Uncertainty Calculations

考纲要求

• Convert absolute uncertainty to fractional/percentage and vice versa
• Show uncertainty estimates beside every value in a table
• Calculate uncertainty estimates in derived quantities
• Estimate absolute uncertainty in gradient: |best - worst|
• Estimate absolute uncertainty in y-intercept
• Express a quantity as value, uncertainty and unit

常见题型

• Derived Quantities — ln, lg, reciprocal uncertainty propagation
• Gradient Uncertainty — best and worst fit lines
• Combined Uncertainty — error propagation in final answers

核心公式

• $\Delta(\ln x) = \frac{\Delta x}{x}$
• $\Delta(\lg x) = \frac{\Delta x}{x \ln 10} \approx 0.434\frac{\Delta x}{x}$
• $\Delta(1/x) = \frac{\Delta x}{x^2}$
• $\Delta(1/\sqrt{h}) = \frac{\Delta h}{2h^{3/2}}$
• Percentage: $\frac{\Delta x}{x} \times 100\%$
• Gradient uncertainty: $|\text{best} - \text{worst}|$

常见错误

• 忘记在表格中给每个 derived value 列出 uncertainty
• 梯度误差线画错（worst line 必须通过所有 error bars）
• 不确定度的有效数字位数不匹配