# 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. ## 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. ## Possible Questions/Answer - **Q:** Why are absolute uncertainties added for subtraction instead of being subtracted? **A:** Uncertainty represents a range of possible error. When you subtract two measurements, the errors could combine in a worst-case scenario. For example, if the first measurement was at its maximum possible value and the second was at its minimum, the error in the difference would be the sum of their individual errors. - **Q:** What is the uncertainty in a constant like π? **A:** Exact numbers and defined constants (like π or the '2' in the formula for the area of a triangle) are considered to have zero uncertainty. They do not contribute to the total uncertainty in a calculation. ## Summary | **Operation** | **Rule for Propagating Uncertainty** | | :--- | :--- | | **Addition / Subtraction** | Add the **absolute uncertainties**. | | **Multiplication / Division** | Add the **percentage uncertainties**. | | **Power (Aⁿ)** | Multiply the **percentage uncertainty** by the power *n*. | These rules are fundamental for correctly stating the results of experiments, ensuring that the final calculated value reflects the precision of the original measurements.