Processing Uncertainties

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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.

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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.

Thêm câu hỏi

9.Which of the following is the definition of reliability?
Easy
- AA measure of the suitability of an experimental procedure to measure what it is intended to measure.
- BA measure of the ability of an experimental procedure to produce the expected results when using the same method and equipment.
- CA measure of how close a measurement is to the true value.
- DA measure of the spread of repeated measurements about the mean.
10.Which of the following is the definition of validity?
Easy
- AA measure of the ability of an experimental procedure to produce the expected results when using the same method and equipment.
- BA measure of the suitability of an experimental procedure to measure what it is intended to measure.
- CA measure of how close a measurement is to the true value.
- DA measure of the spread of repeated measurements about the mean.
11.Which of the following is an example of a systematic error?
Medium
- AA ruler reading varying slightly each time due to temperature changes.
- BA balance that reads 2 g when nothing is on it.
- CA stopwatch reaction time varying between trials.
- DA voltmeter fluctuating due to electrical noise.
12.How can random errors be reduced?
Medium
- ABy recalibrating the instrument.
- BBy taking repeat measurements and calculating an average.
- CBy using a different instrument.
- DBy correcting the experimental technique.
13.How can systematic errors be reduced?
Medium
- ABy taking more readings and averaging.
- BBy recalibrating instruments or using different instruments.
- CBy using a more precise device with smaller intervals.
- DBy ignoring anomalous results.
14.A measurement is given as 5.0 ± 0.2 cm. What is the percentage uncertainty?
Medium
- A0.2%
- B4%
- C2.5%
- D25%
15.When adding two quantities, a and b, how do their uncertainties combine?
Medium
- AAdd the absolute uncertainties.
- BAdd the fractional uncertainties.
- CMultiply the absolute uncertainties.
- DAdd the percentage uncertainties.
16.When multiplying two quantities, a and b, how do their uncertainties combine?
Medium
- AAdd the absolute uncertainties.
- BAdd the fractional uncertainties.
- CMultiply the fractional uncertainties.
- DMultiply the absolute uncertainties.
17.A quantity y is calculated from y = aⁿ. If the fractional uncertainty in a is ε, what is the fractional uncertainty in y?
Medium
- Aε
- Bnε
- Cε/n
- Dn + ε
18.On a graph, what do error bars represent?
Medium
- AThe absolute uncertainty of the plotted values.
- BThe percentage uncertainty of the gradient.
- CThe systematic error of the experiment.
- DThe difference between the best and worst lines of fit.
19.How is the percentage uncertainty in the gradient of a graph calculated?
Medium
- A(best gradient − worst gradient) / best gradient × 100%
- B(best gradient + worst gradient) / best gradient × 100%
- C(worst gradient − best gradient) / worst gradient × 100%
- D(best gradient − worst gradient) / worst gradient × 100%
20.How is the absolute uncertainty in the gradient calculated from the maximum and minimum lines of fit?
Medium
- A(max gradient − min gradient) / 2
- B(max gradient + min gradient) / 2
- Cmax gradient − min gradient
- D(max gradient − min gradient) × 2
21.The accepted value of the acceleration due to gravity is 9.81 m s⁻². An experiment gives 10.35 m s⁻². What is the percentage difference?
Medium
- A5.5%
- B0.55%
- C5.2%
- D54%
22.Which of the following statements about uncertainties are correct? (select all that apply)
Medium
- AAbsolute uncertainties have the same units as the quantity.
- BPercentage uncertainties have no units.
- CThe uncertainty in constants such as π is taken to be zero.
- DFractional uncertainties are always expressed as a percentage.
- EUncertainties in trigonometric functions will be tested in the exam.
23.Which of the following are types of measurement error that lead to uncertainty? (select all that apply)
Medium
- ARandom errors
- BSystematic errors
- CZero errors
- DReading errors
- ECalibration errors
24.Which of the following are ways to reduce random errors? (select all that apply)
Medium
- ARepeat measurements several times and calculate an average.
- BUse a more precise device with smaller measuring intervals.
- CRecalibrate the instrument.
- DUse a different instrument.
- EMake corrections or adjustments to the technique.
25.Which of the following are ways to reduce systematic errors? (select all that apply)
Medium
- ARecalibrate the instrument.
- BUse a different instrument.
- CMake corrections or adjustments to the technique.
- DRepeat measurements several times and calculate an average.
- EUse a more precise device with smaller measuring intervals.
26.Which of the following statements about precision and accuracy are correct? (select all that apply)
Medium
- APrecise measurements have very little spread about the mean value.
- BAccurate measurements are close to the true value.
- CA measurement can be precise but not accurate.
- DA measurement can be accurate but not precise.
- EPrecision is affected by systematic errors.
27.Random errors affect the precision of measurements, causing a wider spread of results about the mean value.
Easy
True or false?
28.Systematic errors affect the precision of measurements by increasing the spread of results.
Easy
True or false?
29.A zero error is a type of systematic error.
Easy
True or false?
30.The uncertainty in a measurement is the difference between the measured value and the true value.
Easy
True or false?
31.Error bars on a graph must all be the same size.
Easy
True or false?
32.The percentage difference is a type of percentage uncertainty.
Easy
True or false?
33.A reliable experiment produces consistent results when repeated many times.
Easy
True or false?
34.Validity is a measure of how close a measurement is to the true value.
Easy
True or false?
35.The uncertainty in a digital reading is ± the last significant digit unless otherwise quoted.
Easy
True or false?
36.The uncertainty in the natural log of a value is equal to the fractional uncertainty in that value.
Easy
True or false?
37.Match each term with its correct definition.
Medium
- Random error
- Systematic error
- Precision
- Accuracy
- Unpredictable fluctuations in readings due to uncontrollable factors
- Consistent error from faulty instruments or flawed method
- Little spread about the mean value
- Closeness to the true value
38.Match each uncertainty calculation to the correct formula.
Medium
- Uncertainty in a reading (analogue)
- Uncertainty in a measurement
- Uncertainty in repeated data
- Uncertainty in digital readings
- ± half the smallest division
- at least ±1 smallest division
- half the range
- ± the last significant digit
39.Arrange the following from smallest uncertainty to largest uncertainty for an analogue ruler with smallest division 1 mm.
Medium
- A single reading
- A measurement
- A reading with a more precise ruler (0.5 mm division)
- A measurement with a less precise ruler (2 mm division)
40.Match each uncertainty representation with its correct description.
Medium
- Absolute uncertainty
- Fractional uncertainty
- Percentage uncertainty
- The uncertainty expressed as a fixed quantity, with the same units as the measurement
- The uncertainty expressed as a fraction of the measurement
- The uncertainty expressed as a percentage of the measurement
41.A digital thermometer reads 25.3 °C. What is the absolute uncertainty in this reading, in °C?
Medium
42.An analogue ruler with a smallest division of 1 mm is used to take a single reading. What is the absolute uncertainty in this reading, in mm?
Medium
43.A measurement is recorded as 20.0 ± 0.5 cm. What is the percentage uncertainty? Give your answer as a percentage to one decimal place.
Medium
44.Complete the definitions of the two types of measurement error.
Easy
Random errors cause unpredictable fluctuations in readings and affect the ____ of measurements. Systematic errors arise from faulty instruments or flawed methods and affect the ____ of readings.
45.Complete the statements about reducing errors.
Easy
To reduce random errors, repeat measurements and calculate an ____. To reduce systematic errors, instruments should be ____.
46.Complete the definitions of reliability and validity.
Easy
Reliability is a measure of the ability of an experimental procedure to produce ____ results when using the same method and equipment. Validity is a measure of the suitability of an experimental procedure to measure what it is ____ to measure.
47.Complete the rules for combining uncertainties.
Medium
When adding or subtracting quantities, add the ____ uncertainties. When multiplying or dividing quantities, add the ____ uncertainties.
48.Complete the rules for uncertainty in powers and graphs.
Medium
When raising a quantity to a power n, multiply the percentage uncertainty by ____. On a graph, the percentage uncertainty in the gradient is calculated using the best and ____ gradients.
49.What is the name of the type of systematic error that occurs when an instrument gives a reading when the true reading is zero?
Easy
50.What term describes a measurement that has very little spread about the mean value?
Easy
51.What term describes a measurement that is close to the true value?
Easy
52.What term describes the ability of an experimental procedure to produce the expected results when using the same method and equipment?
Easy
53.Explain the difference between precision and accuracy, and state how each is affected by random and systematic errors.
Hard
54.Explain how to reduce random errors and systematic errors in an experiment.
Hard
55.Describe how to determine the percentage uncertainty in the gradient of a graph using lines of best and worst fit.
Hard
56.Explain how to determine the percentage uncertainty in the gradient of a graph using lines of best and worst fit.
Hard
57.A digital thermometer reads 25.3 °C. What is the absolute uncertainty in this reading, in °C?
Medium
- A0.1 °C
- B-0.9 °C
- C1.1 °C
- D2.1 °C
58.An analogue ruler with a smallest division of 1 mm is used to take a single reading. What is the absolute uncertainty in this reading, in mm?
Medium
- A0.5 mm
- B-0.5 mm
- C2.5 mm
- D1.5 mm
59.A measurement is recorded as 20.0 ± 0.5 cm. What is the percentage uncertainty? Give your answer as a percentage to one decimal place.
Medium
- A1.5 %
- B4.5 %
- C3.5 %
- D2.5 %

Thẻ ghi nhớ

What are random errors?

Random errors cause unpredictable fluctuations in an instrument's readings due to uncontrollable factors, such as environmental conditions. They affect precision, causing a wider spread of results about the mean.

How can random errors be reduced?

Repeat measurements several times and calculate an average from them.

What is the reading uncertainty for an analogue device?

The uncertainty is ±0.5 the smallest measuring interval.

What is the reading uncertainty for a digital device?

The uncertainty is ±1 the smallest measuring interval.

What are systematic errors?

Systematic errors arise from faulty instruments or flaws in the experimental method. They are repeated consistently, affecting the accuracy of all readings.

How can systematic errors be reduced?

Recalibrate instruments, use different instruments, or make corrections/adjustments to the technique.

What is a zero error?

A zero error is a type of systematic error where an instrument gives a reading when the true reading is zero. For example, a top-pan balance that starts at 2 g instead of 0 g.

How do you account for a zero error?

Subtract the offset from each value. For example, if a scale starts at 2 g, a measurement of 50 g is actually 50 – 2 = 48 g. The offset can be positive or negative.

What is precision?

Precision refers to how close repeated measurements are to each other, i.e., little spread about the mean. A precise measurement has small random uncertainty.

What is accuracy?

Accuracy refers to how close a measurement is to the true value. An accurate measurement has small systematic error.

What is reliability?

Reliability is a measure of the ability of an experimental procedure to produce the expected results when using the same method and equipment. A reliable experiment produces consistent results when repeated.

What is validity?

Validity is a measure of the suitability of an experimental procedure to measure what it is intended to measure. It requires identifying and controlling variables that may affect the outcome.

What is the uncertainty in a reading?

The uncertainty in a reading is ± half the smallest division.

What is the uncertainty in a measurement?

The uncertainty in a measurement is at least ±1 smallest division.

What is the uncertainty in repeated data?

The uncertainty is half the range: ± ½ (largest - smallest value).

What is the uncertainty in digital readings?

The uncertainty is ± the last significant digit unless otherwise quoted.

What is the uncertainty in the natural log of a value?

The absolute uncertainty in ln(x) is equal to the uncertainty in x divided by x.

How do you calculate absolute uncertainty from a set of repeated measurements?

Absolute uncertainty = half the range = ½ (largest - smallest value).

How do you calculate percentage uncertainty from absolute uncertainty?

Percentage uncertainty = (absolute uncertainty / measured value) × 100%.

What is the rule for combining uncertainties in addition and subtraction?

For y = a ± b, the absolute uncertainties add: Δy = Δa + Δb.

What is the rule for combining uncertainties in multiplication and division?

For y = a × b or y = a / b, the fractional uncertainties add: Δy/y = Δa/a + Δb/b.

What is the rule for combining uncertainties when raising to a power?

For y = aⁿ, the fractional uncertainty is multiplied by the power: Δy/y = n (Δa/a).

What are error bars on a graph?

Error bars are drawn above and below a data point (or from side to side) to show the absolute uncertainty of that measurement.

How do you determine the uncertainty in a gradient from a graph?

Draw the best line of best fit and the worst line of best fit (steepest or shallowest within error bars). The percentage uncertainty in the gradient is (|best gradient - worst gradient| / best gradient) × 100%.

What is the formula for percentage difference?

Percentage difference = (|experimental value - accepted value| / accepted value) × 100%.

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