1.2 Limitation of Physical Measurements

Random Errors

• Definition: These errors impact the precision of measurements, meaning that repeated measurements vary around the mean value. They are caused by unpredictable variations in the experimental conditions and cannot be completely eliminated.

• How to reduce random errors:
  • Take multiple readings (at least three) and calculate a mean. This approach averages out random variations and helps detect anomalies.
  • Use computers/data loggers to collect data, which reduces human error and allows for smaller intervals between readings.
  • Use appropriate equipment with higher resolution, such as a micrometre ($0.1$ mm precision) instead of a ruler ($1$ mm precision).

Systematic Errors

• Definition: These errors affect the accuracy of measurements and occur when there's a consistent deviation from the true value. Systematic errors cause results to be consistently too high or too low.

• How to reduce systematic errors:
  • Calibrate instruments using a known value to ensure the correct starting point (e.g., using a standard weight to check a balance).
  • Correct for background factors (e.g., subtracting background radiation in experiments).
  • Read the meniscus at eye level to minimise parallax error.
  • Use controls to compare with known values.

Key Concepts in Measurement

Precision	Consistency of repeated measurements. High precision means measurements are close to each other but not necessarily accurate.
Repeatability	The ability to obtain the same results under the same conditions by the same experimenter.
Reproducibility	Similar results obtained when different experimenters conduct the experiment using different setups.
Resolution	The smallest detectable change in a measurement.
Accuracy	How close a measurement is to the true value.

Uncertainty in Measurements

Uncertainty indicates the range within which the true value lies, reflecting both random and systematic errors.

1. Types of Uncertainty:

• Absolute Uncertainty: Expressed as a fixed amount, e.g., $7.0 ± 0.6 \ V$.
• Fractional Uncertainty: Ratio of the uncertainty to the measured value, e.g., $\frac{0.6}{7.0} = 0.086$.
• Percentage Uncertainty: Fractional uncertainty expressed as a percentage, e.g., $8.6$%.

2. Calculating Uncertainty:

• For digital readings, uncertainty is often assumed to be $±$ the last significant digit.
• For analogue readings, the uncertainty is typically ± half of the smallest division of the measuring instrument (e.g., $±0.5°C$ for a thermometer with $1°C$ divisions).
• For repeated measurements, uncertainty is calculated as half the range (difference between the largest and smallest readings), expressed as $\text{mean} \pm \frac{\text{range}}{2}$.

Combining Uncertainties

• When adding or subtracting quantities, add the absolute uncertainties.
• When multiplying or dividing quantities, add the percentage uncertainties.
• For uncertainties involving powers, multiply the percentage uncertainty by the power.

Uncertainties and Graphs

• Error Bars: Indicate uncertainties on a graph. The line of best fit should pass through all error bars, excluding anomalies.
• Uncertainty in Gradients and Y-Intercepts:
  • Calculate using lines of best and worst fit.
  • Percentage uncertainty in gradient =$\frac{\text{best gradient - worst gradient}}{\text{best gradient}} \times 100.$
  • For y-intercepts, use a similar formula.