📚 Edexcel A-Level Combined Science: Measurement, Errors and Uncertainty | 爱德思A-Level综合科学：测量、误差与不确定度
This revision guide covers the quantitative skills needed for Edexcel A-Level Combined Science practical work, especially the measurement, error and uncertainty material often practised under topic 1.8.4/184. It brings together the common language of uncertainty used in chemistry titrations, physics circuit readings and biology enzyme investigations.
本复习指南涵盖爱德思A-Level综合科学实验工作所需的定量技能，特别是通常在课题1.8.4/184中练习的测量、误差与不确定度内容。它整合了化学滴定、物理电路读数和生物酶研究中通用的不确定度语言。
You are expected not only to record data but also to judge how reliable that data is. This means quoting absolute and percentage uncertainties, propagating errors through calculations, and using uncertainty bars or max-min gradients when analysing graphs.
你不仅要记录数据，还要判断这些数据有多可靠。这意味着要给出绝对和百分不确定度，在计算中传递误差，并在分析图表时使用误差棒或最大-最小斜率法。
1. Why Measurement Uncertainty Matters | 为什么测量不确定度很重要
Every measurement made with a ruler, balance, thermometer or voltmeter has a limit set by the instrument resolution and by the experimenter. In Edexcel Combined Science papers, a raw answer without an uncertainty is often treated as incomplete at higher levels.
用尺子、天平、温度计或电压表进行的每一次测量都会受到仪器分辨率和实验者操作的限制。在爱德思综合科学试卷中，较高层次上，没有不确定度的原始答案通常被视为不完整。
Uncertainty is not a sign of failure. It tells another scientist how much confidence to place in the value. For example, a temperature rise of 6.0 ± 0.5 °C is a more honest statement than 6.0 °C because it shows the precision limit of the thermometer and the two readings used to obtain the change.
不确定度并不代表失败。它告诉其他科学家可以在多大程度上信任该数值。例如，6.0 ± 0.5 °C 的温升比 6.0 °C 更诚实，因为它显示了温度计以及获得温升所用的两次读数的精度极限。
2. Accuracy, Precision, Repeatability, Reproducibility and Resolution | 准确度、精密度、重复性、再现性与分辨率
Accuracy describes how close the average of repeated readings is to the accepted or true value. Precision describes how close repeated readings are to one another; a set of values can be precise but systematically inaccurate.
准确度描述重复读数平均值与公认值或真值的接近程度。精密度描述重复读数之间的接近程度；一组数据可以很精密，但因系统误差而不准确。
Repeatability is the agreement between results obtained by the same experimenter using the same method and equipment within a short time. Reproducibility is agreement between results obtained by different experimenters, different equipment or different laboratories.
重复性是指同一实验者在短时间内使用相同方法和设备所得结果之间的一致性。再现性是指不同实验者、不同设备或不同实验室所得结果之间的一致性。
Resolution is the smallest observable change in the quantity being measured. A digital balance reading to 0.01 g has a resolution of 0.01 g, while a thermometer marked every 0.5 °C has a resolution of 0.5 °C.
分辨率是可观测到的最小变化量。读数至0.01 g的数字天平分辨率为0.01 g，而刻度间隔为0.5 °C的温度计分辨率为0.5 °C。
| Term 术语 | Meaning 含义 | Example 示例 |
|---|---|---|
| Accuracy 准确度 | Closeness to true value 与真值接近程度 | Mean titre within 0.10 cm³ of expected 平均滴定值在预期值±0.10 cm³内 |
| Precision 精密度 | Spread of repeated readings 重复读数分散程度 | Repeated titres within 0.05 cm³ 重复滴定值相差不超过0.05 cm³ |
| Resolution 分辨率 | Smallest detectable change 最小可检测变化 | Digital thermometer reads to 0.1 °C 数字温度计读至0.1 °C |
3. Systematic and Random Errors | 系统误差与随机误差
A systematic error shifts all readings in the same direction. It can be caused by poorly calibrated equipment, a zero error on a balance or ammeter, or a method that consistently loses product or heat. Repeating the experiment does not reduce a systematic error.
系统误差会使所有读数朝同一方向偏移。它可能由校准不良的设备、天平或电流计的零误差，或者持续损失产物或热量的方法引起。重复实验不能减小系统误差。
A random error is caused by unpredictable fluctuations in reading the scale, variations in judging colour change, or small changes in temperature and voltage. Taking many readings and calculating a mean reduces the effect of random errors.
随机误差由读取刻度时不可预测的波动、判断颜色变化时的差异或者温度与电压的微小变化引起。多次读数并取平均值可以减小随机误差的影响。
Zero errors are often systematic. If an ammeter reads 0.2 A when no current flows, every current reading is 0.2 A too high. The correction is to subtract the zero reading or to adjust the instrument before use.
零误差通常是系统性的。如果电流计在无电流时读数为0.2 A，则每个电流读数都偏高0.2 A。校正方法是减去零读数或在使用前调零仪器。
4. Absolute and Percentage Uncertainty | 绝对不确定度与百分不确定度
Absolute uncertainty is the interval around a measurement within which the true value is expected to lie. It has the same unit as the quantity. Percentage uncertainty expresses this interval as a fraction of the measured value.
绝对不确定度是测量值周围真值预期所在的区间。它与该物理量具有相同单位。百分不确定度将此区间表示为测量值的分数。
Percentage uncertainty = (absolute uncertainty ÷ measured value) × 100%
百分不确定度 = (绝对不确定度 ÷ 测量值) × 100%
For a single analogue reading, the absolute uncertainty is usually taken as half the smallest scale division. For a digital instrument, it is often taken as the smallest displayed digit or half of it, depending on the specification guidance.
对于单次模拟读数，绝对不确定度通常取最小刻度间隔的一半。对于数字仪器，通常取最小显示数字或其二分之一，具体取决于课程指南。
If a temperature change is found by subtracting an initial reading from a final reading, the absolute uncertainty of the change is the sum of the two reading uncertainties. For example, if each thermometer reading is ±0.5 °C, then ΔT = T₂ – T₁ has an uncertainty of ±1.0 °C.
如果温度变化由末读数减去初读数得到，则温差的绝对不确定度为两次读数不确定度之和。例如，若每次温度计读数不确定度为±0.5 °C，则ΔT = T₂ – T₁的不确定度为±1.0 °C。
5. Combining Uncertainties | 合并不确定度
When measured values are added or subtracted, add their absolute uncertainties. This gives the largest possible interval for the result.
当测量值相加或相减时，应将其绝对不确定度相加。这样可以得到结果可能的最大区间。
If R = A + B or R = A – B, then ΔR = ΔA + ΔB
若 R = A + B 或 R = A – B，则 ΔR = ΔA + ΔB
When measured values are multiplied or divided, add their percentage uncertainties. This applies to many combined science calculations such as concentration, density, pressure and rate.
当测量值相乘或相除时，应将其百分不确定度相加。这适用于浓度、密度、压强和速率等许多综合科学计算。
If R = A × B or R = A ÷ B, then %ΔR = %ΔA + %ΔB
若 R = A × B 或 R = A ÷ B，则 %ΔR = %ΔA + %ΔB
When a measured value is raised to a power, multiply the percentage uncertainty by that power. For example, if kinetic energy is calculated from Eₖ = ½mv², the percentage uncertainty in v² is twice the percentage uncertainty in v.
当测量值被乘方时，应将百分不确定度乘以该指数。例如，若动能由 Eₖ = ½mv² 计算，则 v² 的百分不确定度是 v 的百分不确定度的两倍。
If R = Aⁿ, then %ΔR = n × %ΔA
若 R = Aⁿ，则 %ΔR = n × %ΔA
6. Uncertainties from Graphs | 图表中的不确定度
When a straight-line graph is used to find a gradient or intercept, you can estimate the uncertainty by drawing the best-fit line and two worst-fit lines. The worst-fit lines should pass through the error bars, not necessarily through every point.
当使用直线图求斜率或截距时，可以通过绘制最佳拟合线和两条最差拟合线来估算不确定度。最差拟合线应通过误差棒，而不必穿过每个数据点。
The best gradient is mbest. The maximum gradient is mmax and the minimum gradient is mmin. The uncertainty in the gradient is half the range.
最佳斜率为 mbest。最大斜率为 mmax，最小斜率为 mmin。斜率的不确定度为该范围的一半。
Δm = (mmax – mmin) ÷ 2
Δm = (mmax – mmin) ÷ 2
Error bars on individual points show the uncertainty in the x or y value. If the error bars are too small to draw on the graph, state this and explain that the uncertainty is too small to be shown at the chosen scale.
单个点上的误差棒表示 x 或 y 值的不确定度。如果误差棒太小而无法在图上画出，应说明这一点，并解释在所选比例下不确定度小到无法显示。
7. Significant Figures and Decimal Places | 有效数字与小数位数
Uncertainties are usually quoted to one significant figure, or occasionally two if the first digit is 1. The measured value should then be quoted to the same decimal place as the absolute uncertainty.
不确定度通常保留一位有效数字，若首位数字为1，偶尔保留两位。测量值则应与绝对不确定度保留到相同的小数位。
For example, if a mean titre is 24.35 cm³ and the half-range uncertainty is 0.07 cm³, the result should be written as 24.35 ± 0.07 cm³. Writing 24.350 ± 0.07 cm³ would suggest a resolution that the burette does not have.
例如，若平均滴定值为24.35 cm³，半距不确定度为0.07 cm³，结果应写作24.35 ± 0.07 cm³。写成24.350 ± 0.07 cm³则暗示滴定管不具备该分辨率。
Percentage uncertainties should normally be given to two significant figures. Always match the final calculated quantity to the uncertainty precision, not just to an arbitrary number of significant figures.
百分不确定度通常保留两位有效数字。最终计算量应与不确定度的精度相匹配，而不是随意保留某几位有效数字。
8. Worked Example: Calorimetry Enthalpy Calculation | 计算示例：量热法焓变计算
In a calorimetry experiment, a digital balance reads the mass of water as 25.0 g with an uncertainty of ±0.1 g. A digital thermometer reads the initial and final temperatures, giving a temperature change ΔT = 12.5 °C with each reading uncertain by ±0.5 °C.
在量热实验中，数字天平测得水的质量为25.0 g，不确定度为±0.1 g。数字温度计读出初始和末温度，得到温度变化ΔT = 12.5 °C，每次读数不确定度为±0.5 °C。
The absolute uncertainty in ΔT is ±1.0 °C because two readings are used. The percentage uncertainty in mass is (0.1 ÷ 25.0) × 100% = 0.40%. The percentage uncertainty in ΔT is (1.0 ÷ 12.5) × 100% = 8.0%.
ΔT的绝对不确定度为±1.0 °C，因为使用了两次读数。质量的百分不确定度为 (0.1 ÷ 25.0) × 100% = 0.40%。ΔT的百分不确定度为 (1.0 ÷ 12.5) × 100% = 8.0%。
Since q = mcΔT, and c is taken as a constant, the percentage uncertainty in q is the sum of the percentage uncertainties in m and ΔT. Therefore %Δq = 0.40% + 8.0% = 8.4%.
由于 q = mcΔT，且 c 视为常数，q 的百分不确定度为 m 和 ΔT 百分不确定度之和。因此 %Δq = 0.40% + 8.0% = 8.4%。
| Quantity 物理量 | Value 数值 | Absolute uncertainty 绝对不确定度 | Percentage uncertainty 百分不确定度 |
|---|---|---|---|
| Mass m 质量 | 25.0 g | ±0.1 g | 0.40% |
| Temperature change ΔT 温度变化 | 12.5 °C | ±1.0 °C | 8.0% |
| Heat energy q 热量 | calculated 计算 | from %Δq 由%Δq求得 | 8.4% |
9. Comparing Results: Percentage Difference | 比较结果：百分差
When comparing an experimental value with an accepted value, use percentage difference or percentage error. This is the absolute difference divided by the accepted value, multiplied by 100%.
当将实验值与公认值进行比较时，使用百分差或百分误差。它是绝对差除以公认值，再乘以100%。
Percentage difference = (|experimental value – accepted value| ÷ accepted value) × 100%
百分差 = (|实验值 – 公认值| ÷ 公认值) × 100%
If the percentage difference is greater than the estimated percentage uncertainty, the result is likely to be inaccurate, indicating a systematic error. If the accepted value lies within the experimental uncertainty range, the result is consistent with the accepted value.
如果百分差大于估算的百分不确定度，结果很可能不准确，表明存在系统误差。如果公认值位于实验不确定度范围内，则结果与公认值一致。
For example, an experimental enthalpy change of -48 kJ mol⁻¹ with ±5 kJ mol⁻¹ uncertainty is consistent with an accepted value of -50 kJ mol⁻¹, because the interval -53 to -43 kJ mol⁻¹ contains the accepted value.
例如，实验焓变为-48 kJ mol
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