📚 Accuracy, Precision and Uncertainty in Chemical Measurements | 化学测量中的不确定度与误差分析
In every quantitative chemistry experiment, a balance, burette, pipette or thermometer provides a numerical result. Yet no measurement is infinitely exact. Understanding the limits of each reading is a core skill in IB Chemistry, because a result without an uncertainty is scientifically incomplete.
在每一个定量化学实验中，天平、滴定管、移液管或温度计都会给出一个数值结果。然而没有任何测量是无限精确的。理解每次读数的极限是 IB 化学的核心技能，因为一个不带不确定度的结果在科学上是不完整的。
1. Why Uncertainty Matters | 为什么不确定度很重要？
Uncertainty defines the range within which the true value is expected to lie. When you write 25.00 cm³ from a burette, the actual volume may be 25.00 ± 0.05 cm³. Ignoring this range can lead to faulty conclusions, especially when comparing experimental values with literature values or theoretical yields.
不确定度界定了真实值可能落在其中的范围。当你在滴定管上读到 25.00 cm³ 时，实际体积可能是 25.00 ± 0.05 cm³。忽略这个范围可能导致错误结论，尤其是在将实验值与文献值或理论产率进行比较时。
IB Chemistry assessments award marks for recording the uncertainty of instruments, propagating uncertainties in calculations, and evaluating the impact of procedural errors on final results. Therefore, mastering this topic directly improves both practical work and examination performance.
IB 化学评估要求记录仪器的不确定度、在计算中传递不确定度，并评估操作误差对最终结果的影响。因此，掌握这一主题既能直接提升实验操作水平，也能提高考试成绩。
2. Precision vs Accuracy | 精度与准确度
Precision refers to how closely repeated measurements agree with each other. Accuracy refers to how closely a measured value agrees with the accepted or true value. A thermometer that always reads 37.5 °C when the true temperature is 37.0 °C is precise but inaccurate.
精度是指多次重复测量结果之间彼此接近的程度。准确度是指测量值与公认值或真实值接近的程度。一支总是显示 37.5 °C 而真实温度为 37.0 °C 的温度计，虽然精度高但不准确。
For example, a student titrates a standard sodium hydroxide solution three times and obtains volumes of 20.10, 20.05 and 20.08 cm³. These results are precise because the spread is small. If the expected volume based on a stoichiometric calculation is 19.20 cm³, then the results are precise but not accurate, indicating a systematic error such as a wrongly labelled concentration.
例如，一位学生用标准氢氧化钠溶液滴定三次，得到体积为 20.10、20.05 和 20.08 cm³。因为数值非常接近，这些结果精度高。如果根据化学计量计算预期体积为 19.20 cm³，那么这些结果精度虽高但准确度差，表明存在系统误差，例如标准溶液浓度标签错误。
| Precision 精度 | Accuracy 准确度 | |
| Related error type 相关误差类型 | Random errors 随机误差 | Systematic errors 系统误差 |
| Effect of repeating 重复测量的影响 | Improves 可改善 | No effect 无影响 |
| Detected by 检测方法 | Looking at spread 观察离散程度 | Comparing to true value 与真实值比较 |
3. Systematic and Random Errors | 系统误差与随机误差
Random errors cause readings to fluctuate unpredictably above and below the true value. They arise from personal judgement, slight temperature changes, or electrical noise in a sensor. Repeated measurements and averaging reduce their effect on the mean.
随机误差使读数在真实值上下不可预测地波动。它们源于个人判断、温度微变或传感器中的电噪声。重复测量并取平均值可降低它们对均值的影响。
Systematic errors push all readings consistently in one direction. Examples include an incorrectly calibrated balance, a burette that leaks slowly, or heat loss from an uninsulated calorimeter. Systematic errors cannot be discovered by repeating the experiment; they require recalibration or a genuinely different method.
系统误差使所有读数一致地偏向某一方向。例如天平校准错误、滴定管缓慢漏液、或热量计未保温导致的热损失。系统误差无法通过重复实验被发现；它们需要重新校准或改用真正不同的方法。
In practice, a common systematic error is the “end-point overshoot” in a titration, where the student adds excess titrant every time. The titres are precise but every value is slightly too large. Identifying that the endpoint colour should be pale pink, not deep pink, can eliminate this problem.
在实操中，一个常见的系统误差是滴定时的“终点过滴”，即每次都多加了滴定剂。滴定体积虽然精确，但每次都偏大。认识到终点应为浅粉色而非深粉色，可以消除这一问题。
4. Absolute, Relative and Percentage Uncertainty | 绝对、相对和百分比不确定度
The absolute uncertainty is the actual uncertainty in a reading, written with the same unit as the measurement, for example 25.00 ± 0.05 cm³. The relative uncertainty is the absolute uncertainty divided by the measured value, and the percentage uncertainty is that ratio multiplied by 100.
绝对不确定度是读数中的实际不确定度，单位与测量值一致，例如 25.00 ± 0.05 cm³。相对不确定度是绝对不确定度除以测量值，百分比不确定度是这个比值乘以 100。
Percentage uncertainty = (Δx / x) × 100%
As the measured quantity increases for a fixed absolute uncertainty, the percentage uncertainty decreases. Weighing 1.00 g of solid on a balance with ±0.01 g uncertainty gives a 1% error, while weighing 10.00 g gives only a 0.1% error. Therefore, chemists prefer larger masses and volumes when possible.
当绝对不确定度固定时，随着测量量增大，百分比不确定度会减小。在不确定度为 ±0.01 g 的天平上称量 1.00 g 固体，误差为 1%；而称量 10.00 g 时误差仅为 0.1%。因此，化学家会尽可能选择较大的质量和体积。
5. Combining Uncertainties | 不确定度的传递
When results are used in calculations, uncertainties must be propagated. A simplified approach is used in IB Chemistry:
当结果用于计算时，不确定度必须被传递。IB 化学采用一套简化方法：
For addition or subtraction, absolute uncertainties add:
对于加减法，绝对不确定度相加：
ΔZ = ΔA + ΔB
For multiplication or division, percentage uncertainties add:
对于乘除法，百分比不确定度相加：
(ΔZ/Z) × 100% = (ΔA/A) × 100% + (ΔB/B) × 100%
For powers such as Z = Aⁿ, the percentage uncertainty is multiplied by n:
对于幂运算如 Z = Aⁿ，百分比不确定度乘以 n：
% uncertainty in Z = n × (% uncertainty in A)
Consider a titration where 25.00 ± 0.05 cm³ of NaOH reacts with 20.00 ± 0.06 cm³ of HCl. The total volume uncertainty when adding them is 0.11 cm³, because absolute uncertainties add. If calculating concentration using volumes in a ratio, you must add percentage uncertainties instead.
考虑一个滴定实验：25.00 ± 0.05 cm³ NaOH 与 20.00 ± 0.06 cm³ HCl 反应。将两者体积相加时，总体积不确定度为 0.11 cm³，因为绝对不确定度相加。如果使用体积比计算浓度，则必须使用百分比不确定度相加。
6. Significant Figures and Uncertainty Reporting | 有效数字与不确定度表达
The number of significant figures in a result should reflect its uncertainty. A value such as 0.2845 ± 0.05 g is misleading, because the uncertainty affects the second decimal place, so the result should be reported as 0.28 ± 0.05 g or 0.284 ± 0.005 g depending on the actual precision.
结果的有效数字位数应反映其不确定度。像 0.2845 ± 0.05 g 这样的表达具有误导性，因为不确定度影响了第二位小数，因此应写成 0.28 ± 0.05 g 或 0.284 ± 0.005 g，具体取决于实际精度。
For a balance reading to 0.01 g, a measured mass of 5.60 g should not be recorded simply as 5.6 g. The trailing zero indicates that the uncertainty is ±0.01 g, which is important information. During calculations, keep one extra digit until the final answer, then round to the appropriate number of significant figures.
对于精确到 0.01 g 的天平，测量质量 5.60 g 不应简单记录为 5.6 g。末尾的零表示不确定度为 ±0.01 g，这是重要信息。在计算过程中保留一位额外数字，直到最终答案再四舍五入到适当的有效数字。
When reporting a final value, the uncertainty should have at most two significant figures, and the measured value should be reported to the same decimal place as its uncertainty. For example, a titre of 20.10 ± 0.05 cm³ is correct, while 20.100 ± 0.1 cm³ is not.
报告最终结果时，不确定度最多保留两位有效数字，测量值的小数位数应与不确定度一致。例如，20.10 ± 0.05 cm³ 是正确的，而 20.100 ± 0.1 cm³ 是不正确的。
7. Repeats, Averages and Outliers | 重复测量、平均值和离群值
Repeating a measurement and calculating a mean reduces the effect of random errors. However, repeating a measurement never removes systematic errors. For titrations, IB students are expected to repeat until concordant titres are obtained, usually within ±0.10 cm³ of each other.
重复测量并计算平均值可以降低随机误差的影响。然而，重复测量永远不能消除系统误差。对于滴定，IB 学生应重复操作直到获得平行滴定数据，通常彼此相差在 ±0.10 cm³ 以内。
An outlier is a measurement that clearly does not fit the pattern of the other results. It should be examined rather than blindly discarded. A known procedural mistake, such as overshooting the endpoint once, is a valid reason to exclude the result. The decision to exclude an outlier should be reported in the evaluation section of the practical work.
离群值是一个明显不符合其他结果模式的测量值。应当对其进行核查，而不是盲目舍弃。已知的操作失误，例如一次过滴，是排除该结果的合理理由。排除离群值的决定应在实验评估部分中报告。
For example, titres of 20.10, 20.05, 20.08 and 21.45 cm³ indicate that 21.45 cm³ is likely an outlier. Discarding it and averaging the remaining values gives 20.08 cm³. But if no error was noticed, the student should discuss why the value may have been anomalous, for example due to a parallax error.
例如，滴定体积为 20.10、20.05、20.08 和 21.45 cm³，表明 21.45 cm³ 很可能是离群值。舍弃后求平均值得到 20.08 cm³。但如果当时没有发现错误，学生应讨论该值异常的可能原因，例如视差误差。
8. Graphical Analysis and Error Bars | 图形分析与误差棒
Graphical analysis often produces more reliable results than reading isolated measurements. When plotting a calibration curve or a rate-concentration graph, each point should include an error bar representing its uncertainty. The error bar can be vertical, horizontal, or both, depending on which variable has significant uncertainty.
图形分析通常比读取孤立的测量值更可靠。在绘制校准曲线或速率-浓度图时，每个点都应包含代表其不确定度的误差棒。误差棒可以是垂直、水平或两者兼有，取决于哪个变量的不确定度较大。
The best-fit line should pass through all error bars if the measurements are consistent. If several error bars do not intersect the line, the data may contain a systematic error, or the uncertainty may have been underestimated. The uncertainty in the gradient can be estimated by drawing the steepest and shallowest plausible lines through the error bars.
如果测量值一致，最佳拟合线应穿过所有误差棒。如果多条误差棒与直线不相交，则数据可能包含系统误差，或者不确定度被低估了。梯度的不确定度可以通过画穿过误差棒的最陡和最平缓的合理直线来估算。
In a first-order rate experiment, a plot of ln[A] against time should be linear. If the y-intercept and slope are used to calculate the rate constant k, the uncertainty in k depends on the uncertainty in both the slope and the intercept. Using a large number of points reduces this uncertainty.
在一级速率实验中，ln[A] 对时间的图应为直线。如果使用截距和斜率计算速率常数 k，k 的不确定度取决于斜率和截距两者的不确定度。使用大量数据点可以减小这一不确定度。
9. Minimising Errors in Practical Work | 实验中的误差最小化
Choosing a suitable measuring instrument is the first step towards high-quality data. A 50 cm³ burette with a calibration uncertainty of ±0.05 cm³ is preferred for accurate titrations, while a 10 cm³ measuring cylinder with ±0.1 cm³ uncertainty is only suitable when approximate volumes are needed. A more precise instrument is not always appropriate: using a 50 cm³ burette to measure 2 cm³ is wasteful and introduces larger relative uncertainty.
选择合适的测量仪器是获得高质量数据的第一步。校准不确定度为 ±0.05 cm³ 的 50 cm³ 滴定管适合精确滴定，而不确定度为 ±0.1 cm³ 的 10 cm³ 量筒仅适合需要近似体积的场合。更精密的仪器并非总是合适：用 50 cm³ 滴定管量取 2 cm³ 液体不仅浪费，还会引入更大的相对不确定度。
Correct technique reduces procedural errors. Reading liquid volumes at eye level eliminates parallax error. Ensuring that a pipette drains fully and that the burette jet is filled before titration removes trapped-air errors. Using a white tile makes colour changes at the endpoint easier to judge.
正确的操作技巧可以减少操作误差。平视读取液体体积可消除视差误差；确保移液管完全排空、滴定管尖嘴在滴定前充满液体，可消除气泡误差；使用白瓷板可使终点颜色变化更容易判断。
For thermochemistry experiments, heat losses to the surroundings can be minimised with insulation and lids on the calorimeter. For gravimetric analysis, allowing the precipitate to cool in a desiccator prevents absorption of moisture. In all experiments, checking the calibration of instruments with a standard reference before beginning is a valuable habit.
在热化学实验中，可以通过在热量计上加保温层和盖子来减少向周围的热损失。在重量分析中，让沉淀在干燥器中冷却可防止吸收水分。在所有实验中，在开始前用标准参照物检查仪器校准是一个有价值的习惯。
10. Conclusion | 结论
Uncertainty and error analysis is not a formality in chemistry; it is the language scientists use to describe the reliability of their data. Distinguishing precision from accuracy, propagating uncertainties correctly, and evaluating experimental procedures critically are abilities that IB Chemistry continuously assesses. A thoughtful presentation of uncertainties turns a collection of readings into a convincing scientific conclusion.
不确定度与误差分析在化学中不是一种形式，而是科学家用来描述数据可靠性的语言。区分精度与准确度、正确传递不确定度、批判性评估实验流程，正是 IB 化学持续考查的能力。对不确定度的严谨表达，能把一组读数转化为令人信服的科学结论。
When planning experiments, students should consider which step contributes the largest uncertainty and then decide whether improving that step is possible. Practical work becomes much more meaningful when results are understood not as single points but as ranges containing the truth, with the width of each range telling us how much confidence we may place in the data.
在规划实验时，学生应思考哪一步对总不确定度的贡献最大，然后判断该步骤是否可以被改进。当我们不以单个点来理解结果，而是将其视为包含真实值的区间，并且每个区间的宽度告诉我们对该数据可以抱有多大信心时，实验工作就会变得更有意义。
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