# Propagation of Errors

Physics I · Principles of Laboratory Practice · https://tryals.app/learn/physics-i/propagation-of-errors

## General Error Propagation

When a calculated result $R = f(x, y)$ depends on measured values, uncertainty propagates. The combined absolute uncertainty $\delta R$ is found using partial derivatives:

$$\delta R = \sqrt{\left(\frac{\partial f}{\partial x}\delta x\right)^2 + \left(\frac{\partial f}{\partial y}\delta y\right)^2}$$

**Common algebraic cases** simplify this general rule into handy formulas for sums, products, and powers.

| Operation | Formula | Type |
| --- | --- | --- |
| **Sum / Difference** | $R = x \pm y$ | $\delta R = \sqrt{(\delta x)^2 + (\delta y)^2}$ |
| **Product / Quotient** | $R = xy$ or $x/y$ | $\frac{\delta R}{R} = \sqrt{(\delta x/x)^2 + (\delta y/y)^2}$ |
| **Power** | $R = x^n$ | $\frac{\delta R}{R} = |n|\frac{\delta x}{x}$ |

## Worked Example & Pitfalls

**Example**: Calculate velocity $v = d/t$ where $d = 2.50 \pm 0.02$ m and $t = 1.20 \pm 0.05$ s. Applying the quotient rule yields $v = 2.08 \pm 0.09$ m/s.

**Key insight**: The quantity with the largest relative uncertainty dominates the final error. Always improve that measurement first.

**Common pitfall**: Independent uncertainties combine in quadrature ($\sqrt{a^2 + b^2}$), **not** by simple addition. Adding errors linearly severely overstates uncertainty, while ignoring exponents understates it.

## Practice questions

6 of this lesson's 13 practice questions, with answers. The full set is in the app.

### 1. Order the steps for propagating uncertainty through a calculation.

**Answer:**

1. Write the formula relating measured quantities to the result
2. Compute partial derivatives with respect to each variable
3. Multiply each partial derivative by the corresponding uncertainty
4. Add the squared terms in quadrature (square root of sum of squares)
5. Report the result with the propagated uncertainty

**Why:** Error propagation uses partial derivatives and quadrature addition for independent uncertainties.

Page: https://tryals.app/practice/physics-i/propagation-of-errors/order-the-steps-for-propagating-uncertainty-through-a-calculation

### 2. You measure a square plate’s side as $L = 10.0 \pm 0.1$ cm (1% uncertainty) and compute the area $A = L^{2}$. What is the uncertainty of the area?

A. About 0.01%, the square of 1%
B. Exactly 0.1 cm², matching the ruler
C. About 2%: powers multiply relative error
D. About 1%, squaring changes nothing

**Answer:** C. About 2%: powers multiply relative error

**Why:** $(10.1)^2 = 102.01$, about 2% above 100. Powers are error amplifiers: exponent × relative error. This is why a period measured to 1% gives $g$ (which involves $T^{2}$) only to 2%, and why precise exponents demand very precise inputs.

Page: https://tryals.app/practice/physics-i/propagation-of-errors/you-measure-a-square-plates-side-as-l-10-0-0-1-cm-1

### 3. You add two measured lengths: $a = 50.0 \pm 0.3$ cm and $b = 50.0 \pm 0.4$ cm. Assuming independent errors, the uncertainty of $a + b$ is…

A. 0.5 cm, combining in quadrature
B. 0.35 cm, the average of the two
C. 0.4 cm, the larger one wins
D. 0.7 cm, simply add the uncertainties

**Answer:** A. 0.5 cm, combining in quadrature

**Why:** $\sqrt{0.09 + 0.16} = 0.5$ cm. Quadrature is the mathematics of partial cancellation: uncorrelated wobbles add as areas, not lengths. Straight addition (0.7) is the pessimistic bound for perfectly correlated errors.

Page: https://tryals.app/practice/physics-i/propagation-of-errors/you-add-two-measured-lengths-a-50-0-0-3-cm-and-b-50-0-0-4

### 4. When adding two independent measurements, their absolute uncertainties add directly.

**Answer:** False

**Why:** False. Independent uncertainties add **in quadrature**, $\sqrt{\sigma_1^2 + \sigma_2^2}$, because they are as likely to cancel as to reinforce. Adding directly assumes the worst case every time and systematically overstates the error; direct addition is right only for *correlated* systematic effects.

Page: https://tryals.app/practice/physics-i/propagation-of-errors/when-adding-two-independent-measurements-their-absolute

### 5. Independent errors add in quadrature rather than linearly. Why does that make the total smaller than the plain sum?

A. Because quadrature is an approximation that underestimates the true error
B. Because some errors are systematic and cancel exactly
C. Because independent errors cancel as often as they reinforce
D. Because squaring makes each error smaller before they are added

**Answer:** C. Because independent errors cancel as often as they reinforce

**Why:** Random errors partially cancel. That is also why the rule fails for SYSTEMATIC errors, those are correlated, so they do add linearly.

Page: https://tryals.app/practice/physics-i/propagation-of-errors/independent-errors-add-in-quadrature-rather-than-linearly-why-does

### 6. Which statements about error propagation are correct?

A. For $f = a - b$, uncertainties subtract
B. Multiplying by a constant k scales the uncertainty by |k|
C. Uncertainties always add in quadrature for independent variables
D. The uncertainty of a product equals the product of uncertainties

**Answer:** B. Multiplying by a constant k scales the uncertainty by |k|; C. Uncertainties always add in quadrature for independent variables

**Why:** Subtraction also adds uncertainties in quadrature; product rules use relative uncertainties.

Page: https://tryals.app/practice/physics-i/propagation-of-errors/which-statements-about-error-propagation-are-correct