Errors and uncertainties
Introduced| English | Meaning |
|---|---|
| uncertainty/ʌnˈsɜːtənti/ | An estimate of the possible range of error in a measurement. |
| micrometer/maɪˈkrɒmɪtə/ | A precision measuring instrument that uses a screw mechanism to measure small lengths or thicknesses. |
| vernier calipers/ˈvɜːnɪə ˈkælɪpəz/ | vernier calipers |
| systematic error/ˌsɪstəˈmætɪk ˈerə/ | A consistent measurement bias caused by the method or instrument. |
| zero error/ˈzɪərəʊ ˈerə/ | An instrument reading that is nonzero when the true quantity is zero. |
| calibration/ˌkælɪˈbreɪʃn/ | Comparing and, where appropriate, adjusting a measuring instrument against a known reference. |
| parallax/ˈpærəlæks/ | The apparent change in an object's position when viewed from different locations or angles. |
| random error/ˈrændəm ˈerə/ | Unpredictable variation between repeated measurements. |
| precision/prɪˈsɪʒn/ | How closely repeated measurements agree. |
| accuracy/ˈækjʊrəsi/ | How close a measurement is to the true value. |
| absolute uncertainty/ˈæbsəluːt ʌnˈsɜːtənti/ | absolute uncertainty |
| significant figures/sɪɡˈnɪfɪkənt ˈfɪɡəz/ | The meaningful digits in a measured or rounded number, beginning with its first non-zero digit. |
How sure are you?
- Measure a wire twice and you might get $5.77\ \text{mm}$, then $5.79\ \text{mm}$.
- Neither is "wrong" — every measurement has some uncertainty 不确定度.
- Good physics means knowing how much you are unsure.
Uncertainty lab
Classify measurement problems by how they affect data.
Reading an instrument
- Every scale can only be read so finely.
- A micrometer 螺旋测微器 reads to $0.01\ \text{mm}$; vernier calipers 游标卡尺 to $0.1\ \text{mm}$.
- The smallest step the instrument shows sets a limit on your reading.
A vernier caliper measures length precisely, with a small uncertainty
Read the whole millimetres from the main scale, then add the tenths where the sliding scale lines up
A careful enough measurement can be exactly correct, with zero uncertainty.
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No real measurement is exact — every reading has some uncertainty. The skill is estimating how big it is.
Systematic errors 系统误差
- A systematic error shifts every reading by the same amount, the same way.
- Causes: a zero error 零点误差, a wrong calibration 校准, or parallax 视差 (reading from the side).
- Repeating the measurement does not remove it.
A micrometer screw gauge measures to the nearest 0.01 mm
Parallax: viewing a scale from an angle shifts the reading — a systematic error
Which of these is a systematic error?
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A zero error shifts every reading by the same amount in the same direction — that is systematic. The others vary randomly from reading to reading.
Random errors 随机误差
- A random error makes readings jump above and below, with no pattern.
- Repeat the measurement many times and take the mean — the scatter partly cancels.
Precision 精密度: how narrow the distribution is around the true value T
To reduce a random error, repeat the measurement and take the ____.
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Random errors scatter both ways, so the mean of many repeats partly cancels them. (A systematic error survives averaging.)
Precision vs accuracy 准确度
- Precision = how tightly repeated readings cluster.
- Accuracy = how close they are to the true value.
Match each term to the definition the examiner marks.
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Repeating fixes only the random half. A zero error makes readings precise and wrong, which is why the two words are not interchangeable.
Which error hurts which?
- A systematic error moves the whole cluster off-target → worse accuracy.
- A random error spreads the cluster out → worse precision.
Match each pattern of repeated readings to its description.
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Precision is how tight the cluster is; accuracy is how near the true value (the bullseye) it sits.
Writing an uncertainty
- We write a result as $x \pm \Delta x$, where $\Delta x$ is the absolute uncertainty 绝对不确定度.
- The percentage uncertainty is $\dfrac{\Delta x}{|x|} \times 100\%$.
A length is $(20.0 \pm 0.5)\ \text{cm}$. What is its percentage uncertainty?
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$\dfrac{\Delta x}{|x|} \times 100\% = \dfrac{0.5}{20.0} \times 100\% = 2.5\%$.
Combining: add or subtract
- If $y = a + b$ or $y = a - b$, add the absolute uncertainties.
- $\Delta y = \Delta a + \Delta b$.
Combining: multiply or divide
- If $y = \dfrac{a \cdot b}{c}$, add the percentage uncertainties.
- $\dfrac{\Delta y}{|y|} = \dfrac{\Delta a}{|a|} + \dfrac{\Delta b}{|b|} + \dfrac{\Delta c}{|c|}$.
For $y = \dfrac{a \cdot b}{c}$, how do you combine the uncertainties?
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For multiplying and dividing, add the percentage uncertainties: $\dfrac{\Delta y}{|y|} = \dfrac{\Delta a}{|a|} + \dfrac{\Delta b}{|b|} + \dfrac{\Delta c}{|c|}$.
Which rules for combining uncertainties are correct? Select all that apply.
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Uncertainties never cancel. Subtracting two similar values keeps the sum of their uncertainties over a much smaller result, which is why such a difference is so imprecise.
Combining: powers
- If $y = a^{n}$, multiply the percentage uncertainty by the power.
- $\dfrac{\Delta y}{|y|} = |n| \cdot \dfrac{\Delta a}{|a|}$.
A radius is measured with a $0.4\%$ uncertainty. A circle's area is $A = \pi r^{2}$. What is the percentage uncertainty in $A$?
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For a power, multiply by the power: $A \propto r^{2}$, so the % uncertainty is $2 \times 0.4\% = 0.8\%$.
A period T is measured with a 2.0% uncertainty. What is the percentage uncertainty in T squared?
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Squaring doubles the percentage uncertainty; a square root halves it. Quoting 2.0% for T^2 is one of the commonest errors in this topic.
Worked example: volume of a ball
- Diameter $d = (5.26 \pm 0.02)\ \text{cm}$, and volume $V \propto d^{3}$.
- % uncertainty in $d$: $\dfrac{0.02}{5.26} \times 100\% \approx 0.38\%$.
- Power rule: % uncertainty in $V = 3 \times 0.38\% \approx 1.1\%$.
- So $V = (76.2 \pm 0.9)\ \text{cm}^{3}$.
Significant figures 有效数字
- Give an answer the same significant figures as the least precise measurement.
- Usually 2 or 3 in this course — too many makes it look more exact than it is.
You've got it
- systematic error → shifts every reading (worse accuracy); random → scatter (worse precision)
- repeat and take the mean to beat random error
- combine uncertainties: add absolute for $+\,-$, add % for $\times\,\div$, × the power for $a^{n}$