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Error in a sum or difference
Intermediate
Absolute errors add when quantities are added or subtracted.
Δ(A±B) = ΔA + ΔBWhen quantities are added or subtracted, their absolute errors add (worst case). For Z = A ± B, ΔZ = ΔA + ΔB. Subtraction does not cancel errors, which is why subtracting two nearly equal measured numbers is risky.
Two lengths 10.0 ± 0.1 cm and 4.0 ± 0.1 cm: sum 14.0 ± 0.2 cm, difference 6.0 ± 0.2 cm.
Errors never subtract. A common slip is to write ΔZ = ΔA − ΔB.
In more detail
When you add or subtract quantities, their absolute uncertainties add. For a difference of 5.0 ± 0.1 and 2.0 ± 0.2, the result is 3.0 ± 0.3.
Real-world uses
- Finding the length of a gap by subtracting two measured positions gives an uncertainty equal to the sum of both errors.
- Net mass of a product is found by subtracting the container mass, which doubles the uncertainty of the balance reading.
- When measuring the length of a rod by subtracting two scale readings, each with its own error, the uncertainty in the difference is the sum of both errors.
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Frequently asked questions
Do uncertainties add in quadrature?
For independent random errors, the more accurate rule is to add squares and take the square root, which gives a smaller result than the simple sum.
Why do subtractions often have large relative errors?
When two values are nearly equal, their difference is small, so the same absolute errors become a large fraction of the result.
Why does subtracting two similar numbers give a big relative error?
The absolute errors add, but the result is small. A small result divided into a sum of errors produces a large relative error.
Related topics in Measurement & Math Tools
Random error · Absolute error · Relative and percent error · Error in product or quotient · Error in powers · Mean and standard deviation