Error Propagation

Updated 15 Sep 2026

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What it is
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  estimating how uncertainty in a set of measured input variables carries through to uncertainty in a computed result
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Maximum (worst-case) error formula
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  $\Delta y \approx \sum_{i=1}^{n} \left|\dfrac{\partial f}{\partial x_i}\right|\Delta x_i$
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Statistical (independent random errors) formula
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  $\sigma_y = \sqrt{\sum_{i=1}^{n} \left(\dfrac{\partial f}{\partial x_i}\right)^2 \sigma_{x_i}^2}$
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Worked example: Rectangle area
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  $L = 20.0\pm0.1$ cm, $W=10.0\pm0.05$ cm → $A = 200.0 \pm 2.0\ \text{cm}^2$ ( $1.0\%$ relative error)
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worked example: kinetic energy ( $E_k = \frac12 mv^2$ )
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  $m=1000\pm10$ kg, $v=20\pm0.5$ m/s → mass contributes $1\%$ relative error, velocity contributes $5\%$
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  key insight: a squared variable ( $v^2$ ) doubles the sensitivity of error propagation compared to a linear variable ( $m$ ) — small uncertainty in a squared term matters much more
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Worked example: cylinder volume
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  $r = 5.0\pm0.04$ cm, $h=20.0\pm0.1$ cm → $V = 1570.8 \pm 26.3\ \text{cm}^3$ via the RSS (root-sum-square) formula
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Related
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  Catastrophic Cancellation

Linked references 2