1.8 Assessment of Total Uncertainty in the Final Result

When physical measurements are used in calculations, their individual uncertainties combine and carry through to the final result. Understanding how to calculate this "propagated" uncertainty is essential for determining the reliability of any calculated quantity. The rules for propagating uncertainty depend on the mathematical operations being performed.

Precision And Accuracy→

1. Uncertainty in Sums and Differences

Rule: For addition and subtraction, the absolute uncertainties are always added.

Formula

If $x=A\pm B$, then the absolute uncertainty $\Delta x$ is: $\Delta x=\Delta A+\Delta B$ The final result is expressed as $x\pm (\Delta A+\Delta B)$.

Explanation: Whether you are adding or subtracting the measurements, the potential for error from each measurement contributes to the total uncertainty. You add the uncertainties because the error from one measurement could be in the opposite direction to the error in the other, creating the maximum possible error in the final result.

Example: Subtraction

Let the initial length be $L_1=15.5\pm 0.2$ cm and the final length be $L_2=12.0\pm 0.3$ cm. The change in length is: $\Delta L=L_1-L_2=(15.5-12.0)\text{ cm}=3.5\text{ cm}$ The uncertainty is the sum of the absolute uncertainties: $\text{Uncertainty}=0.2\text{ cm}+0.3\text{ cm}=0.5\text{ cm}$ Result: The change in length is 3.5 ± 0.5 cm.

Example: Addition

Two masses, $m_1=5.0\pm 0.1$ kg and $m_2=2.5\pm 0.1$ kg, are added together. The total mass is: $M=m_1+m_2=(5.0+2.5)\text{ kg}=7.5\text{ kg}$ The uncertainty is the sum of the absolute uncertainties: $\text{Uncertainty}=0.1\text{ kg}+0.1\text{ kg}=0.2\text{ kg}$ Result: The total mass is 7.5 ± 0.2 kg.

2. Uncertainty in Products and Quotients

Rule: For multiplication and division, the percentage (or fractional) uncertainties are added.

Formula

If $x=A\times B$ or $x=A/B$, then the percentage uncertainty in $x$ is: $\%\text{ Uncertainty in }x=\%\text{ Uncertainty in }A+\%\text{ Uncertainty in }B$ Where the percentage uncertainty for a quantity is calculated as: $\%\text{ Uncertainty}=\frac{\text{Absolute Uncertainty}}{\text{Measured Value}}\times 100\%$

Example: Multiplication

Let's find the area of a rectangle with length $L=5.0\pm 0.1$ cm and width $W=2.0\pm 0.1$ cm.

1. Calculate the Area: $A=L\times W=5.0\text{ cm}\times 2.0\text{ cm}=10.0{\text{ cm}}^2$
2. Calculate Percentage Uncertainties: $\%\text{ Uncertainty in }L=\frac{0.1}{5.0}\times 100\%=2\%$ $\%\text{ Uncertainty in }W=\frac{0.1}{2.0}\times 100\%=5\%$
3. Add Percentage Uncertainties: $\%\text{ Uncertainty in }A=2\%+5\%=7\%$
4. Calculate Absolute Uncertainty in Area: $\Delta A=7\%\text{ of }10.0{\text{ cm}}^2=0.07\times 10.0=0.7{\text{ cm}}^2$ Result: The area is 10.0 ± 0.7 cm².

3. Uncertainty in a Power

Rule: For a quantity raised to a power, the percentage uncertainty is multiplied by the power.

Formula

If $x=A^n$, then the percentage uncertainty in $x$ is: $\%\text{ Uncertainty in }x=n\times (\%\text{ Uncertainty in }A)$

Example: Volume of a Sphere

The volume of a sphere is $V=\frac{4}{3}\pi r^3$. The numbers $\frac{4}{3}$ and $\pi$ are exact and have no uncertainty. The uncertainty in volume comes from the radius, $r$, raised to the power of 3.

Let the radius be $r=2.25\pm 0.01$ cm.

1. Calculate the Volume: $V=\frac{4}{3}\pi (2.25{)}^3\approx 47.7{\text{ cm}}^3$
2. Calculate Percentage Uncertainty in Radius: $\%\text{ Uncertainty in }r=\frac{0.01}{2.25}\times 100\%\approx 0.44\%$
3. Apply the Power Rule: $\%\text{ Uncertainty in }V=3\times (\%\text{ Uncertainty in }r)=3\times 0.44\%=1.32\%$
4. Calculate Absolute Uncertainty in Volume: $\Delta V=1.32\%\text{ of }47.7{\text{ cm}}^3=0.0132\times 47.7\approx 0.63{\text{ cm}}^3$ Result: The volume is 47.7 ± 0.6 cm³.

4. Uncertainty in Average Value

To find the uncertainty in the average value of several measurements:

1. Find the average value.
2. Find the deviation of each value from the average.
3. The mean deviation is the uncertainty.

5. Uncertainty in Timing Experiments

The uncertainty in the time period of a vibrating body is found by dividing the least count of the timing device by the number of vibrations.

$\Delta T=\frac{\text{Least Count of stopwatch}}{\text{Number of vibrations}}$

Timing multiple oscillations reduces the per-period uncertainty, improving the precision of the result.

Summary

These rules are fundamental for correctly stating the results of experiments, ensuring that the final calculated value reflects the precision of the original measurements.