Error Analysis: Some Practical Considerations | 误差分析：若干实用考量

📚 Error Analysis: Some Practical Considerations | 误差分析：若干实用考量

Errors are unavoidable in numerical work and measurement. A practical understanding of absolute error, relative error, rounding, truncation, and error propagation is essential for making reliable conclusions in AQA A-level Mathematics.

在数值计算和测量中，误差是不可避免的。对绝对误差、相对误差、舍入误差、截断误差以及误差传播有实用的理解，对于在 AQA A-level 数学中得出可靠结论至关重要。

---

1. What Is Error in Numerical Work? | 数值计算中的误差是什么？

cuses on the size of this error using absolute error, relative error, and percentage error.

在数值计算中，我们很少知道精确的真值 x₀；通常只有近似值 x。带符号误差为 e = x − x₀。误差分析通常利用绝对误差、相对误差和百分误差来关注这个误差的大小。

E_abs = |x − x₀|

E_rel = |x − x₀| ÷ |x₀|, x₀ ≠ 0

E_percent = E_rel × 100%

Absolute error has the same units as the quantity being measured, while relative error is dimensionless. Relative error is therefore more useful when comparing accuracy across very different scales.

绝对误差与被测量具有相同的单位，而相对误差没有量纲。因此，在比较尺度差异很大的量时，相对误差更为有用。

---

2. Significant Figures and Decimal Places | 有效数字与小数位数

ounded to 4 significant figures has an error bound of 0.0005, because the fourth significant figure lies in the thousandths column.

当一个数舍入到 n 位有效数字时，最后保留的数字带有该数位的半个单位不确定度。例如，x = 3.142 舍入到 4 位有效数字，其误差界为 0.0005，因为第四位有效数字位于千分位。

Decimal places are counted from the decimal point, not from the first non-zero digit. Thus 0.00456 to 2 decimal places is 0.00, which is not meaningful for error analysis unless significant figures are also considered.

小数位数是从小数点开始计数的，而不是从第一个非零数字开始计数。因此，0.00456 保留 2 位小数得到 0.00，如果不结合有效数字，这对误差分析没有意义。

error bound = ½ × 10⁻ᵏ for k decimal places

For n significant figures, the error bound is half the value of the last significant digit. This simple interval is very useful when stating measurement precision.

对于 n 位有效数字，误差界是最后一位有效数字所代表数值的一半。在说明测量精度时，这个简单的区间非常有用。

---

3. Rounding and Truncation Errors | 舍入误差与截断误差

Rounding error arises when an exact or longer decimal is replaced by a nearby value with fewer digits, usually using a round-to-nearest rule. Truncation error arises when an infinite process is stopped early, such as using only the first three terms of a Taylor series or stopping an iteration before full convergence.

舍入误差产生于将精确值或更长的小数替换为按四舍五入规则取较少位数得到的近似值。截断误差产生于无限过程提前终止，例如只取泰勒级数的前三项，或在完全收敛之前停止迭代。

A calculator may store more digits than it displays, so displayed values can hide internal precision. In exam work, always state the precision you are using and avoid premature rounding in intermediate steps.

计算器存储的位数可能多于显示位数，因此显示值可能隐藏内部精度。在考试作答时，始终说明所用精度，并避免在中间步骤过早舍入。

---

广告 / Sponsored

4. Error Bounds and Interval Estimates | 误差界与区间估计

expected to lie in the interval [x − δ, x + δ]. The quantity δ is an absolute error bound.

如果测量量报告为 x ± δ，则真值预期位于区间 [x − δ, x + δ] 内。δ 是绝对误差界。

x − δ ≤ x_true ≤ x + δ

When combining measurements, you can find the minimum and maximum possible values of a derived quantity by using the end points of each interval. This gives a rigorous worst-case bound, which is often preferred in practical error analysis.

当组合多个测量量时，可以利用每个区间的端点求出导出量的最小值和最大值。这给出严格的最坏情况误差界，在实际误差分析中通常更受欢迎。

---

5. Propagation of Errors in Sums and Differences | 和与差的误差传播

ndividual absolute errors. If z = x + y or z = x − y, then Δz ≤ Δx + Δy in the worst case.

对于加法或减法，结果的最大绝对误差等于各绝对误差之和。如果 z = x + y 或 z = x − y，则最坏情况下 Δz ≤ Δx + Δy。

Δz ≈ Δx + Δy

This rule is especially important when subtracting nearly equal numbers: a small absolute error in each number can produce a very large relative error in the difference, because the denominator |x − y| is small.

这一规则在两个近乎相等的数相减时尤为重要：每个数的微小绝对误差可能在差值中产生很大的相对误差，因为分母 |x − y| 很小。

---

6. Propagation of Errors in Products and Quotients | 积与商的误差传播

to handle. If z = xy or z = x ÷ y, then to a first approximation the relative errors add.

对于乘法和除法，相对误差更容易处理。如果 z = xy 或 z = x ÷ y，则在一阶近似下相对误差相加。

Δz/|z| ≈ Δx/|x| + Δy/|y|

This is particularly convenient because percentage errors can be added directly. The worst-case maximum relative error is found by adding the individual relative errors without allowing any cancellation.

这特别方便，因为百分误差可以直接相加。最坏情况的最大相对误差是通过将各个相对误差直接相加得到的，不允许任何抵消。

---

7. Errors in Powers and Roots | 幂与方根的误差传播

If z = xⁿ, the relative error in z is approximately |n| times the relative error in x. For example, if a radius has a 2% relative error, the area r² has about a 4% relative error.

如果 z = xⁿ，则 z 的相对误差约为 x 的相对误差的 |n| 倍。例如，若半径的相对误差为 2%，则面积 r² 的相对误差约为 4%。

Δz/|z| ≈ |n| × Δx/|x|

For square roots, n = ½, so the relative error is halved. For inverse

Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）