Uncertainties and Errors in Measurements and Results

Understanding Uncertainties

• Uncertainty refers to the estimate of how much a measured or calculated value may deviate from the true value.
• Uncertainty may be due to systematic errors (errors that consistently affect results in the same way) or random errors (errors that vary in an unpredictable manner).
• Common sources of uncertainty include limitations of measuring instruments, operator error, and changes in the experimental environment.

Expressing and Quantifying Uncertainty

• Uncertainty is usually expressed as a range of values, e.g., 7.5 ± 0.2, where the reading is 7.5 and the uncertainty is ± 0.2.
• Absolute uncertainty refers to the magnitude of the uncertainty, without considering the size of the measurement. It is usually expressed as ± some value.
• Percentage uncertainty is absolute uncertainty expressed as a percentage of the measured value. It helps to compare uncertainties of different measurements.

Propagation of Uncertainty

• Uncertainty propagation refers to how uncertainties of individual measurements combine to affect the uncertainty of the final result.
• In addition or subtraction, absolute uncertainties add up.
• In multiplication or division, percentage uncertainties add up.
• In raising to a power, the percentage uncertainty multiplies by the power.

Systematic and Random Errors

• Systematic errors consistently affect outcomes in a predictable direction. Causes include poorly calibrated equipment, biased observers, or environmental conditions.
• Random errors vary unpredictably, balancing out in the long term. Causes include fluctuations in readings, or environmental changes.
• Systematic errors cause a lack of accuracy, while random errors cause a lack of precision. Accuracy reflects how close measurements are to the true value, while precision reflects how close repeated measurements are to each other.

Reducing and Managing Uncertainty and Errors

• Minimising uncertainties and errors can be achieved by repeating measurements, improving experimental techniques, or using better measuring equipment.
• Statistical analysis can also help manage uncertainties, such as calculating the mean and standard deviation of multiple measurements.
• It’s important to consider uncertainties and errors when interpreting results, and always express final results with the appropriate level of uncertainty.