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Approximation error bounds
SL AI 1.6
A measurement is always accurate to ±21 ("plus or minus one half") of the smallest division on the scale. The same idea applies to rounding numbers: the rounded value is accurate to ±21 the value of the digit "place" (tens, ones, tenths, hundredths, etc.) it is rounded to.
Typically, we express the error bounds of a number x using inequalities:
x−21u≤x<x+21u
where x is the measured value, u is the smallest unit on the scale we used. Note that some people equivalently write
x∈[x−21u,x+21u)
The upper bound, x+21u, is not included since it would be rounded up to the next increment.
Absolute and percentage error
SL AI 1.6
The actual size of an error is the difference between the approximated value (VA) and the exact value (VE). We call this the absolute error, and can calculate it with the equation
absolute error =∣VA−VE∣
Expressing the error as a percentage of the exact value allows us to see how much the error "matters" in a specific scenario. This is the percentage error, which we calculate with the equation
percentage error =VE∣VA−VE∣×100%
Bounds of calculation based on other approximations
SL AI 1.6
The error bounds of values like length and width impact the error bounds of any secondary calculation, like area, that we use them to find. For example, if an object of length l and width w has error bounds 0.9≤l≤1.1 and 2.6≤w≤2.9, then the object's minimum and maximum area are
Amin=0.9(2.6)=2.34
Amax=1.1(2.9)=3.19
i.e., the error bounds are 2.34≤A<3.19.
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