Random and systematic errors

• Random errors cause unpredictable variation between repeated measurements and mainly reduce precision.

• Repeating measurements and calculating a mean reduces the effect of random variation on the final result.

• Systematic errors shift measurements consistently away from the true value and mainly reduce accuracy.

• Repeating measurements does not remove a systematic error because the same bias remains.

• A zero error or incorrect calibration can produce a systematic offset.

• Identify the error type before suggesting how it should be reduced or removed.

Systematic uncertainty produces a consistent displacement, whereas random uncertainty produces a spread of measurements. Repetition reduces the effect of random variation but does not remove a systematic offset. Source

Repeatability and reproducibility

• Repeatability means obtaining similar results when the same method, investigator and equipment are used under the same conditions.

• Reproducibility means obtaining similar results when the investigation is repeated using different investigators or equipment.

• Good repeatability shows that repeated measurements under closely matched conditions have little unexplained variation.

• Good reproducibility gives stronger evidence that the result does not depend strongly on one particular setup or investigator.

Expressing uncertainty

• Absolute uncertainty can be written with a measurement as $x\pm\Delta x$ and has the same unit as the measured quantity.

• Fractional uncertainty is $\frac{\Delta x}{x}$.

• Percentage uncertainty is $\frac{\Delta x}{x}\times100$.

• A larger fractional or percentage uncertainty means the measurement is less precise relative to its size.

• Use uncertainty to communicate the reasonable range within which the measured value may lie.

• Uncertainty must be carried through calculations to represent uncertainty in the final derived quantity.

Significant figures and uncertainty

• The number of reported significant figures must be consistent with the measurement's uncertainty.

• Quote the measured value to the same decimal place as its absolute uncertainty.

• Do not report extra digits that imply greater precision than the measurement supports.

• Smaller uncertainty may justify more significant digits, but additional digits cannot make an inaccurate experiment accurate.

Accuracy, precision and resolution

• Accuracy describes how close a measurement is to the true or accepted value.

• Precision describes how closely repeated measurements agree with each other.

• High precision does not guarantee high accuracy because a systematic error can shift every reading.

• Resolution is the smallest change in a quantity that a measuring instrument can distinguish.

• Better resolution can allow smaller changes to be detected, but it does not automatically remove other measurement errors.

Reducing measurement errors

• Reduce random error by taking repeated readings and using their mean where appropriate.

• Improve control of experimental conditions when uncontrolled variation causes readings to fluctuate.

• Reduce systematic error by checking calibration, correcting zero errors or replacing faulty equipment.

• Compare measuring equipment with a known reference where this can reveal a consistent offset.

• Do not claim that repetition removes systematic error; repeated biased readings remain biased.

• Choose measuring equipment with sufficient resolution for the change being investigated.

Combining uncertainties

• For addition or subtraction, add the absolute uncertainties: $\Delta Q=\Delta a+\Delta b$.

• For multiplication or division, add fractional uncertainties: $\frac{\Delta Q}{Q}=\frac{\Delta a}{a}+\frac{\Delta b}{b}$.

• Percentage uncertainties may be added instead when quantities are multiplied or divided.

• If $Q=x^n$, the fractional uncertainty is $\frac{\Delta Q}{Q}=|n|\frac{\Delta x}{x}$.

• Therefore, raising a measured quantity to a power multiplies its percentage uncertainty by the magnitude of that power.

• Combinations involving trigonometric or logarithmic functions are not required for this syllabus section.

Error bars and graph uncertainties

• Error bars represent uncertainty in individual plotted data values.

• Error bars may be vertical, horizontal or both, depending on which plotted quantities have uncertainty.

• Individual points on a graph may or may not have associated error bars.

• Use steepest and shallowest acceptable straight lines to determine the possible range of gradient.

• A common gradient uncertainty calculation is $\Delta m=\frac{m_{\max}-m_{\min}}{2}$.

• Similarly, the intercept uncertainty can be obtained from $\Delta c=\frac{c_{\max}-c_{\min}}{2}$.

• Interpret larger uncertainty ranges as weaker constraints on the gradient or intercept.

Orders of magnitude and estimation

• An order of magnitude represents the scale of a quantity using the nearest power of ten.

• Write an approximate value in the form $A\times10^n$ to identify its scale.

• Use physically sensible assumptions rather than attempting unjustified precision.

• Round an approximate physical quantity to its nearest order of magnitude when required.

• For a derived estimate, combine reasonable approximate values using an appropriate physical relationship.

• Check that the final estimate has a sensible scale and does not contain false precision.

Each successive order of magnitude differs by a factor of $10$. Thinking in factors of $10$ helps judge whether an estimated physical quantity has a sensible scale. Source