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Error Analysis and Uncertainty

24 min Sensors & Measurement Math

Error Analysis

Every measurement has uncertainty. Understanding how errors propagate through calculations is essential for reliable engineering.

Types of Error

Absolute error: $\Delta x$ — the magnitude of uncertainty.

Relative error: $\frac{\Delta x}{x} \times 100\%$

Error Propagation (basic rules)

Addition/Subtraction: Add absolute errors. $\Delta(a \pm b) = \Delta a + \Delta b$
Multiplication/Division: Add relative errors. $\frac{\Delta(ab)}{ab} = \frac{\Delta a}{a} + \frac{\Delta b}{b}$
Powers: $\frac{\Delta(a^n)}{a^n} = n \cdot \frac{\Delta a}{a}$

Example 1

A ruler measures length as $25.3 \pm 0.1$ cm. Find the relative error.

Relative error $= 0.1/25.3 \times 100\% = 0.40\%$.

Example 2

Area of a rectangle: $L = 10.0 \pm 0.2$ cm, $W = 5.0 \pm 0.1$ cm. Find $A$ and its uncertainty.

$A = 10.0 \times 5.0 = 50.0$ cm².

$\frac{\Delta A}{A} = \frac{0.2}{10.0} + \frac{0.1}{5.0} = 0.02 + 0.02 = 0.04 = 4\%$.

$\Delta A = 0.04 \times 50 = 2.0$ cm².

$A = 50.0 \pm 2.0$ cm².

Example 3

$v = d/t$ where $d = 100 \pm 2$ m and $t = 10.0 \pm 0.5$ s. Find $v$ and its uncertainty.

$v = 100/10 = 10.0$ m/s.

$\frac{\Delta v}{v} = \frac{2}{100} + \frac{0.5}{10} = 0.02 + 0.05 = 0.07 = 7\%$.

$\Delta v = 0.7$ m/s. $v = 10.0 \pm 0.7$ m/s.

1. Measurement: $50.0 \pm 0.5$ g. Relative error?

2. $a = 20 \pm 1$, $b = 30 \pm 2$. Find $a + b$ with error.

3. $V = L \times W \times H$. If each has 2% relative error, what is the error in $V$?

4. $r = 5.0 \pm 0.1$ cm. Error in $A = \pi r^2$?

5. Which contributes more to error: a 1% error in time or a 1% error in distance when computing $v = d/t$?

6. Repeated measurements: 10.2, 10.4, 10.1, 10.3, 10.5. Mean and estimate of uncertainty?

Show Answer Key

1. $0.5/50 = 1\%$

2. $50 \pm 3$

3. $2\% + 2\% + 2\% = 6\%$

4. $\Delta A/A = 2 \times 0.1/5.0 = 4\%$. $A = 78.5$ cm², $\Delta A = 3.1$ cm².

5. Equal — division adds relative errors equally.

6. Mean = 10.3. Range ≈ 0.4, estimate $\Delta \approx 0.2$.