Errors and uncertainties

When making measurements in physics, it is important to consider how accurate and reliable your results are. All measurements have some degree of error and uncertainty, which can affect conclusions drawn from experiments.

Types of Errors

Systematic Errors

Systematic errors are consistent, repeatable errors caused by faults in the measurement system or procedure. They affect all measurements in the same way, making results consistently too high or too low.

Common causes:

• Faulty or incorrectly calibrated instruments
• Zero errors (instrument does not read zero when it should)
• Poor experimental design

Random Errors

Random errors are unpredictable variations that occur when making repeated measurements. They cause results to scatter around the true value.

Common causes:

• Human reaction time
• Fluctuations in environmental conditions
• Limitations in reading instruments

Zero Error

A zero error is a specific type of systematic error where an instrument does not read zero when it should. This leads to all readings being offset by a fixed amount.

Precision and Accuracy

• Precision refers to how close repeated measurements are to each other (consistency).
• Accuracy refers to how close a measurement is to the true or accepted value.

Assessing Uncertainties

Absolute Uncertainty

The absolute uncertainty is the margin of uncertainty associated with a measurement, expressed in the same units as the measurement.

Example: $l = 50.0 \pm 0.1$ cm (absolute uncertainty is 0.1 cm)

Percentage (or fractional) Uncertainty

The percentage uncertainty expresses the uncertainty as a percentage of the measured value.

$$\text{Percentage uncertainty} = \frac{\text{absolute uncertainty}}{\text{measured value}} \times 100\%$$

Combining Uncertainties

When calculating a value from several measurements, the uncertainties combine in specific ways:

• Addition or subtraction: Add absolute uncertainties.
• Multiplication or division: Add percentage uncertainties.

Reducing Errors and Uncertainties

• Use more precise instruments.
• Repeat measurements and calculate the mean.
• Calibrate instruments and check for zero errors.
• Use appropriate experimental techniques.