WHY INDEPENDENT UNCERTAINTIES ADD IN QUADRATURE
INTRO - Addition in quadrature is a description in words of the Pythagorean addition c2 = a2 + b2 for a right
triangle, and is a characteristic of the net result of moving along orthogonal directions in a space. So we go a
certain distance y north and then x east. These displacements are independent in the sense that moving north
doesn’t change our east-west position: the x and y axes are orthogonal. Mathematically this shows up in the
calculation of the magnitude of the net displacement vector c = x + y:
This magnitude is
√c•c = √c2 = c = √(x + y)•(x + y) = √(x2 + y2 + 2x•y).
Since x and y are orthogonal and their displacements independent, then x•y = 0 and c = √(x2 + y2) is given by
addition in quadrature. A similar idea applies to fluctuations contributing to a sum or product. If they are
independent their net effect is obtained by addition in quadrature:
IN A SUM – Suppose a and b are some measured quantities. Measurement processes for determining a and b
give well-defined averages and uncertainties <a> ± δa and <b> ± δb. If we repeat the measurement process
for a one more time we get ai, the “i th” measurement of ai, and we can define the fluctuation δai as
δai = ai - <a>. Similarily for the j th measurement of b:
ai = <a> + δai
bj = <b> + δbj
where, by definition <δai> = <δbj> = 0. If the fluctuations in a and b are independent, then the average of the
product of the fluctuations is zero: <δaiδbi> = 0 (definition of independence). The quantities √<δai2> and
√<δbj2>, always positive, are a measure of how big the fluctuations in a and b are in a single measurement.
Now we calculate the fluctuations in the sum c = a + b. Treating fluctuations like differentials, we have
<c> = <(a + b)> = <a> + <b>,
and
δc = δai + δbj.
<δc> = <(δai + δbj)> = 0, so the average of δc doesn’t reveal anything about the magnitude of the δc’s. To
probe this we calculate <δci2>:
<δc2> = <δai2> + 2<δaiδbj> + <δbj2> = <δa2> + <δb2>.
Since the cross term <δaiδbi> = 0 independent uncertainties in a sum add in quadrature:
δc = √<δc2> = √(<δa2> + <δb2>)
MULTIPLE INDEPENDENT MEASUREMENTS OF THE SAME QUANTITY – Let c = (a1 + a2 + .. + aN)/N,
where ai are independent measurements of a. Then c = 〈a〉 and <δc2> = [<δai2> + <δa22> + ... +
<δaN2>]/N2 = <δai2>/N, where all of the cross terms <δaiδaj> = 0. Thus, after averaging N independent
measurements of a, we have:
a = 〈a〉,
δa = δ〈a〉 = δai/√N, where δai is the uncertainty for a single measurement
IN A PRODUCT – Let c = ab. Then, using the chain rule dc = a db + b da or dc/c = da/a + db/b
δck/<c> = δai/<a> + δbi/<b> and (δck/<c>)2 = (δai/<a>)2 + (δbi/<b>)2 + 2δaiδbi /<a><b>,
or
<(δc/<c>)2> = <(δai/<a>)2> + <(δbi/<b>)2> = (δa/<a>)2 + (δb/<b>)2,
so independent fractional uncertainties in a product add in quadrature:
δc/<c> = √ (<δck2>/<c>2) = √[(δa/<a>)2 + (δb/<b>)2]
IN THE GENERAL CASE – Suppose c = f(a,b). Then
δci = [∂f/∂a]δai +[∂f/∂b]δbi
and
so
<δci2> = [∂f/∂a]2<δai2> +[∂f/∂b]2<δbi2>
δc = √<δci2> = √([∂f(<a>,<b>)/∂<a>]2<δai2> +[∂f(<a>,<b>)/∂<b>]2<δbi2>)