When do calculations based on measured values, the calculated value will “inherit” the measured values’ uncertainties. The proper lingo is to say that the measured values’ uncertainties will propagate to become the calculated value’s uncertainty. There are three “rules” to help us evaluate the resultant uncertainty quickly.

Summation Rule

If *X* is calculated from the measured values of *A*, *B* and *C* by the equation

then

Basically, when measurements are summed, their **absolute** uncertainties also sum up to be the calculated value’s **absolute** uncertainty. Do note that

- Each measurement’s uncertainty is “weighted” by their coefficients.
- Even if a measurement is being subtracted, its uncertainty is added to, not subtracted from, the resultant uncertainty.

Product Rule

If *X* is calculated from the measured values of *A*, *B* and *C* by the equation

then

Basically, when measurements are multiplied together, their **percentage** uncertainties also sum up to be the calculated value’s **percentage** uncertainty. Why? Because ! Do note that

- Each measurement’s fractional/percentage uncertainty is “weighted” by its power.
- Even if the measurement is being divided, its (percentage) uncertainty is still added to, not subtracted from, the resultant (percentage) uncertainty.

Coefficients do not affect the resultant fractional/percentage uncertainty.

Max-Minus-Min-Divide-by-Two Rule

What if the calculation contains a mixture of summing and multiplying measurements, e.g. ? Or what if the calculation involves special functions such as sine or log? For such cases, there is no short-cut available. We simply have to figure out the maximum and minimum possible values before calculating the uncertainty.

For example, let’s suppose that *L* and *θ* are measured to be

and *x* is calculated using the formula

We can figure out the extreme outcomes for *x* based on the extreme values of *L* and *θ*.

This allows us to calculate the uncertainty of *x* as

–

**Video Explanation **

Absolute vs Percentage Uncertainties. When to Use Which? (xmphysics)

Plus Minus Times Divide. Uncertainties Always Add Up (xmphysics)

Uncertainty Propagation Rules Summary. Does a Coefficient Scale Up the Uncertainty? (xmphysics)

**Concept Test **

QQ0039

QQ0043

**Beyond the Syllabus**

Error Propagation with Uncensored Math (Glacierfilm.com)

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