550 Chapter 9. Essential Statistics for Data Analysis
These general relations can be used to derive formulae for specific functions as
shown below.
C.1 AdditionofParameters
Suppose we haveu=x 1 +x 2 +....+xN. In this case the derivatives ofu=f(x)
will be given by
∂f
∂x 1
=
∂f
∂x 2=....=
∂f
∂xN=1. (9.5.4)
Equation 9.5.3, then reduces to
δu=[
δx^21 +δx^22 +....+δx^2 N] 1 / 2
, (9.5.5)
which states that the total error in the measurement will simply be equal to the
square root of the sum of individual errors squared.
Note that the above formula also holds if the some or all of the parameters have
negative signs. In other words, the formula remains the same whether the parameters
are added or subtracted in the function.
C.2 MultiplicationofParameters
Let us now see how the errors propagate if the function has the multiplicative form.
For simplicity we will restrict ourselves to two variables, that is, we will assume that
u=x 1 x 2. In this case the derivatives ofu=f(x) will be given by
∂f
∂x 1= x 2 (9.5.6)and∂f
∂x 2= x 1. (9.5.7)Substituting these values into equation 9.5.3 gives
δu
u=
[(
δx 1
x 1) 2
+
(
δx 2
x 2) 2 ]^1 /^2
. (9.5.8)
The generalized form of this equation forNparameters is given by
δu
u=
[(
δx 1
x 1) 2
+
(
δx 2
x 2) 2
+....+
(
δxN
xN) 2 ]^1 /^2
. (9.5.9)
Note that hereδu/urefers to relative error. For absolute error this must be multiplied
byu. What the above formula tells us is that the relative error in the measurement of
Nindependent measurements is simply the square root of the sum of the individual
relative errors squared.
The reader can verify that the above formula does not change its form in case of
division. For example, the error inu=x 1 x 2 /x 3 can be determined from the above
formula without any modifications.