Processing Uncertainties

边玩边学

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Random & Systematic Errors

Measurements aim to find the true value, but uncertainty is always present.
Random errors cause unpredictable fluctuations in readings due to uncontrollable factors, affecting precision.
To reduce random errors, repeat measurements and calculate an average.
Systematic errors arise from faulty instruments or flawed methods, affecting accuracy consistently.
To reduce systematic errors, recalibrate instruments or use different ones, and correct techniques.
Zero errors are a type of systematic error where an instrument gives a reading when the true reading is zero.
To account for zero errors, subtract the offset from each measurement.

Precision, Accuracy, Reliability & Validity

Precision refers to how close repeated measurements are to each other; small random uncertainty means high precision.
Accuracy refers to how close a measurement is to the true value; small systematic error means high accuracy.
Repeating measurements and taking a mean can increase accuracy and help identify anomalies.
Reliability is the ability of an experiment to produce consistent results when repeated.
Validity is the suitability of the experimental procedure to measure what it intends to measure.
Variables that may affect the outcome must be identified and controlled for valid results.

Calculating Uncertainties

Uncertainty is a range of values around a measurement within which the true value is expected to lie.
Uncertainties are not the same as errors; errors are issues that cause a reading to differ from the true value.
Absolute uncertainty is given as a fixed quantity (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.
For a reading, uncertainty is ± half the smallest division; for a measurement, at least ±1 smallest division.
For repeated data, uncertainty is half the range: ± ½ (largest - smallest value).
For digital readings, uncertainty is ± the last significant digit unless otherwise quoted.

Combining Uncertainties

When adding or subtracting quantities, add the absolute uncertainties.
When multiplying or dividing quantities, add the fractional (or percentage) uncertainties.
When raising a quantity to a power, multiply the fractional uncertainty by the power.
Absolute uncertainties have the same units as the quantity; percentage uncertainties have no units.
The uncertainty in constants such as π is taken to be zero.
Uncertainties in trigonometric and logarithmic functions will not be tested in the exam.

Determining Uncertainties from Graphs

Error bars are plotted on graphs to show the absolute uncertainty of values.
To find the uncertainty in a gradient, draw the best line of best fit and the worst line of best fit (steepest or shallowest that fits within all error bars).
Percentage uncertainty in gradient = |(best gradient − worst gradient) / best gradient| × 100%.
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.
Error bars do not need to be the same size for all data points.

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 smaller the percentage difference, the more accurate the results.

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练习题

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1.What is the name for unpredictable fluctuations in an instrument's readings caused by uncontrollable factors such as environmental conditions?
Easy
- ARandom errors
- BSystematic errors
- CZero errors
- DReading errors
2.Which type of error affects the accuracy of all readings obtained?
Easy
- ARandom errors
- BSystematic errors
- CReading errors
- DPrecision errors
3.Random errors can be reduced by taking repeat measurements and calculating an average.
Easy
True or false?
4.When measuring a quantity using an analogue device such as a ruler, what is the uncertainty in the measured quantity?
Easy
- A±1 the smallest measuring interval
- B±0.5 the smallest measuring interval
- C±2 the smallest measuring interval
- D±0.1 the smallest measuring interval
5.When measuring a quantity using a digital device such as a digital scale, what is the uncertainty in the measured quantity?
Easy
- A±0.5 the smallest measuring interval
- B±1 the smallest measuring interval
- C±2 the smallest measuring interval
- D±0.1 the smallest measuring interval
6.A top-pan balance starts at 2 g instead of 0 g. A measurement of 50 g is taken. What is the true mass?
Medium
- A50 g
- B52 g
- C48 g
- D25 g
7.Which statement best describes precision?
Medium
- AHow close a measurement is to the true value
- BHow little spread there is about the mean value
- CThe difference between the experimental and accepted value
- DThe suitability of an experiment to measure what it intends to measure
8.Which statement best describes accuracy?
Medium
- AHow close a measurement is to the true value
- BHow little spread there is about the mean value
- CThe ability to reproduce results consistently
- DThe smallest division on a measuring instrument

更多题目

9.Which of the following is the correct definition of reliability?
Medium
- 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
- CHow close a measurement is to the true value
- DHow little spread there is about the mean value
10.Which of the following is the correct definition of validity?
Medium
- 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
- CHow close a measurement is to the true value
- DHow little spread there is about the mean value
11.Systematic errors can be reduced by recalibrating instruments or using different instruments.
Easy
True or false?
12.Random errors affect the accuracy of measurements, while systematic errors affect the precision.
Easy
True or false?
13.Which of the following is a type of systematic error?
Medium
- AZero error
- BReading error
- CRandom fluctuation
- DEnvironmental variation
14.How can zero errors be accounted for?
Medium
- ABy adding the offset to each value
- BBy taking the difference of the offset from each value
- CBy multiplying each value by the offset
- DBy ignoring the offset
15.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
16.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 mainly by systematic errors
17.Match each term with its correct definition.
Medium
- Random error
- Systematic error
- Zero error
- Reading error
- Unpredictable fluctuations in readings
- Consistent error from faulty instruments or method
- Instrument gives a reading when true reading is zero
- Uncertainty due to smallest measuring interval
18.Match each term with its correct description.
Medium
- Precision
- Accuracy
- Reliability
- Validity
- Little spread about the mean value
- Close to the true value
- Consistent results when repeated
- Suitability of the experiment to measure what it intends to measure
19.Match each type of uncertainty to the correct way it is expressed.
Medium
- Absolute uncertainty
- Fractional uncertainty
- Percentage uncertainty
- A fixed quantity with the same units as the measurement
- A fraction of the measurement
- A percentage of the measurement
20.Which of the following statements about the uncertainty in a single reading are correct? (select all that apply)
Medium
- AFor an analogue device, it is ± half the smallest division
- BFor a digital device, it is ± the last significant digit
- CFor a digital device, it is ± half the smallest division
- DFor an analogue device, it is ± the smallest division
- EIt is always zero for a digital device
21.A student measures the length of a pendulum five times and obtains the values 20.1 cm, 20.0 cm, 20.2 cm, 20.1 cm, 20.0 cm. What is the uncertainty in the mean length due to the spread of the readings?
Medium
- A±0.1 cm
- B±0.2 cm
- C±0.05 cm
- D±0.5 cm
22.A digital voltmeter displays a reading of 3.45 V. What is the absolute uncertainty in this reading?
Medium
- A±0.01 V
- B±0.005 V
- C±0.1 V
- D±0.05 V
23.A quantity y is calculated from y = a × b. The percentage uncertainty in a is 2% and in b is 3%. What is the percentage uncertainty in y?
Hard
- A5%
- B1%
- C6%
- D2.5%
24.Which of the following are correct rules for combining uncertainties? (select all that apply)
Medium
- AWhen adding or subtracting quantities, add the absolute uncertainties
- BWhen multiplying or dividing quantities, add the percentage uncertainties
- CWhen raising a quantity to a power n, multiply the percentage uncertainty by n
- DWhen adding or subtracting quantities, add the percentage uncertainties
- EWhen multiplying or dividing quantities, add the absolute uncertainties
25.On a graph, the best-fit line has a gradient of 0.25 and the worst-fit line has a gradient of 0.28. What is the percentage uncertainty in the gradient?
Medium
- A12%
- B0.12%
- C10.7%
- D3%
26.Error bars on a graph must all be the same size.
Medium
True or false?
27.The accepted value for 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%
- B5.2%
- C0.54%
- D55%
28.Which of the following statements about uncertainty in graphs are correct? (select all that apply)
Medium
- AError bars show the absolute uncertainty of plotted values
- BThe worst-fit line is the steepest or shallowest line that fits within all error bars
- CThe percentage uncertainty in the y-intercept can be found using best and worst y-intercepts
- DThe line of best fit must pass through every data point
- EError bars are only drawn for the first and last points
29.The uncertainty in a measurement is an estimate of the range within which the true value is expected to lie.
Easy
True or false?
30.A student records five repeated measurements of a time interval. Which quantity is found by calculating half the range of the values?
Medium
- AThe uncertainty in the repeated data
- BThe percentage uncertainty in the data
- CThe fractional uncertainty in the data
- DThe absolute uncertainty in the mean, divided by the mean
31.A voltage V = 6.0 V is measured with an absolute uncertainty of ±0.1 V. A current I = 0.50 A is measured with an absolute uncertainty of ±0.01 A. The resistance R is calculated from R = V / I. What is the percentage uncertainty in R?
Medium
- A2%
- B3.7%
- C1.7%
- D0.2%
32.A student measures the diameter of a sphere as (4.00 ± 0.05) cm. The volume is calculated using V = (π/6)d³. What is the percentage uncertainty in the volume?
Medium
- A1.25%
- B3.75%
- C0.375%
- D0.125%
33.In an experiment, the period T of a simple pendulum is measured for different lengths L. A graph of T² against L is plotted. The gradient is found to be 4.05 s² m⁻¹ with an uncertainty of ±0.15 s² m⁻¹. The accepted value of 4π²/g is 4.03 s² m⁻¹. What is the percentage difference between the experimental and accepted values?
Hard
- A0.50%
- B0.49%
- C0.02%
- D3.7%
34.A graph of force F (y-axis) against extension x (x-axis) for a spring is plotted. The line of best fit has gradient 12.0 N m⁻¹. The steepest line of worst fit has gradient 13.2 N m⁻¹. Which of the following statements are correct? (select all that apply)
Hard
- AThe absolute uncertainty in the gradient is 0.6 N m⁻¹.
- BThe percentage uncertainty in the gradient is 10%.
- CThe percentage uncertainty in the gradient is 5%.
- DThe absolute uncertainty in the gradient is 1.2 N m⁻¹.
- EThe worst gradient is the steepest line.
35.The uncertainty in a digital reading is taken as ±1 in the last significant digit unless otherwise quoted.
Medium
True or false?
36.A micrometer screw gauge has a smallest division of 0.01 mm. What is the uncertainty in a single reading taken with this instrument?
Medium
- A±0.01 mm
- B±0.005 mm
- C±0.05 mm
- D±0.1 mm
37.In an experiment, the period T of a simple pendulum is measured for different lengths L. A graph of T² against L is plotted. The gradient is found to be 4.05 s² m⁻¹ with a percentage uncertainty of 2.5%. What is the absolute uncertainty in the gradient?
Medium
- A±0.10 s² m⁻¹
- B±0.01 s² m⁻¹
- C±0.41 s² m⁻¹
- D±1.62 s² m⁻¹
38.A graph of force F (y-axis) against extension x (x-axis) for a spring is plotted. The line of best fit has gradient 12.0 N m⁻¹. The steepest line of worst fit has gradient 13.2 N m⁻¹. What is the percentage uncertainty in the gradient?
Medium
- A10%
- B1.2%
- C9.1%
- D12%
39.A student uses a ruler with a smallest division of 1 mm to measure the length of a wire once. What is the uncertainty in this single reading?
Medium
- A±0.5 mm
- B±1 mm
- C±2 mm
- D±0.1 mm
40.A digital ammeter has a smallest measuring interval of 0.01 A and displays a current of 1.25 A. What is the percentage uncertainty in this reading?
Medium
- A0.8%
- B0.08%
- C1.0%
- D0.4%
41.Complete the following sentences about uncertainties.
Medium
The uncertainty in a reading is ±____ the smallest division. The uncertainty in a measurement is at least ±____ smallest division.
42.Complete the following sentences about combining uncertainties.
Medium
When adding or subtracting data, add the ____ uncertainties. When multiplying or dividing data, add the ____ uncertainties.
43.What is the name for the type of systematic error that occurs when an instrument gives a reading when the true reading is zero?
Easy
44.What is the term for the measure of the ability of an experimental procedure to produce the expected results when using the same method and equipment?
Easy
45.What is the term for the measure of the suitability of an experimental procedure to measure what it is intended to measure?
Easy
46.A digital stopwatch has a smallest measuring interval of 0.01 s. What is the uncertainty in a single reading, in seconds?
Medium
47.A ruler has a smallest measuring interval of 1 mm. What is the uncertainty in a single reading, in mm?
Medium
48.The length of a rectangle is measured as (10.0 ± 0.1) cm and its width as (5.0 ± 0.1) cm. Calculate the absolute uncertainty in the perimeter, in cm.
Hard
49.A cube has side length (2.0 ± 0.1) cm. Calculate the percentage uncertainty in its volume.
Hard
50.Complete the sentence about percentage difference.
Medium
Percentage difference = |experimental value − ____ value| / ____ value × 100%.
51.A digital stopwatch has a smallest measuring interval of 0.01 s. A time of 12.34 s is recorded. What is the absolute uncertainty in this reading, in seconds?
Medium
52.The uncertainty in a measurement is quoted as ±0.3 cm on a length of 15.0 cm. Calculate the percentage uncertainty in the length. Give your answer to 1 decimal place.
Medium
53.Explain the difference between random errors and systematic errors, and state one way to reduce each.
Hard
54.Explain what is meant by the term 'zero error' and how it can be accounted for.
Hard
55.Explain how the percentage uncertainty in the gradient of a graph is determined using the lines of best and worst fit.
Hard
56.Explain how you would determine the percentage uncertainty in the y-intercept of a graph using the line of best fit and the line of worst fit.
Hard
57.Explain how you would determine the percentage uncertainty in the gradient of a graph using the lines of best and worst fit.
Hard
58.A digital stopwatch has a smallest measuring interval of 0.01 s. What is the uncertainty in a single reading, in seconds?
Medium
- A1.01 s
- B0.01 s
- C-0.99 s
- D2.01 s
59.A ruler has a smallest measuring interval of 1 mm. What is the uncertainty in a single reading, in mm?
Medium
- A1.5 mm
- B-0.5 mm
- C0.5 mm
- D2.5 mm
60.The length of a rectangle is measured as (10.0 ± 0.1) cm and its width as (5.0 ± 0.1) cm. Calculate the absolute uncertainty in the perimeter, in cm.
Hard
- A0.4 cm
- B1.4 cm
- C2.4 cm
- D-0.6 cm
61.A cube has side length (2.0 ± 0.1) cm. Calculate the percentage uncertainty in its volume.
Hard
- A14 %
- B17 %
- C15 %
- D16 %
62.A digital stopwatch has a smallest measuring interval of 0.01 s. A time of 12.34 s is recorded. What is the absolute uncertainty in this reading, in seconds?
Medium
- A-0.99 s
- B1.01 s
- C2.01 s
- D0.01 s
63.The uncertainty in a measurement is quoted as ±0.3 cm on a length of 15.0 cm. Calculate the percentage uncertainty in the length. Give your answer to 1 decimal place.
Medium
- A3 %
- B2 %
- C4 %
- D1 %

闪卡

What is the uncertainty of a measurement?

The uncertainty is an estimate of the difference between a measurement reading and the true value. It is a range of values around a measurement within which the true value is expected to lie.

What are random errors and how do they affect measurements?

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 when using an analogue device?

The uncertainty is ±0.5 times the smallest measuring interval.

What is the reading uncertainty when using a digital device?

The uncertainty is ±1 times the smallest measuring interval.

How can reading errors be reduced?

Use a more precise device with smaller measuring intervals and therefore less uncertainty.

What are systematic errors and how do they affect measurements?

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 or use different instruments, and make corrections or 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?

Take the difference of the offset from each value. For example, if a scale starts at 2 g instead of 0 g, a measurement of 50 g would actually be 50 – 2 = 48 g.

What is meant by precision?

Precise measurements have very little spread about the mean value; the measured values are very close to each other. If the random uncertainty is small, the measurement is precise.

What is meant by accuracy?

A measurement is accurate if it is close to the true value. If the systematic error is small, the measurement is accurate.

How can accuracy be increased?

Repeat measurements and find a mean of the results. Repeating also helps identify anomalies that can be omitted.

What is reliability in the context of experiments?

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 in the context of experiments?

Validity is a measure of the suitability of an experimental procedure to measure what it is intended to measure. Variables that may affect the outcome must be identified and controlled.

What is absolute uncertainty?

Absolute uncertainty is the uncertainty given as a fixed quantity. It has the same units as the quantity.

What is fractional uncertainty?

Fractional uncertainty is the uncertainty given as a fraction of the measurement.

What is percentage uncertainty?

Percentage uncertainty is the uncertainty given as a percentage of the measurement. It has no units.

How do you find the uncertainty in repeated data?

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

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 a quantity is raised to a power?

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

What is the uncertainty in a constant such as π?

The uncertainty in constants such as π is taken to be zero.

What are error bars on a graph?

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

How do you find the percentage uncertainty in a gradient from a graph?

Draw the best and worst lines of best fit, then use: percentage uncertainty = |(best gradient − worst gradient) / best gradient| × 100%.

What is the percentage difference and how is it calculated?

Percentage difference indicates how close an experimental value is to the accepted value: percentage difference = |(experimental value − accepted value) / accepted value| × 100%.

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