Error Propagation

Guide to Error Propagation

1  Why Quote Errors?

• Every measurement has a limit (± value) that brackets where the actual value can lie:contentReference[oaicite:1]{index=1}.
• Instrument error is set by device design (e.g.\ ruler ±0.5 mm, micrometer ±0.001 mm) and user skill:contentReference[oaicite:3]{index=3}.

2  Types of Error

Type	Definition	Typical Size
Instrument (reading)	Finite resolution of the device	Specified by manufacturer
Statistical	Scatter of repeated readings (includes instrument + natural variation)	Use sample standard deviation

These notes follow the worst-case (maximum) method; the same algebra works with 1 σ uncertainties if you treat σ like Δx:contentReference[oaicite:5]{index=5}.

3  Expressing Error

• Absolute error: $Y\pm \Delta y$ (same units).
• Relative error: $\frac{\Delta y}{Y}$ (dimensionless).
• Percentage error: $\frac{\Delta y}{Y}\times 100\%$:contentReference[oaicite:7]{index=7}.

Example – ruler reading: $t=6.5\text{mm}\pm 0.5\text{mm}$ gives an 8 % relative error.

4  Propagation Rules

Rule 1 Addition & Subtraction

If $Z=X\pm Y$ with $X\pm \Delta x$ and $Y\pm \Delta y$ then
$\boxed{\Delta z=\Delta x+\Delta y}$
Example: $8.5\text{mm}\pm 0.6\text{mm}$ from $6.5\pm 0.5$ mm + $2.0\pm 0.1$

Subtractions use the same formula; relative error often grows (4.5 mm ± 0.6 mm → 13 %):contentReference[oaicite:11]{index=11}.

Rule 2 Multiplication & Division

For $B=XY$ or $C=Y/X$ add relative errors:

$$\frac{\Delta B}{B}=\frac{\Delta x}{X}+\frac{\Delta y}{Y},\frac{\Delta C}{C}=\frac{\Delta x}{X}+\frac{\Delta y}{Y}$$

Example volume: $V=WHL$ → 17 % error from adding three relative

Rule 3 General Function $G=f(X)$

For small $\Delta x$
$\boxed{\Delta G\approx \left|\frac{df}{dX}\right|\Delta x}$
With $X=5.0\pm 0.2$ m and $G=X^{10}$,
$\Delta G=10X^9\Delta x\approx 3.9\times 10^6$:contentReference[oaicite:17]{index=17}.
For multivariable $G=f(X,Y,\dots )$ sum the partial-derivative contributions:contentReference[oaicite:19]{index=19}.

Special Cases

1. Compare two results: subtract and add errors (Rule 1):contentReference[oaicite:21]{index=21}.
2. Multiply/divide by an exact constant: scale the absolute error by that constant:contentReference[oaicite:23]{index=23}.
3. Trig/log functions: convert angles to radians before using Rule 3:contentReference[oaicite:25]{index=25}.

5  Confidence Levels (Assuming Gaussian Errors)

• About 68 % of readings lie within $\pm 1\Delta$,
• 95 % within $\pm 2\Delta$,
• 99.7 % within $\pm 3\Delta$:contentReference[oaicite:27]{index=27}.

6  Reporting Guidelines

1. Quote one, at most two, significant figures in the error:contentReference[oaicite:29]{index=29}.
2. Round the measured value to the same decimal place as its error.
3. Use % errors when specifications are given that way (common in electronics):contentReference[oaicite:31]{index=31}.

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Quick Reference Sheet

Operation	Propagation Formula	When to Use
$Z=X\pm Y$	$\Delta z=\Delta x+\Delta y$	Adding lengths, masses, voltages
$B=XY$ or $C=Y/X$	$\frac{\Delta B}{B}=\frac{\Delta x}{X}+\frac{\Delta y}{Y}$	Areas, volumes, ratios
$G=f(X)$	$\Delta G \approx	f’(X)
$G=f(X,Y)$	$\Delta G \approx	f_X

Tip: Always check whether worst-case propagation is appropriate; for large datasets, root-sum-of-squares (RSS) with σ may be more realistic.