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How do you square a number with uncertainty?

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EducationScienceBiologySimple Error Propagation Formulas for Simple Expressions

Simple Error Propagation Formulas for Simple Expressions

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By John Pezzullo

Even though some general error-propagation formulas are very complicated, the rules for propagating SEs through some simple mathematical expressions are much easier to work with. Here are some of the most common simple rules.

All the rules that involve two or more variables assume that those variables have been measured independently; they shouldn’t be applied when the two variables have been calculated from the same raw data.

Adding or subtracting a constant doesn’t change the SE

Adding (or subtracting) an exactly known numerical constant (that has no SE at all) doesn’t affect the SE of a number. So if x = 38 ± 2, then x + 100 = 138 ± 2. Likewise, if x = 38 ± 2, then x – 15 = 23 ± 2.

Multiplying (or dividing) by a constant multiplies (or divides) the SE by the same amount

Multiplying a number by an exactly known constant multiplies the SE by that same constant. This situation arises when converting units of measure. For example, to convert a length from meters to centimeters, you multiply by exactly 100, so a length of an exercise track that’s measured as 150 ± 1 meters can also be expressed as 15,000 ± 100 centimeters.

For sums and differences: Add the squares of SEs together

When adding or subtracting two independently measured numbers, you square each SE, then add the squares, and then take the square root of the sum, like this:

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For example, if each of two measurements has an SE of ± 1, and those numbers are added together (or subtracted), the resulting sum (or difference) has an SE of

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