) uncertainties.
4Raising to a power n
- Raising to a power n multiply the percentage uncertainty by n, so a squared quantity doubles it and a square root halves it.
5Constants such as π or 2
- Constants such as π or 2 contribute no uncertainty.
6
For g = 4π2l ÷ T2 the percentage uncertainty is (Δl ÷ l) + 2(ΔT ÷ T).
Section 5 of 10Why the rules work
Why the rules work
Sums
- Sums both quantities may be at their extremes together, (a + Δa) + (b + Δb) = (a + b) + (Δa + Δb).
Products
- (a + Δa)(b + Δb) ≈ ab when the small cross term is ignored.
- (1 + Δa ÷ a + Δb ÷ b)
Powers
- Powers repeating the same quantity n times adds its fractional uncertainty n times.
This is a worst-case estimate
- This is a worst-case estimate simple, safe and expected at this level.
A dominant uncertainty
- A dominant uncertainty if one uncertainty is more than about three times the others, improve that measurement.
Section 6 of 10Uncertainty from a graph
1
Draw error bars on the points when they carry uncertainties.
2Gradient
- Gradient use a large triangle on the best-fit line and take points from the line, not from the data.
3Uncertainty in the gradient
- Uncertainty in the gradient draw the worst acceptable line through all the error bars and find Δm = |mbest − mworst|.
4
The same method gives the uncertainty in an intercept.
Section 7 of 10Reporting and comparing results
1State the final value with uncertainty and unit
- State the final value with uncertainty and unit for example g = 9.6 ± 0.4 m s-2.
2Consistent with an accepted value
- Consistent with an accepted value if the accepted value lies within the range of the uncertainty.
3If the difference is larger than the uncertainty
- If the difference is larger than the uncertainty a systematic error is likely.
4Percentage difference alone does not show quality
- Percentage difference alone does not show quality compare it with the percentage uncertainty.
Section 8 of 10More from the specification
More from the specification
Precision vs Accuracy
- Precision refers to the degree of agreement or closeness among repeated measurements obtained under identical conditions,
- whereas accuracy denotes how close a measured value is to the true value of the quantity.
Random and Systematic Errors
- Random errors cause readings to scatter unpredictably about the mean and can be minimised by taking repeated measurements and finding the average,
- whereas systematic errors shift readings by a consistent offset due to instrument calibration or zero error.
Instrumental Precision and Uncertainty
- The absolute uncertainty of an instrument is determined by its resolution or smallest scale division, indicating the degree of uncertainty in each measurement taken.
Precision vs Accuracy
- Precision refers to the degree of agreement or closeness among repeated measurements obtained under identical conditions,
- whereas accuracy denotes how close a measured value is to the true value of the quantity.
Random and Systematic Errors
- Random errors cause readings to scatter unpredictably about the mean and can be minimised by taking repeated measurements and finding the average,
- whereas systematic errors shift readings by a consistent offset due to instrument calibration or zero error.
Instrumental Precision and Uncertainty
- The absolute uncertainty of an instrument is determined by its resolution or smallest scale division, indicating the degree of uncertainty in each measurement taken.
Precision
- How close repeated measurements are to each other.
Accuracy
- How close a measurement is to the true value.
Random
- Without a pattern in speed or direction.
Random error
- An unpredictable variation in readings, reduced by repeating and averaging.
Section 9 of 10Key terms
| Term | Meaning |
|---|---|
| Absolute uncertainty | The uncertainty in the same unit as the measurement, ±Δx. |
| Fractional uncertainty | The absolute uncertainty divided by the measured value. |
| Percentage uncertainty | The fractional uncertainty multiplied by 100%. |
| Half-range method | Taking the uncertainty of repeated data as half the difference between the largest and smallest values. |
| Error bar | A line on a graph point showing the uncertainty in that value. |
| Worst acceptable line | The steepest or shallowest straight line that still passes through all the error bars. |
| Consistent | Describes a result whose uncertainty range includes the accepted value. |
| Resultant uncertainty | The uncertainty of a result calculated from several measurements. |
Section 10 of 10Exam technique
1
Adding absolute uncertainties when multiplying — use percentage or fractional uncertainties for products and quotients.
2
Subtracting uncertainties when quantities are subtracted — uncertainties always add.
3
Forgetting to multiply the percentage uncertainty by the power.
4
Quoting the uncertainty to several significant figures or a different decimal place from the value.
5
Including exact constants such as π in the uncertainty sum.