📚 A-Level Biology: Estimating Measurement Uncertainty | A-Level 生物：测量不确定度的估算方法
In A-Level Biology, practical work is assessed not only by the quality of your results but also by how honestly you evaluate their reliability. Every measurement you take, whether with a ruler, a balance, a pipette, or a colorimeter, carries some degree of uncertainty. Understanding how to estimate and express that uncertainty is a core skill in CIE practical examinations.
在 A-Level 生物考试中，实验操作不仅考查你实验结果的质量，还考查你是否诚实地评估了结果的可靠性。无论是用尺子、天平、移液管还是比色计，你所做的每一次测量都带有一定程度的不确定性。理解如何估算并表达这种不确定度，是 CIE 实验考试的一项核心技能。
1. What Is Measurement Uncertainty? | 什么是测量不确定度？
Measurement uncertainty is the range within which the true value of a measured quantity is expected to lie. It is not a mistake or an error in the sense of a blunder; rather, it is an inherent feature of all measuring instruments and human observation. For example, if you measure the length of a leaf as 5.2 cm with a ruler marked in millimetres, the actual length could be anywhere from 5.15 cm to 5.25 cm.
测量不确定度是指被测量的真实值预期所在的区间。它并不是错误或失误，而是所有测量仪器和人类观察所固有的特性。例如，如果你用毫米刻度的尺子测量一片叶子的长度为 5.2 cm，其真实长度可能在 5.15 cm 到 5.25 cm 之间。
The uncertainty of a measurement is usually expressed with a ± sign. In the example above, the reading would be reported as 5.2 ± 0.05 cm. The value 0.05 cm is called the absolute uncertainty. It gives you a sense of how precise the measurement is: the smaller the absolute uncertainty relative to the measurement, the more precise the result.
测量不确定度通常用 ± 符号表示。在上面的例子中，读数应记为 5.2 ± 0.05 cm。其中 0.05 cm 称为绝对不确定度。它让你感受到测量的精密程度：绝对不确定度相对于测量值越小，结果就越精密。
2. Types of Uncertainty | 不确定度的类型
There are two main categories of uncertainty in biological measurements: systematic and random. Systematic uncertainty shifts all measurements in one direction, often caused by a poorly calibrated instrument or a consistent observational bias. For example, a balance that reads 0.02 g too high gives every mass measurement an identical positive error. This type of uncertainty cannot be reduced by repeating measurements.
生物测量中的不确定度主要分为两大类：系统不确定度和随机不确定度。系统不确定度会使所有测量值朝同一方向偏移，通常由仪器校准不当或观察者一致的偏见引起。例如，一台读数偏高 0.02 g 的天平会使每次质量测量都有相同的正误差。这类不确定度无法通过重复测量来减小。
Random uncertainty varies unpredictably from one reading to the next. It may arise from parallax when reading a meniscus, small fluctuations in temperature, or slight differences in how you cut a piece of tissue. Random uncertainty can be reduced by taking repeated readings and calculating a mean, which is why CIE biology practicals often require you to repeat measurements at least three times.
随机不确定度在每次读数之间不可预测地变化。它可能来自读取弯月面时的视差、温度的微小波动，或切割组织块时手法的细微差异。随机不确定度可以通过重复读数并计算平均值来减小，这就是为什么 CIE 生物实验通常要求你至少重复测量三次的原因。
3. Absolute vs Percentage Uncertainty | 绝对不确定度与百分比不确定度
Absolute uncertainty has the same units as the measurement itself. For instance, a thermometer with graduations of 1 °C has an absolute uncertainty of ±0.5 °C if you can read to half a division. Percentage uncertainty is calculated by dividing the absolute uncertainty by the measured value and multiplying by 100. It allows you to compare the precision of different measurements regardless of their units or magnitudes.
绝对不确定度与测量值本身具有相同的单位。例如，分度为 1 °C 的温度计，如果读数能读到半格，其绝对不确定度为 ±0.5 °C。百分比不确定度通过将绝对不确定度除以测量值再乘以 100 来计算。它使你能够比较不同测量之间的精密度，而不受单位或数值大小的影响。
Consider measuring 10 cm³ of liquid with a 10 cm³ measuring cylinder. If the cylinder has divisions of 0.2 cm³, the absolute uncertainty is ±0.1 cm³, giving a percentage uncertainty of (0.1 ÷ 10) × 100 = 1%. Now imagine you measure only 2 cm³ with the same cylinder. The percentage uncertainty becomes (0.1 ÷ 2) × 100 = 5%. This demonstrates why you should always choose an instrument whose capacity closely matches the volume you need to measure.
考虑用 10 cm³ 的量筒量取 10 cm³ 液体。如果量筒的分度为 0.2 cm³，绝对不确定度为 ±0.1 cm³，则百分比不确定度为 (0.1 ÷ 10) × 100 = 1%。现在想象用同一个量筒只量取 2 cm³ 液体，百分比不确定度变为 (0.1 ÷ 2) × 100 = 5%。这说明为什么你应始终选择量程与所需测量体积相匹配的仪器。
percentage uncertainty = (absolute uncertainty ÷ measured value) × 100%
百分比不确定度 = (绝对不确定度 ÷ 测量值) × 100%
4. Reading Uncertainty for Common Instruments | 常见仪器的读数不确定度
In CIE biology practicals, you will encounter several standard instruments. For a ruler with millimetre markings, the uncertainty is usually ±0.5 mm if you can estimate between divisions, but CIE mark schemes often accept ±1 mm. For a thermometer with 1 °C graduations, the uncertainty is ±0.5 °C. For a 10 cm³ measuring cylinder with 0.2 cm³ divisions, the uncertainty is ±0.1 cm³. A balance reading to two decimal places has an uncertainty of ±0.005 g, though ±0.01 g is often accepted in exams.
在 CIE 生物实验中，你会遇到几种标准仪器。对于毫米刻度的尺子，如果你能在刻度之间估读，不确定度通常为 ±0.5 mm，但 CIE 评分标准通常也接受 ±1 mm。对于分度为 1 °C 的温度计，不确定度为 ±0.5 °C。对于分度为 0.2 cm³ 的 10 cm³ 量筒，不确定度为 ±0.1 cm³。读数精确到小数点后两位的天平，其不确定度为 ±0.005 g，但考试中通常也接受 ±0.01 g。
It is essential to record the uncertainty of each instrument you use in your planning and analysis. In CIE Paper 3 and Paper 5, candidates who state the uncertainty and explain its effect on the final result are rewarded. A helpful table is shown below for quick revision.
在规划和数据分析中，记录你所使用的每种仪器的不确定度至关重要。在 CIE 试卷 3 和试卷 5 中，写出不确定度并解释其对最终结果影响的考生会获得分数。下表有助于快速复习。
| Instrument | Graduation | Typical absolute uncertainty |
| Ruler / 尺子 | 1 mm | ±0.5 mm or ±1 mm |
| Thermometer / 温度计 | 1 °C | ±0.5 °C |
| 10 cm³ measuring cylinder / 10 cm³ 量筒 | 0.2 cm³ | ±0.1 cm³ |
| 25 cm³ pipette / 25 cm³ 移液管 | single mark | ±0.06 cm³ (usually ±0.1 cm³ in A-Level) |
| Top-pan balance / 托盘天平 | 0.01 g | ±0.005 g or ±0.01 g |
| Colorimeter / 比色计 | 0.01 absorbance | ±0.005 absorbance |
5. Combining Uncertainties in Addition and Subtraction | 加减运算中不确定度的合成
When you add or subtract measurements, the absolute uncertainties are added together. For example, if you measure the initial length of a root as 30.0 ± 0.5 mm and the final length as 42.0 ± 0.5 mm, the growth is 12.0 mm. The combined absolute uncertainty is 0.5 + 0.5 = 1.0 mm. The growth should therefore be reported as 12.0 ± 1.0 mm.
当你对测量值进行加法或减法运算时，绝对不确定度应相加。例如，如果你测得根系的初始长度为 30.0 ± 0.5 mm，最终长度为 42.0 ± 0.5 mm，则生长量为 12.0 mm。合成的绝对不确定度为 0.5 + 0.5 = 1.0 mm。因此，生长量应报告为 12.0 ± 1.0 mm。
This rule is particularly important when you calculate the change in mass, the change in length, or the decrease in colour intensity. Many students forget that subtracting two readings does not cancel out the uncertainties; instead, it makes the percentage uncertainty of the difference much larger than that of the individual readings.
这条规则在计算质量变化、长度变化或颜色强度下降时尤为重要。许多学生忘记，两个读数相减并不会抵消不确定度；相反，它会使差值的百分比不确定度远大于单个读数的百分比不确定度。
6. Combining Uncertainties in Multiplication and Division | 乘除运算中不确定度的合成
For multiplication and division, you must add the percentage uncertainties, not the absolute uncertainties. Suppose you measure the mass of a potato chip as 5.00 ± 0.01 g and its volume as 4.00 ± 0.20 cm³. The density is 5.00 ÷ 4.00 = 1.25 g cm⁻³. The percentage uncertainty in mass is (0.01 ÷ 5.00) × 100 = 0.2%. The percentage uncertainty in volume is (0.20 ÷ 4.00) × 100 = 5.0%. The total percentage uncertainty is 5.2%, so the absolute uncertainty in density is (5.2 ÷ 100) × 1.25 = 0.065 g cm⁻³. The density is therefore 1.25 ± 0.07 g cm⁻³.
对于乘法和除法，你必须将百分比不确定度相加，而不是绝对不确定度。假设你测得一块土豆片的质量为 5.00 ± 0.01 g，体积为 4.00 ± 0.20 cm³。密度为 5.00 ÷ 4.00 = 1.25 g cm⁻³。质量的百分比不确定度为 (0.01 ÷ 5.00) × 100 = 0.2%。体积的百分比不确定度为 (0.20 ÷ 4.00) × 100 = 5.0%。总百分比不确定度为 5.2%，因此密度的绝对不确定度为 (5.2 ÷ 100) × 1.25 = 0.065 g cm⁻³。所以密度为 1.25 ± 0.07 g cm⁻³。
Notice that the largest source of uncertainty dominates the final result. Reducing the smaller uncertainty would have little effect on the total. In your practical exam, you should identify which measurement contributes most to the total uncertainty and suggest why, or propose an improvement such as using a larger volume or a more precise instrument.
注意，最大的不确定度来源主导了最终结果。减小较小的一项对总和几乎无影响。在实验考试中，你应识别哪个测量对总不确定度贡献最大，并解释原因，或提出改进建议，例如使用更大的体积或更精密的仪器。
7. Uncertainty in Serial Dilutions | 连续稀释中的不确定度
Serial dilutions are common in biology practicals, such as when preparing a glucose calibration curve or a bacterial dilution series. Each dilution step introduces its own uncertainty, and these uncertainties accumulate. For a dilution where 1 cm³ of solution is added to 9 cm³ of distilled water, both volumes contribute to the uncertainty of the dilution factor.
连续稀释在生物实验中非常常见，例如制备葡萄糖标准曲线或细菌稀释系列。每一步稀释都会引入自身的不确定度，且这些不确定度会累积。对于将 1 cm³ 溶液加入 9 cm³ 蒸馏水的稀释步骤，两个体积都对稀释因子的不确定度有贡献。
If a 1 cm³ pipette has an uncertainty of ±0.02 cm³ and a 10 cm³ measuring cylinder has an uncertainty of ±0.1 cm³, the percentage uncertainties are 2% and 1% respectively. The dilution factor has a total percentage uncertainty of 3%. After three serial dilutions, the total percentage uncertainty is approximately 9%, which may be significant. To minimise this, use a larger initial volume or a burette with finer graduations.
如果 1 cm³ 移液管的不确定度为 ±0.02 cm³，10 cm³ 量筒的不确定度为 ±0.1 cm³，则百分比不确定度分别为 2% 和 1%。稀释因子的总百分比不确定度为 3%。经过三次连续稀释后，总百分比不确定度约为 9%，这可能相当显著。为尽量减少这一影响，可使用较大的初始体积或分度更细的滴定管。
8. Using Range to Estimate Uncertainty | 用极差估算不确定度
When you repeat a measurement several times, the spread of your results provides an estimate of random uncertainty. A simple method is to use half the range. For example, if your three repeats for the time taken for a reaction are 12.0 s, 12.5 s and 13.0 s, the range is 13.0 − 12.0 = 1.0 s. Half the range is 0.5 s. The mean is 12.5 s, so the result is reported as 12.5 ± 0.5 s.
当你重复测量多次时，结果的离散程度可用来估算随机不确定度。一种简单的方法是使用极差的一半。例如，如果某反应时间的三次重复测量为 12.0 s、12.5 s 和 13.0 s，极差为 13.0 − 12.0 = 1.0 s。极差的一半为 0.5 s。平均值为 12.5 s，因此结果应报告为 12.5 ± 0.5 s。
This method is acceptable in CIE examinations and is often faster than calculating standard deviation. However, it is important to remember that the range only uses the two extreme values, so it is sensitive to outliers. If one reading is clearly anomalous, you should investigate possible causes and, if justified, discard it before estimating the uncertainty.
这种方法在 CIE 考试中是可接受的，通常比计算标准差更快。但重要的是要记住，极差只使用两个极端值，因此对异常值敏感。如果某个读数明显异常，你应调查可能的原因，并在合理的情况下将其剔除后再估算不确定度。
9. Uncertainty in Counting and Enzyme Assays | 计数和酶活性测定中的不确定度
Counting colonies on an agar plate or cells in a haemocytometer involves a different kind of uncertainty. The uncertainty here is not from an instrument but from sampling and randomness. For a count of N, a common estimate of the standard uncertainty is √N. For example, if you count 100 colonies, the uncertainty is approximately √100 = 10. This means the true count is likely between 90 and 110.
在琼脂平板上计数菌落或在血细胞计数板上计数细胞时，会涉及不同类型的不确定度。这里的不确定度并非来自仪器，而是来自取样和随机性。对于计数 N，常见的标准不确定度估算为 √N。例如，如果你数到 100 个菌落，不确定度约为 √100 = 10。这意味着真实计数很可能在 90 到 110 之间。
This square-root relationship explains why counting a larger number reduces the percentage uncertainty. If you count only 25 colonies, the percentage uncertainty is (5 ÷ 25) × 100 = 20%. If you count 400 colonies, it is (20 ÷ 400) × 100 = 5%. In dilution plating, you should aim to count plates with between 30 and 300 colonies to keep the uncertainty within reasonable bounds.
这种平方根关系解释了为什么计数数量越大，百分比不确定度越小。如果你只数到 25 个菌落，百分比不确定度为 (5 ÷ 25) × 100 = 20%。如果数到 400 个菌落，则为 (20 ÷ 400) × 100 = 5%。在稀释涂布法中，你应尽量选择菌落数在 30 到 300 之间的平板，以将不确定度控制在合理范围内。
10. Presenting Results with Uncertainty | 带有不确定度的结果表达
In CIE biology, you are expected to present your final result with the appropriate uncertainty. The number of decimal places in the uncertainty should match the number of decimal places in the measurement. For example, 12.5 ± 0.5 s is correct, but 12.5 ± 0.50 s or 12.50 ± 0.5 s is not. The uncertainty should also be rounded to one significant figure unless you have a strong reason not to.
在 CIE 生物考试中，你应当以适当的不确定度表达最终结果。不确定度的小数位数应与测量值的小数位数一致。例如，12.5 ± 0.5 s 是正确的，但 12.5 ± 0.50 s 或 12.50 ± 0.5 s 不正确。除非有充分理由，否则不确定度应四舍五入为一位有效数字。
When drawing graphs, uncertainties can be shown as error bars that extend above and below each data point. In CIE Paper 5, you may be asked to draw error bars and then determine whether a line of best fit passes through them. If the error bars overlap between two treatments, the difference between the treatments may not be significant. This is a key skill that links measurement uncertainty to biological conclusions.
在绘制图表时，不确定度可以用误差线表示，误差线在每个数据点的上下延伸。在 CIE 试卷 5 中，你可能会被要求绘制误差线，并判断最佳拟合线是否穿过这些误差线。如果两个处理之间的误差线重叠，则处理之间的差异可能不显著。这是一项关键技能，将测量不确定度与生物学结论联系起来。
11. Reducing Uncertainty | 减小不确定度
You can reduce measurement uncertainty in several practical ways. First, use an instrument with smaller graduations. For example, use a 25 cm³ burette instead of a 50 cm³ measuring cylinder for accurate dispensing. Second, take repeated readings and calculate the mean, which reduces the effect of random uncertainty. Third, increase the size of the quantity being measured. Measuring 50 cm³ instead of 10 cm³ with the same instrument reduces the percentage uncertainty by a factor of five.
你可以通过几种实际方法来减小测量不确定度。第一，使用分度更小的仪器。例如，使用 25 cm³ 滴定管代替 50 cm³ 量筒进行精确加液。第二，重复读数并计算平均值，这可以减少随机不确定度的影响。第三，增大被测物理量的大小。用同一仪器测量 50 cm³ 而非 10 cm³，百分比不确定度会缩小到原来的五分之一。
It is also important to control the environment. Temperature fluctuations, air currents, or uneven lighting can all increase the variability of measurements. In enzyme experiments, maintain a constant water bath temperature. In photosynthesis experiments, use a consistent light source and distance. These measures do not change the instrument uncertainty, but they reduce the biological variability that contributes to the overall spread of your data.
控制实验环境同样重要。温度波动、气流或光照不均都会增加测量的变异性。在酶实验中，保持恒温水浴温度。在光合作用实验中，使用一致的光源和距离。这些措施不会改变仪器不确定度，但会减少导致数据整体离散的生物学变异性。
12. Common Mistakes in CIE Practical Exams | CIE 实验考试中的常见错误
A frequent mistake is confusing uncertainty with percentage uncertainty. Absolute uncertainty is expressed in the same units as the measurement, while percentage uncertainty is dimensionless. Another common error is adding absolute uncertainties when multiplying or dividing; you must convert to percentage uncertainties first, then add, and finally convert back to absolute values if required.
一个常见错误是混淆绝对不确定度与百分比不确定度。绝对不确定度的单位与测量值相同，而百分比不确定度无量纲。另一个常见错误是在乘法或除法中直接相加绝对不确定度；你必须先转换为百分比不确定度，再相加，最后在需要时再转换回绝对值。
Many candidates also forget to quote the uncertainty at all, or write ‘±0.5’ without specifying the unit. Always include units. Finally, do not overstate precision. If your timer only records to the nearest second, you cannot claim a time of 12.34 s with an uncertainty of ±0.01 s. Your result must be consistent with the instrument’s actual resolution.
许多考生还忘记写出不确定度，或写“±0.5”但不注明单位。始终要包含单位。最后，不要过度声称精密度。如果你的计时器只能记录到秒，你就不能声称时间为 12.34 s，且不确定度为 ±0.01 s。你的结果必须与仪器的实际分辨率一致。
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