Errors and uncertainty
Section: Physical Quantities and Units | Syllabus: Cambridge AS Level Physics 9702
Errors and uncertainty In all measurements, there is always some uncertainty in the measured value. We cannot be sure that we have made a completely accurate measurement. This can be due to different causes, generally called experimental errors .
Experimental Errors Experimental errors lead to measurements that are different from the true values. There can be different reasons for these errors. For example, the reading on an ammeter or a voltmeter can fluctuate and not remain constant.
The pointer on an analogue meter may not be exactly on one of the scale divisions. Experimental errors can be categorized into two main types: Random or Systematic . Type of Error Description Examples How to Reduce Random Error Not predictable or constant.
Readings scatter around the mean value. Fluctuations in meter readings, variation in room temperature, parallax error from varying angles. Take repeat readings, discard anomalies, and calculate a mean.
Systematic Error Predictable effects that affect all results in a similar way (consistently higher or lower). Zero errors (e.g., damaged ruler end), poorly calibrated instruments, constant parallax error.
Repeat readings will not reduce these. Instrument must be recalibrated or zeroed. Figure 1.3: Parallax Error Diagram Placeholder (Visualizing how viewing a scale from different angles causes random parallax error) Systematic Errors A balance or analogue meter that has not been set exactly to zero gives rise to a zero error .
A metre rule with a damaged end can also give a zero error. All measurements in this case will be too large (or too small), unless measurements are made between two parts of the scale that do not include zero.
Figure 1.4: Zero Error Diagram Placeholder (Damaged metre rule showing zero error) A systematic error may also arise if an experimenter does not consider all the variables that might need to be controlled, such as the temperature of the surroundings.
Accuracy vs Precision Accuracy The accuracy of a measurement describes how close the measured value is to the true value of the quantity. Accepted Values: Sometimes the true value is a published accepted value (e.g., g = 9.81 m s^-2).
Unknown Values: Often the true value is not known (e.g., acceleration of a specific trolley), as variables vary by experiment. Precision Precision is shown by the agreement between repeat readings . A set of results that vary by less than 1% may be considered precise.
Possible precision depends on the uncertainty of the measuring equipment. Measuring Equipment Precision Instrument Calibration Uncertainty Standard Ruler 1 mm ± 0.5 mm Vernier Calliper 0.1 mm ± 0.05 mm Micrometer Screw Gauge 0.01 mm ± 0.005 mm Reading a Vernier Calliper Figure 1.6: Vernier Calliper Internal/External Scales (Showing 24.5 mm reading: main scale at 24, sliding scale aligned at 5) Reading a Micrometer Screw Gauge Figure 1.7: Micrometer Screw Gauge Components (Showing 12.66 mm reading: sleeve at 12.5, revolving thimble at 16) Important: Experimental Conditions Precision is also limited by conditions.
A stopwatch can read to 0.01s, but human reaction time (0.2-0.3s) makes such precision impossible for manual timing. Similarly, measuring a moving trolley with a ruler may have an uncertainty of 0.5cm or 1cm rather than 1mm.
Significant Figures Rule Use the same number of significant figures as in the original measurements when recording a mean value. Results should never be given to a greater uncertainty than can be inferred from the measurements.
The Target Analogy Figure 1.8: Target Darts Analogy (a) Precise, but not accurate (Clumped away from center) (b) Accurate and precise (Clumped at center) (c) Accurate, but not precise (Spread around center) (d) Neither accurate nor precise (Spread away from center) Calculating Uncertainties When quantities are used in calculations, their uncertainties must be combined.
Rules for Combining Uncertainties Addition / Subtraction: Add the absolute uncertainties (Δ y = Δ a + Δ b). Multiplication / Division: Add the percentage (fractional) uncertainties (fraction = fraction + fraction).
Powers (y = a^n): Multiply the percentage uncertainty by the power (fraction = n × fraction). Example: Density Uncertainty Density = Mass / Volume Mass = 100 g ± 1 g (1% uncertainty) Volume = 50 cm^3 ± 1 cm^3 (2% uncertainty) Percentage uncertainty in Density = 1% + 2% = 3%
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