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Error Propagation is the process of estimating how errors in the inputs of a calculation can affect the output. An example of this would be if a calculation involves a value that has a 5% error, the output of the calculation will also have an error of 5%. This is because the error has been propagated through the calculation.
Error propagation occurs when you measure some quantities a, b, c, … with uncertainties δa, δb, δc … and you then want to calculate some other quantity Q using the measurements of a, b, c, etc.
It turns out that the uncertainties δa, δb, δc will propagate (i.e. “extend to”) to the uncertainty of Q.
To calculate the uncertainty of Q, denoted δQ, we can use the following formulas.
Note: For each of the formulas below, it’s assumed that the quantities a, b, c, etc. have errors that are random and uncorrelated.
Addition or Subtraction
If Q = a + b + … + c – (x + y + … + z)
Then δQ = √(δa)2 + (δb)2 + … + (δc)2 + (δx)2 + (δy)2 + … + (δz)2
Example: Suppose you measure the length of a person from the ground to their waist as 40 inches ± .18 inches. You then measure the length of a person from their waist to the top of their head as 30 inches ± .06 inches.
Suppose you then use these two measurements to calculate the total height of the person. The height would be calculated as 40 inches + 30 inches = 70 inches. The uncertainty in this estimate would be calculated as:
- δQ = √(δa)2 + (δb)2 + … + (δc)2 + (δx)2 + (δy)2 + … + (δz)2
- δQ = √(.18)2 + (.06)2
- δQ = 0.1897
This gives us a final measurement of 70 ± 0.1897 inches.
Multiplication or Division
If Q = (ab…c) / (xy…z)
Then δQ = |Q| * √(δa/a)2 + (δb/b)2 + … + (δc/c)2 + (δx/x)2 + (δy/y)2 + … + (δz/z)2
Example: Suppose you want to measure the ratio of the length of item a to item b. You measure the length of a to be 20 inches ± .34 inches and the length of b to be 15 inches ± .21 inches.
The ratio defined as Q = a/b would be calculated as 20/15 = 1.333. The uncertainty in this estimate would be calculated as:
- δQ = |Q| * √(δa/a)2 + (δb/b)2 + … + (δc/c)2 + (δx/x)2 + (δy/y)2 + … + (δz/z)2
- δQ = |1.333| * √(.34/20)2 + (.21/15)2
- δQ = 0.0294
Measured Quantity Times Exact Number
If A is known exactly and Q = Ax
Then δQ = |A|δx
Example: Suppose you measure the diameter of a circle as 5 meters ± 0.3 meters. You then use this value to calculate the circumference of the circle c = πd.
The circumference would be calculated as c = πd = π*5 = 15.708. The uncertainty in this estimate would be calculated as:
- δQ = |A|δx
- δQ = |π| * 0.3
- δQ = 0.942
Thus, the circumference of the circle is 15.708 ± 0.942 meters.
Uncertainty in a Power
If n is an exact number and Q = xn
Then δQ = |Q| * |n| * (δx/x)
Example: Suppose you measure the side of a cube to be s = 2 inches ± .02 inches. You then use this value to calculate the volumne of the cube v = s3.
The volume would be calculated as v = s3 = 23 = 8 in.3. The uncertainty in this estimate would be calculated as:
- δQ = |Q| * |n| * (δx/x)
- δQ = |8| * |3| * (.02/2)
- δQ = 0.24
Thus, the volume of the cube is 8 ± .24 in.3.
General Formula for Error Propagation
If Q = Q(x) is any function of x then the general formula for error propagation can be defined as:
δQ = |dQ/dX|δx
Note that you’ll rarely have to derive these formulas from scratch, but it can be good to know the general formula used to derive them.