Percentage Uncertainty | 百分比不确定度

📚 Percentage Uncertainty | 百分比不确定度

When you measure a physical quantity, no measurement is perfectly exact. The uncertainty tells you the range within which the true value is likely to lie. Percentage uncertainty expresses this uncertainty as a fraction of the measured value, making it easier to compare measurements of different sizes and to combine uncertainties in calculations.

当你测量一个物理量时，没有任何测量是绝对精确的。不确定度告诉你真值可能所在的范围。百分比不确定度将这个不确定度表示为测量值的分数，便于比较不同大小的测量值，并在计算中合成不确定度。

1. Understanding Percentage Uncertainty | 理解百分比不确定度

Percentage uncertainty is a way of stating how large an uncertainty is compared with the measurement itself. It is calculated by dividing the absolute uncertainty by the measured value and multiplying by 100%. A small percentage uncertainty means the measurement is relatively precise; a large percentage uncertainty means the measurement is relatively imprecise.

百分比不确定度是一种表示不确定度相对于测量值本身有多大的方法。用绝对不确定度除以测量值再乘以100%计算。百分比不确定度小，说明测量相对精密；百分比不确定度大，说明测量相对不精密。

%U = (Δx / x) × 100%

In this formula, Δx is the absolute uncertainty and x is the measured value. For example, if a length is measured as 20.0 cm with an absolute uncertainty of 0.2 cm, the percentage uncertainty is 1%.

在这个公式中，Δx 是绝对不确定度，x 是测量值。例如，测得长度为 20.0 cm，绝对不确定度为 0.2 cm，则百分比不确定度为 1%。

2. Absolute Uncertainty vs Percentage Uncertainty | 绝对不确定度与百分比不确定度

Absolute uncertainty has the same unit as the quantity. For example, a length may be written as (25.0 ± 0.2) mm. The absolute uncertainty is 0.2 mm. Percentage uncertainty has no unit; it is a ratio expressed as a percentage. The same length has a percentage uncertainty of (0.2 / 25.0) × 100% = 0.8%.

绝对不确定度与物理量具有相同的单位。例如长度可写成 (25.0 ± 0.2) mm，绝对不确定度是 0.2 mm。百分比不确定度没有单位，它是表示为百分比的比值。同一长度的百分比不确定度为 (0.2 / 25.0) × 100% = 0.8%。

It is important not to confuse the two. In calculations, absolute uncertainty is used for addition and subtraction, while percentage uncertainty is used for multiplication, division, and powers.

重要的是不要混淆两者。在计算中，加减法使用绝对不确定度，而乘除法、幂运算使用百分比不确定度。

3. Instrument Precision and Reading Uncertainty | 仪器精密度与读数不确定度

For a single reading taken from a digital instrument, the minimum absolute uncertainty is usually ± the smallest scale division or the manufacturer’s stated precision. If the reading fluctuates, the uncertainty may be larger. For analogue instruments, the reading uncertainty is often taken as ± half the smallest division, but CIE questions usually give the uncertainty unless you are asked to estimate it.

对于数字仪器读取的单次读数，最小绝对不确定度通常为 ± 最小刻度或制造商标明的精确度。如果读数波动，不确定度可能更大。对于模拟仪器，读数不确定度常取 ± 最小分度的一半，但 CIE 题目通常会给出不确定度，除非要求你估算。

Instrument	Typical precision	Example reading uncertainty
Digital voltmeter	±0.01 V	1.23 V ± 0.01 V
Metre rule	±1 mm	500 ± 1 mm
Micrometer screw gauge	±0.01 mm	3.45 ± 0.01 mm

The percentage uncertainty depends on both the instrument precision and the size of the reading. A small reading on a coarse instrument gives a large percentage uncertainty.

百分比不确定度取决于仪器精密度和读数大小。在粗糙仪器上读取小数值会产生较大的百分比不确定度。

4. Calculating Percentage Uncertainty for a Single Measurement | 计算单次测量的百分比不确定度

To find the percentage uncertainty, write down the absolute uncertainty Δx and the measured value x, then calculate (Δx / x) × 100%. For example, if a current is measured as I = 0.45 A ± 0.01 A, the percentage uncertainty is (0.01 / 0.45) × 100% = 2.2%. The result is usually quoted to two significant figures for uncertainties.

求百分比不确定度时，先写下绝对不确定度 Δx 和测量值 x，然后计算 (Δx / x) × 100%。例如测得电流 I = 0.45 A ± 0.01 A，百分比不确定度为 (0.01 / 0.45) × 100% = 2.2%。不确定度通常保留两位有效数字。

Make sure you use the measured value, not the true value or the mean of a different set of data. If you have repeated measurements, first calculate the best estimate of the quantity and the relevant absolute uncertainty before converting to a percentage.

确保使用测量值，而不是真值或另一组数据的平均值。如果你有重复测量数据，先计算该物理量的最佳估计值和相应的绝对不确定度，再转换为百分比。

5. Combining Uncertainties: Addition and Subtraction | 合成不确定度：加法和减法

When quantities are added or subtracted, the absolute uncertainties add. If R = A + B or R = A – B, then ΔR = ΔA + ΔB. This gives the worst-case uncertainty because the true errors could act in the same direction.

当物理量相加或相减时，绝对不确定度相加。若 R = A + B 或 R = A – B，则 ΔR = ΔA + ΔB。这给出最不利情况下的不确定度，因为真实误差可能朝同一方向作用。

ΔR = ΔA + ΔB

For example, if A = 6.0 ± 0.2 and B = 4.0 ± 0.1, then R = A – B = 2.0 with ΔR = 0.2 + 0.1 = 0.3. The result is R = 2.0 ± 0.3. Do not add percentage uncertainties in addition or subtraction.

例如，若 A = 6.0 ± 0.2，B = 4.0 ± 0.1，则 R = A – B = 2.0，ΔR = 0.2 + 0.1 = 0.3。结果为 R = 2.0 ± 0.3。在加减法中不要相加百分比不确定度。

6. Combining Uncertainties: Multiplication and Division | 合成不确定度：乘法和除法

When quantities are multiplied or divided, the percentage uncertainties add. If R = A × B or R = A / B, then %U_R = %U_A + %U_B. This rule is central to CIE practical and theory questions.

当物理量相乘或相除时，百分比不确定度相加。若 R = A × B 或 R = A / B，则 %U_R = %U_A + %U_B。这一规则是 CIE 实验题和理论题的核心。

%U_R = %U_A + %U_B

For example, a voltage V = 5.0 V has a 2% uncertainty and a current I = 0.50 A has a 4% uncertainty. The resistance R = V / I has a percentage uncertainty of 2% + 4% = 6%. Convert to an absolute uncertainty only if the question requires it.

例如，电压 V = 5.0 V 的不确定度为 2%，电流 I = 0.50 A 的不确定度为 4%。电阻 R = V / I 的百分比不确定度为 2% + 4% = 6%。仅在题目要求时才转换为绝对不确定度。

7. Combining Uncertainties: Powers and Roots | 合成不确定度：幂和方根

For a quantity raised to a power, the percentage uncertainty is multiplied by the magnitude of the power. If R = Aⁿ, then %U_R = |n| × %U_A. For example, R = A² gives %U_R = 2 × %U_A. The same rule applies to roots, so if R = √A, the percentage uncertainty is half the percentage uncertainty in A.

对于物理量的幂次，百分比不确定度乘以幂的绝对值。若 R = Aⁿ，则 %U_R = |n| × %U_A。例如 R = A² 时 %U_R = 2 × %U_A。同样规则适用于方根，因此若 R = √A，其百分比不确定度为 A 的百分比不确定度的一半。

%U_R = n × %U_A

If a sphere has a measured radius r = 2.0 cm ± 2%, its volume V = (4/3)πr³ has a percentage uncertainty of 3 × 2% = 6%. The constant (4/3)π does not contribute to the uncertainty.

如果球的半径测量值为 r = 2.0 cm ± 2%，其体积 V = (4/3)πr³ 的百分比不确定度为 3 × 2% = 6%。常数 (4/3)π 不贡献不确定度。

8. Repeated Readings and Half-Range Uncertainty | 重复读数与半范围不确定度

Repeating a measurement allows you to estimate random uncertainty. A simple method used in CIE practical work is the half-range: Δx = (x_max – x_min) / 2. The mean value is then quoted as mean ± half-range. More readings reduce the effect of random errors but do not eliminate systematic errors.

重复测量可以估计随机不确定度。CIE 实验中常用的简单方法是半范围：Δx = (x_max – x_min) / 2。然后平均值表示为平均值 ± 半范围。更多读数能减小随机误差的影响，但不能消除系统误差。

Δx = (x_max – x_min) / 2

For example, five timings are 1.22 s, 1.24 s, 1.26 s, 1.23 s and 1.25 s. The mean is 1.24 s. The half-range is (1.26 – 1.22) / 2 = 0.02 s, so the result is 1.24 ± 0.02 s. The percentage uncertainty is (0.02 / 1.24) × 100% = 1.6%.

例如五次计时为 1.22 s、1.24 s、1.26 s、1.23 s 和 1.25 s。平均值为 1.24 s。半范围为 (1.26 – 1.22) / 2 = 0.02 s，因此结果为 1.24 ± 0.02 s。百分比不确定度为 (0.02 / 1.24) × 100% = 1.6%。

9. Percentage Difference vs Percentage Uncertainty | 百分差与百分比不确定度

Percentage difference compares two values, often an experimental result and an accepted value. It is calculated as |(experimental – accepted) / accepted| × 100%. Percentage uncertainty describes the precision of a single measurement or calculated result. A small percentage difference may still occur within a large percentage uncertainty.

百分差用于比较两个值，通常是实验结果与公认值。计算公式为 |(实验值 – 公认值) / 公认值| × 100%。百分比不确定度描述单次测量或计算结果的精密度。百分差很小也可能仍在较大的百分比不确定度范围内。

% difference = |(x – x_acc) / x_acc| × 100%

If an experimental value for g is 9.6 m s⁻² and the accepted value is 9.81 m s⁻², the percentage difference is |(9.6 – 9.81) / 9.81| × 100% = 2.1%. This is not the same as the percentage uncertainty in the measurement.

如果重力加速度的实验值为 9.6 m s⁻²，公认值为 9.81 m s⁻²，则百分差为 |(9.6 – 9.81) / 9.81| × 100% = 2.1%。这与测量的百分比不确定度不是一回事。

10. Uncertainties from Graphs: Best and Worst Lines | 图线中的不确定度：最佳拟合线和最差拟合线

In CIE practical papers, you may need to determine the uncertainty in a gradient or intercept. Draw the line of best fit, then draw the worst acceptable line, which is the steepest or shallowest reasonable line through the error bars. The uncertainty in gradient is often half the difference between the maximum and minimum gradients

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Percentage Uncertainty | 百分比不确定度