Topic 3 of 4
Uncertainty in calculated results
A calculated value cannot be more certain than its inputs justify. The operation in the equation determines how the input uncertainties contribute.
Write a measurement as x ± Δx. The absolute uncertainty Δx is nonnegative and has the same unit as x. For a value well away from zero:
Fractional uncertainty = Δx / |x|
Percentage uncertainty = (Δx / |x|) × 100%
For example, (5.0 ± 0.1) cm has fractional uncertainty 0.1/5.0 = 0.02 and percentage uncertainty 2%. These estimates describe the stated measurement limits; they do not, by themselves, specify a statistical confidence level.
The condition that a fractional uncertainty is small may be written Δx/|x| ≪ 1, meaning much less than 1. Here 0.02 ≪ 1; equivalently, the ratio |x|/Δx = 50 ≫ 1 is much greater than 1. These symbols express a scale comparison, not a universal numerical cut-off for every approximation.
Lowercase δx often denotes a small change. If a length increases from 2.000 m to 2.003 m, δx = +0.003 m and its percentage change is (0.003/2.000) × 100% = +0.15%. A decrease gives a negative change.
Read the stated convention: here Δx in x ± Δx is a nonnegative absolute uncertainty, whereas in a motion equation Δx commonly means the signed final-minus-initial change. Some texts use δx for uncertainty too. The Greek letter alone does not determine the meaning or sign.
Select the rule from the operation
- Addition or subtraction: add absolute contributions
- For q = x + y or q = x - y, a conservative estimate is Δq = Δx + Δy, for separately bounded inputs with no shared effect being cancelled.
- Multiplication or division: add fractional contributions
- For q = xy or q = x/y, with small relative uncertainties, Δq/|q| ≈ Δx/|x| + Δy/|y|. Percentage contributions can be added in the same way.
- A power: multiply by the absolute exponent
- For q = xn, the small-uncertainty estimate is Δq/|q| ≈ |n| Δx/|x|. A squared diameter contributes twice its fractional uncertainty, even if it is in the denominator.
- An exact factor: no extra measured contribution
- For q = kx with exact k, the fractional uncertainty is unchanged. The absolute uncertainty scales by |k|. Constants such as 4 and π in a supplied geometric formula do not add measurement uncertainty.
Worked difference
Subtract values, add their uncertainty contributions
Two separately bounded length readings are (31.6 ± 0.1) cm and (12.4 ± 0.1) cm.
Difference = 31.6 - 12.4 = 19.2 cm.
Absolute uncertainty = 0.1 + 0.1 = 0.2 cm.
The result is (19.2 ± 0.2) cm. The greatest possible difference uses the first reading high and the second low: 31.7 - 12.3 = 19.4 cm. The smallest is 31.5 - 12.5 = 19.0 cm.
Check for shared effects before applying a rule. If the same fixed zero offset z affects both readings on one unchanged scale, (x + z) - (y + z) = x - y. That offset cancels. It should not be counted twice as unrelated uncertainty, although other reading uncertainties may remain.
Optional check Two separately bounded readings are (43.2 +/- 0.2) cm and (11.8 +/- 0.1) cm. With no shared offset to cancel, what is their difference and conservative uncertainty?
Worked quotient
Speed from a distance and a time
A distance L = (10.00 ± 0.02) m is travelled in t = (2.50 ± 0.05) s. Find the average speed v = L/t and its estimated uncertainty.
- Calculate the value: v = 10.00/2.50 = 4.00 m/s.
- Add fractional contributions: Δv/v ≈ 0.02/10.00 + 0.05/2.50 = 0.002 + 0.020 = 0.022.
- Return to an absolute uncertainty: Δv ≈ 4.00 × 0.022 = 0.088 m/s.
- Report sensibly: approximately (4.00 ± 0.09) m/s.
The time contributes 2.0%, compared with the distance's 0.2%. Reducing the timing uncertainty is the more effective priority in this example.
Numerical substitution checks the bounds
For positive v = L/t, increasing L increases v while increasing t decreases it. Choose the input extremes accordingly:
vmax = 10.02/2.45 = 4.0898 m/s
vmin = 9.98/2.55 = 3.9137 m/s
The bounds are close to 4.00 ± 0.09 m/s, but not exactly symmetric about 4.00. This supports the small-uncertainty approximation here. Putting every input at its maximum would not give the maximum of a quotient.
If a denominator's possible range includes zero, or relative uncertainties are large, the simple fractional rule may be unsuitable. Inspect how the supplied expression changes over the allowed input ranges and use numerical bounds where appropriate. For a function that turns within the range, its maximum or minimum need not occur at an endpoint.
Worked power and quotient
Which cylinder measurement matters most?
For a uniform cylinder, volume = πd2L/4 and density is mass divided by volume, so use the supplied model ρ = 4m/(πd2L).
| Quantity | Value and absolute uncertainty |
|---|---|
| Mass m | (0.0500 ± 0.0001) kg |
| Diameter d | (0.0200 ± 0.0002) m |
| Length L | (0.100 ± 0.001) m |
Using the central values gives ρ = 1591.55 kg/m3 before rounding.
The fractional contributions are:
- Mass: 0.0001/0.0500 = 0.002.
- Diameter squared: 2(0.0002/0.0200) = 0.020.
- Length: 0.001/0.100 = 0.010.
Total fractional estimate = 0.032, or 3.2%. The absolute estimate is 1591.55 × 0.032 = 50.93 kg/m3. A suitable reported result is (1.59 ± 0.05) × 103 kg/m3.
The diameter contributes 2.0%, the length 1.0% and the mass 0.2%. Halving the diameter's percentage uncertainty reduces the total more than halving either of the other percentage uncertainties. Improving an already precise mass reading has little effect here.
Keep guard digits during calculation. Round the final uncertainty sensibly, then give the value to a corresponding place. The appropriate precision depends on the evidence; one rule about decimal places does not apply to every raw instrument reading.