Uncertainties & Error Analysis

Absolute Uncertainty

• For a ruler with mm divisions: uncertainty = $\pm 0.5$ mm.
• For a protractor: uncertainty = $\pm 0.5°$.
• For a stopwatch: although resolution is 0.01 s, human reaction time ($\sim 0.2$ s) dominates, so uncertainty is typically $\pm 0.1$ to $\pm 0.25$ s.

Percentage Uncertainty

• Percentage uncertainty converts the absolute uncertainty into a proportion of the measured value: $$\text{percentage uncertainty} = \frac{\text{absolute uncertainty}}{\text{measured value}} \times 100\%$$
• For the same instrument, measuring a larger value gives a smaller percentage uncertainty.
• Example: measuring 4.8 cm with a ruler ($\pm 0.05$ cm) gives $\frac{0.05}{4.8} \times 100 = 1.0\%$. Measuring 42.3 cm gives only $0.12\%$.

Combining Uncertainties

• Addition or subtraction -- add the absolute uncertainties: $$\Delta(A + B) = \Delta A + \Delta B$$
• Multiplication or division -- add the percentage uncertainties: $$\%\text{uncertainty in } \frac{A}{B} = \%\text{uncertainty in } A + \%\text{uncertainty in } B$$
• Raising to a power -- multiply the percentage uncertainty by the power: $$\text{if } y = x^n, \quad \%\Delta y = n \times \%\Delta x$$

Percentage Difference

• Used to compare an experimental result with a known or accepted value: $$\text{percentage difference} = \frac{|\text{experimental} - \text{accepted}|}{\text{accepted}} \times 100\%$$
• A small percentage difference indicates the experiment was accurate.
• If the percentage difference is smaller than the total percentage uncertainty, the result is consistent with the accepted value.

Evaluating Results

• Before drawing conclusions, evaluate the impact that experimental limitations could have had on the data.
• Identify anomalous resultsData points that do not fit the overall trend. Often defined as results that differ from the mean by more than 10%. They should be excluded from mean calculations. -- these are points that do not fit the trend and should be excluded from mean calculations.
• If limitations have negligible impact, conclusions can be drawn with confidence.
• If limitations are significant, conclusions carry a greater chance of being incorrect.
• Results are made more reliable by repeating the experiment, removing anomalies, and calculating means.