Error and Propagation: Study Notes (Bullet Points)

Key Concepts

• Error terminology
  • Absolute error (Δx): an estimate of the uncertainty in a measured quantity x. Often reported as x ± Δx.
  • Relative error: Δx/x (dimensionless), often expressed as a percentage: (Δx/x)×100%.
  • Percentage error: the relative error expressed as a percent.
  • Mean absolute deviation (MAD) as a method to estimate experimental error from repeated measurements: MAD ≈ average of |vi − mean|.
• Rounding and reporting
  • Report values with an uncertainty that reflects the measurement’s precision (e.g., to the least count or appropriate MAD).
  • For a quantity V measured with a voltmeter of least count 0.01 V, the final value is typically reported as Value ± MAD or ± least count depending on method used.
• Propagation of errors: general guidelines used in the transcript
  • Addition/Subtraction: absolute errors add (for a maximum-case estimate) or propagate via standard rules; in the transcript, absolute errors were added for sums/differences when estimating Δ(z).
  • Multiplication/Division: relative (percentage) errors add.
  • Powers: relative error multiplies by the absolute value of the exponent, when the quantity is raised to a power.
  • Products/Quotients of multiple quantities: sum the weighted relative errors according to exponents.
  • Logarithmic quantities: error in log x is Δ(log x) = Δx / (x ln 10) for base-10 logarithms; i.e., a relative error in x translates to an absolute error in log x scaled by 1/ln 10.
  • Trigonometric quantities: for θ with small Δθ (in radians), Δ(sin θ) ≈ cos θ × Δθ.
  • Derived units: the relative error in a derived unit (product/quotient) is the sum of the relative errors of its components, each scaled by its exponent.
• Why subtraction can inflate relative error
  • When subtracting quantities with similar magnitudes, the net result is small, so the ratio Δabs/|result| can be large, yielding large percentage error even if absolute errors are modest.
• Error in geometry and volume/area calculations
  • For volume V = π r^2 h, the relative error is approximately ΔV/V ≈ 2(Δr/r) + (Δh/h).
  • For area A = L × W, the relative error is often approximated as ΔA/A ≈ (ΔL/L) + (ΔW/W) when treating percent errors additively for products.
• Common problem types in the transcript
  • Absolute/percentage error computation for measurements with given uncertainties.
  • Relative error propagation in algebraic expressions with exponents, products, and quotients.
  • Substitution of errors into more complex expressions, including powers and roots.
  • Handling of logarithmic and trigonometric error propagation.
• Examples of formulas used (key references)
  • Volume: V =
    0A \; [=] \; \pi r^2 h and relative error: $\frac{\Delta V}{V} \approx 2\frac{\Delta r}{r} + \frac{\Delta h}{h}\,.$
  • For a product/quotient with exponents: if $Z = A^p B^q C^r \dots$ then
    $\frac{\Delta Z}{Z} \approx \left|p\right|\frac{\Delta A}{A} + \left|q\right|\frac{\Delta B}{B} + \left|r\right|\frac{\Delta C}{C} + \dots$
  • For a square of a sum: if $f = (A+B)^2$, then approximately
    $\Delta f \approx 2(A+B)(\Delta A + \Delta B).$
  • For a sum inside a quotient: if $Q = \frac{(A+B)^2}{C D^3}$, then
    $\frac{\Delta Q}{Q} \approx \frac{2(A+B)(\Delta A + \Delta B)}{(A+B)^2} + \frac{\Delta C}{C} + 3\frac{\Delta D}{D}.$
  • For a logarithm base-10: $\Delta(\log_{10} x) = \frac{\Delta x}{x \ln 10}.$
  • For a trigonometric function: $\Delta(\sin \theta) \approx \cos \theta \; \Delta \theta$ with Δθ in radians.
• Quick check strategy
  • Identify the structure of the expression (sum, product, quotient, power).
  • Determine which variables carry uncertainty and apply the corresponding rule.
  • Convert all uncertainties to relative form when dealing with products/quotients, then sum with weights (exponents).
  • When possible, compare results to least count or MAD-based estimates to ensure consistency.

Problems 1–5: Problem statements, answers, and key notes

• Problem 1
  • Statement: A student measures the voltage across a resistor with a voltmeter of least count 0.01 V. Readings: 2.50 V, 2.52 V, 2.48 V, 2.51 V, 2.49 V. What is the absolute error and final voltage with error? Options: (a) 2.50 ± 0.02 V, (b) 2.50 ± 0.01 V, (c) 2.50 ± 0.015 V, (d) 2.50 ± 0.005 V
  • Solution note: Mean = 2.50 V. Absolute deviations from mean: 0.00, 0.02, 0.02, 0.01, 0.01. Mean absolute deviation ≈ (0.00+0.02+0.02+0.01+0.01)/5 = 0.012 V ≈ 0.01 V.
  • Answer: (b) 2.50 ± 0.01 V.
• Problem 2
  • Statement: Q = A^2 B / √C. % errors: ΔA/A = 1%, ΔB/B = 2%, ΔC/C = 4%. Find % error in Q.
  • Solution note: Relative error contributions: A^2 → 2(ΔA/A) = 2%, B → 2%, √C → (1/2)(ΔC/C) = 2%; Total ≈ 6%.
  • Answer: (a) 6%.
• Problem 3
  • Statement: Assertion: When two physical quantities are subtracted, relative error may increase significantly. Reason: Subtraction reduces net value, increasing the ratio of absolute error to measured value.
  • Solution note: Both A and R true; R correctly explains A.
  • Answer: (a) Both A and R are true, and R is the correct explanation of A.
• Problem 4
  • Statement: Given r = 3.00 ± 0.05 cm, h = 10.0 ± 0.2 cm, volume V = π r^2 h. Calculate % error in V.
  • Solution note: ΔV/V ≈ 2(Δr/r) + (Δh/h) = 2(0.05/3.00) + (0.2/10) ≈ 0.0333 + 0.02 = 0.0533 → ≈ 5.3%.
  • Answer: (c) 5.3%.
• Problem 5
  • Statement: A/B with % errors in A and B; Assertion: If % error in B is larger than in A, the % error in Z = A/B is mainly due to B. Reason: In division, % errors are added.
  • Solution note: Both A and R true; R correctly explains A.
  • Answer: (a) Both A and R are true, and R explains A.

Problems 6–8: Problem statements, answers, and key notes

• Problem 6
  • Statement: A = 10.0 ± 0.1; B = 2.00 ± 0.01; C = 5.0 ± 0.2. Find the absolute error in Q = (A^2 √B)/C.
  • Solution note: ΔA/A = 0.1/10.0 = 0.01; ΔB/B = 0.01/2.00 = 0.005; ΔC/C = 0.2/5.0 = 0.04. Overall relative error: ΔQ/Q = 2(0.01) + 0.005 + 0.04 = 0.02 + 0.0025 + 0.04 = 0.0625. Q ≈ (10)^2 × √2 / 5 = 100 × 1.414 / 5 ≈ 28.28. ΔQ ≈ Q × 0.0625 ≈ 1.77.
  • Answer: (d) 1.77.
• Problem 7
  • Statement: L = 20.0 ± 0.1 cm; W = 10.0 ± 0.2 cm. Find % error in area A = L × W.
  • Solution note: % error ≈ %ΔL + %ΔW = (0.1/20)×100 + (0.2/10)×100 = 0.5% + 2% = 2.5%.
  • Answer: (b) 2.5%.
• Problem 8
  • Statement: A = A^2 / B^2 with % errors ΔA = 4%, ΔB = 1%. Find % error in Z = A^2 / B^2.
  • Solution note: Relative error ΔZ/Z = 2(4%) + 2(1%) = 8% + 2% = 10%. The Assertion is false; the Reason is true.
  • Answer: (d) A is false, R is true.

Problems 9–12: Problem statements, answers, and key notes

• Problem 9
  • Statement: Lengths LA = 3.25 ± 0.01 cm; LB = 4.19 ± 0.01 cm. Find how much longer B is than A.
  • Solution note: Difference D = LB − LA = 0.94 cm. Uncertainty ΔD = √((0.01)^2 + (0.01)^2) = √(2 × 10^−4) ≈ 0.0141 cm ≈ 0.01 cm after rounding to the least count.
  • Answer: (b) 0.94 ± 0.01 cm.
• Problem 10
  • Statement: y = m^2 r^−4 g x l^−3/2; % errors: m=1%, r=0.5%, l=4%, g=p, y has 18% error. Find x and p.
  • Solution note: Δy/y ×100 = [2(Δm/m) + 4(Δr/r) + x(Δg/g) + (3/2)(Δl/l)] × 100. Substituting gives 18 = 2×1 + 4×0.5 + x p + (3/2)×4. So 18 = 2 + 2 + x p + 6 → 18 = 10 + x p → x p = 8. From options, x = 16/3 and p = ±3/2.
  • Answer: (d) 16/3 and ±3/2.
• Problem 11
  • Statement: % error in a product of quantities depends on the sum of individual % errors. Reason: Absolute errors are added in multiplication.
  • Solution note: The statement is true; the reason is false (we add % errors, not absolute errors).
  • Answer: (d) A is false, R is true.
• Problem 12
  • Statement: Z = A − BZ; % error may become large when subtracting even if absolute errors are small.
  • Solution note: Subtraction can amplify relative error because the net result is small.
  • Answer: (a) Both true, R explains A.

Problems 13–18: Problem statements, answers, and key notes

• Problem 13
  • Statement: Cube with edge l = 5.00 ± 0.02 cm. Find % error in volume V = l^3.
  • Solution note: Using ΔV/V ≈ 3(Δl/l) with Δl/l = 0.02/5.00 = 0.004 → ΔV/V ≈ 3 × 0.004 = 0.012 → 1.2%.
  • Answer: (a) 1.2%.
• Problem 14
  • Statement: Z = (A^3 √B)/C^{1/2} D^2 with % errors: A=2%, B=1%, C=3%, D=4%. Find % error in Z.
  • Solution note: Relative error: 3(2%) + (1/2)(1%) + (1/2)(3%) + 2(4%) = 6% + 0.5% + 1.5% + 8% = 16%. They report 16% as the correct value in the solution set.
  • Answer: (d) 16%.
• Problem 15
  • Statement: Error in log x: Δ(log x) = Δx / (x ln 10). Reason: Error in logarithmic quantities is absolute and not relative.
  • Solution note: A is true; R is false (logarithmic errors are relative in form, not an absolute quantity).
  • Answer: (c) A is true, R is false.
• Problem 16
  • Statement: θ = 30° ± 2°. Calculate the maximum possible error in sin θ.
  • Solution note: Δ(sin θ) ≈ cos θ · Δθ with Δθ in radians; Δθ = 2° = 2π/180 rad ≈ 0.0349 rad; cos 30° ≈ 0.866. So Δ(sin θ) ≈ 0.866 × 0.0349 ≈ 0.0302.
  • Answer: (b) 0.034? (The numeric option chosen in the transcript is 0.034; the computed value ~0.030 may be presented as 0.034 in that key.)
• Problem 17
  • Statement: Z = x^2 y^3 / z^4 with x=2.00±0.02, y=1.00±0.01, z=4.00±0.04. Find relative error in Z.
  • Solution note: ΔZ% ≈ 2(Δx/x) + 3(Δy/y) + 4(Δz/z). Evaluate: 2(0.02/2.00) + 3(0.01/1.00) + 4(0.04/4.00) = 0.02 + 0.03 + 0.04 = 0.09 = 9%.
  • Answer: (c) 9%.
• Problem 18
  • Statement: Error in a derived unit depends on relative errors of fundamental quantities. Reason: Derived units are products or ratios of fundamentals, so their relative errors add up.
  • Solution note: A true; R true; R explains A.
  • Answer: (a) Both A and R are true, and R is the correct explanation of A.

Problems 19–20: Problem statements, answers, and key notes

• Problem 19
  • Statement: L1 = 12.125 ± 0.005 cm; L2 = 11.970 ± 0.005 cm. Difference L = L1 − L2 with proper error and significant figures.
  • Solution note: L = 0.155 cm; ΔL = √[(0.005)^2 + (0.005)^2] = √(2 × 0.005^2) ≈ 0.0071 cm ≈ 0.007 cm. Round as appropriate to two sig figs: 0.014 ≈ 0.01 cm; but the presented option uses 0.007 as the uncertainty.
  • Answer: (b) 0.155 ± 0.007 cm.
• Problem 20
  • Statement: Q = (A + B)^2 / (C × D^3) with A = 5.00 ± 0.02, B = 3.00 ± 0.01, C = 2.00 ± 0.02, D = 4.00 ± 0.04. Statements I–III about error propagation; which are correct?
  • Solution note: (I) True: Δf for f = (A+B)^2 is 2(A+B)(ΔA + ΔB). (II) True: ΔQ/Q formula as given. (III) True: Dominant contribution from the term with the highest relative error raised to the highest power.
  • Answer: (c) All statements are correct.

Worked examples and cross-checks (summary of key solution patterns)

• Mean absolute deviation (MAD) for repeated measurements as a practical absolute error estimator.
• Relative-error addition rules for products/quotients and exponents; example: if Z = A^p B^q, then $\frac{\Delta Z}{Z} \approx |p|\frac{\Delta A}{A} + |q|\frac{\Delta B}{B}.$
• When subtracting similar quantities, watch for large relative error due to small net result; see Problems 3, 12.
• For geometric measurements, propagate uncertainty through volume/area formulas by differentiating existing expressions.
• Error in logarithms and trigonometric functions follows standard calculus-based approximations.
• In Problem 11, beware the common misconception: percentage errors add for products, but the stated justification about absolute errors is not correct; the correct statement involves addition of percentage errors, not absolute errors.

Quick reference: key formulas repeated

• Volume: $V = \pi r^2 h; \quad \frac{\Delta V}{V} \approx 2\frac{\Delta r}{r} + \frac{\Delta h}{h}.$
• Product/quotient with exponents: $Z = \prod<em>i A</em>i^{p<em>i} \Rightarrow \frac{\Delta Z}{Z} \approx \sum</em>i |p<em>i| \frac{\Delta A</em>i}{A_i}.$
• Logarithm error (base 10): $\Delta(\log_{10} x) = \frac{\Delta x}{x\ln 10}.$
• Sin error: $\Delta(\sin \theta) \approx \cos\theta \; \Delta\theta\quad (\Delta\theta \text{ in radians}).$
• For f = (A+B)^2: $\Delta f \approx 2(A+B)(\Delta A + \Delta B).$
• For a sum inside a quotient: $Q = \frac{(A+B)^2}{C D^3} \Rightarrow \frac{\Delta Q}{Q} \approx \frac{2(A+B)(\Delta A + \Delta B)}{(A+B)^2} + \frac{\Delta C}{C} + 3\frac{\Delta D}{D}.$