User:JHuwaldt/sandbox

The following are taken from, An Introduction to Error Analysis by John Robert Taylor, pgs 77-79.

The rules for error propagation refer to a situation in which we have found various quantities, $$x, \ldots, w$$ with uncertainties $$\delta x, \ldots, \delta w$$ and then use these values to calculate a quantity $$q$$. The uncertainties in $$x, \ldots, w$$ 'propagate' through the calculation to cause an uncertainty in $$q$$ as follows:

Sums and Differences
If
 * $$q = x + \cdots + z - (u + \cdots + w)$$,

then
 * $$\delta q = \sqrt{ \left( \delta x \right)^2 + \cdots + \left( \delta z \right)^2 + \left( \delta u \right)^2 + \cdots + \left( \delta w \right)^2}$$
 * (provided all errors are independent and random)

and
 * $$\delta q \leq \delta x + \cdots + \delta z + \delta u + \cdots + \delta w$$
 * (always).

Products and Quotients
If
 * $$q = \frac{x \times \cdots \times z}{u \times \cdots \times w}$$,

then
 * $$\frac{\delta q}{\left| q \right|} = \sqrt{\left( \frac{\delta x}{x} \right)^2 + \cdots + \left( \frac{\delta z}{z} \right)^2 + \left( \frac{\delta u}{u} \right)^2 + \cdots + \left( \frac{\delta w}{w} \right)^2}$$
 * (provided all errors are independent and random)

and
 * $$\frac{\delta q}{\left| q \right|} \leq \frac{\delta x}{\left| x \right|} + \cdots + \frac{\delta z}{\left| z \right|} + \frac{\delta u}{\left| u \right|} + \cdots + \frac{\delta w}{\left| w \right|}$$
 * (always).

Measured Quantity Times Exact Number
If $$B$$ is known exactly and
 * $$q = Bx$$

then
 * $$\delta q = \left| B \right| \delta x$$ or, equivalently, $$\frac{\delta q}{\left| q \right|} = \frac{\delta x}{\left| x \right|}$$.

Uncertainty in a Power
If $$n$$ is an exact number and
 * $$q = x^n$$

then
 * $$\frac{\delta q}{\left| q \right|} = \left| n \right| \frac{\delta x}{\left| x \right|}$$.

Uncertainty in a Function of One Variable
If $$q = q(x)$$ is a function of $$x$$, then
 * $$\delta q = \left| \frac{dq}{dx} \right| \delta x$$.

Sometimes, if $$q(x)$$ is complicated and if you have written a program to calculate $$q(x)$$ then, instead of differentiating $$q(x)$$, you may find it easier to use the equivalent formula,
 * $$\delta q = \left| q(x_{best} + \delta x) - q(x_{best}) \right|$$.

General Formula for Error Propagation
If $$q = q(x, \ldots, z)$$ is any function of $$x, \ldots, z$$, then
 * $$\delta q = \sqrt{\left( \frac{\partial q}{\partial x} \delta x \right)^2 + \cdots + \left( \frac{\partial q}{\partial z} \delta z \right)^2 }$$

and
 * $$\delta q \leq \left| \frac{\partial q}{\partial x} \right| \delta x + \cdots + \left| \frac{\partial q}{\partial z} \right| \delta z$$
 * (always).