Processing Uncertainties
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Lesson notes
Random & Systematic Errors
- Random errors cause unpredictable fluctuations in readings due to uncontrollable factors like environmental conditions.
- Random errors affect precision, causing a wider spread of results about the mean.
- To reduce random errors, repeat measurements several times and calculate an average.
- Systematic errors arise from faulty instruments or flaws in the experimental method.
- Systematic errors are repeated consistently, affecting the accuracy of all readings.
- To reduce systematic errors, recalibrate instruments or use different instruments, and correct the technique.
- On graphs, systematic errors are shown by the offset of the line from the origin.
Reading Errors and Zero Errors
- For an analogue device (e.g., ruler), the uncertainty is ±0.5 the smallest measuring interval.
- For a digital device (e.g., digital scale or stopwatch), the uncertainty is ±1 the smallest measuring interval.
- To reduce reading errors, use a more precise device with smaller measuring intervals.
- A zero error is a type of systematic error where an instrument gives a reading when the true reading is zero.
- To account for zero error, subtract the offset from each value (e.g., if a scale starts at 2 g, a 50 g reading is actually 48 g).
- The offset could be positive or negative.
Precision, Accuracy, Reliability, and Validity
- Precision refers to how close repeated measurements are to each other; small random uncertainty means high precision.
- Measurements to a greater number of decimal places are said to be more precise.
- Accuracy refers to how close a measurement is to the true value; small systematic error means high accuracy.
- Accuracy can be increased by repeating measurements and finding a mean, which also helps identify anomalies.
- Reliability is the ability of an experiment to produce consistent results when repeated using the same method and equipment.
- Validity is the suitability of an experimental procedure to measure what it is intended to measure.
- For valid results, all variables that may affect the outcome must be identified and controlled.
Calculating Uncertainties
- Uncertainty is a range of values around a measurement within which the true value is expected to lie; it is an estimate.
- Absolute uncertainty is given as a fixed quantity and has the same units as the measurement.
- Fractional uncertainty is the uncertainty as a fraction of the measurement.
- Percentage uncertainty is the uncertainty as a percentage of the measurement.
- Uncertainty in a reading: ± half the smallest division.
- Uncertainty in a measurement: at least ±1 smallest division.
- Uncertainty in repeated data: half the range, i.e., ± ½ (largest − smallest value).
- Uncertainty in digital readings: ± the last significant digit unless otherwise quoted.
- Uncertainty in the natural log of a value: absolute uncertainty in ln(x) = uncertainty in x divided by x.
Combining Uncertainties
- For addition and subtraction (y = a ± b), add the absolute uncertainties: Δy = Δa + Δb.
- For multiplication and division (y = a × b or y = a/b), add the fractional uncertainties: Δy/y = Δa/a + Δb/b.
- For powers (y = an), multiply the fractional uncertainty by the power: Δy/y = n(Δa/a).
- When adding or subtracting data, add the absolute uncertainties.
- When multiplying or dividing data, add the percentage or fractional uncertainties.
- When raising to a power, multiply the percentage uncertainty by the power.
- Absolute uncertainties (Δ) have the same units as the quantity; percentage uncertainties have no units.
- The uncertainty in constants such as π is taken to be zero.
Determining Uncertainties from Graphs
- The uncertainty in a measurement can be shown on a graph as an error bar.
- Error bars are drawn above and below the point (or side to side) and show the absolute uncertainty.
- To find the uncertainty in a gradient, draw the best line of best fit and the worst line of best fit.
- The best line passes as close as possible to all points; the worst line is the steepest or shallowest line that fits within all error bars.
- Percentage uncertainty in gradient = |(best gradient − worst gradient) / best gradient| × 100%.
- The worst gradient is the one with the greatest difference in magnitude from the best line.
- Alternatively, absolute uncertainty in gradient = (max gradient − min gradient) / 2.
- Percentage uncertainty in y-intercept = |(best y-intercept − worst y-intercept) / best y-intercept| × 100%.
- Absolute uncertainty in y-intercept = (max y-intercept − min y-intercept) / 2.
Percentage Difference
- Percentage difference indicates how close an experimental value is to the accepted value; it is not a percentage uncertainty.
- Percentage difference = |(experimental value − accepted value) / accepted value| × 100%.
- The experimental value is sometimes called the measured value; the accepted value is the true value.
- The accepted value may be labelled on a component or taken from a reputable source such as a peer-reviewed data booklet.
- For example, the accepted value of g is 9.81 m s⁻²; if an experiment gives 10.35 m s⁻², the percentage difference is 5.5%.
- The smaller the percentage difference, the more accurate the results.
Examiner Tips and Tricks
- Always ensure absolute or percentage uncertainty is given to the same number of significant figures as the reading.
- A common misconception is that error bars must all be the same size; in physics, each data point can have different error bar sizes.
- Uncertainties in trigonometric and logarithmic functions will not be tested in the exam.
Slides
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Practice questions
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1.Which type of error causes unpredictable fluctuations in an instrument's readings due to uncontrollable factors such as environmental conditions?
Easy- ARandom error
- BSystematic error
- CZero error
- DParallax error
2.Which of the following best describes a systematic error?
Easy- AIt causes a wider spread of results about the mean.
- BIt arises from faulty instruments or flaws in the experimental method.
- CIt can be reduced by taking more readings and averaging.
- DIt is due to unpredictable fluctuations in readings.
3.What is the uncertainty when taking a single reading from an analogue device such as a ruler?
Easy- A± the smallest measuring interval
- B± half the smallest measuring interval
- C± twice the smallest measuring interval
- D± one tenth of the smallest measuring interval
4.What is the uncertainty when taking a single reading from a digital device such as a digital stopwatch?
Easy- A± half the smallest measuring interval
- B± the smallest measuring interval
- C± twice the smallest measuring interval
- D± zero, as digital devices are exact
5.A student takes repeated measurements of a length. Which expression gives the uncertainty in the mean value?
Medium- AThe largest value minus the smallest value
- BHalf the range, i.e. ± ½ (largest − smallest)
- CThe sum of all values divided by the number of values
- DThe standard deviation divided by the mean
6.A top-pan balance has a zero error and reads 2 g when nothing is on it. A measurement of 50 g is taken. What is the corrected mass?
Medium- A52 g
- B50 g
- C48 g
- D2 g
7.Which statement about precision is correct?
Easy- APrecise measurements are close to the true value.
- BPrecise measurements have very little spread about the mean value.
- CPrecision is reduced by taking more readings.
- DPrecision is a measure of the suitability of the experimental procedure.
8.Which statement about accuracy is correct?
Easy- AAccurate measurements have very little spread about the mean.
- BAccurate measurements are close to the true value.
- CAccuracy is a measure of the ability to reproduce results.
- DAccuracy is increased by using a more precise device.
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