**Error propagation** occurs when you measure some quantities *a*, *b*, *c*, … with uncertainties δ*a*, δ*b*, *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 δ*a*, δ*b*, *δc* will **propagate **(i.e. “extend to”) to the uncertainty of *Q.*

To calculate the uncertainty of *Q*, denoted δ*Q*, 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 δ*Q* = √(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:

- δ
*Q*= √(δa)^{2}+ (δb)^{2}+ … + (δc)^{2}+ (δx)^{2}+ (δy)^{2}+ … + (δz)^{2} - δ
*Q*= √(.18)^{2}+ (.06)^{2} - δ
*Q*=**0.1897**

This gives us a final measurement of **70 ± 0.1897** inches.

**Multiplication or Division**

If *Q* = (ab…c) / (xy…z)

Then δ*Q* = |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:

- δ
*Q*= |Q| * √(δa/a)^{2}+ (δb/b)^{2}+ … + (δc/c)^{2}+ (δx/x)^{2}+ (δy/y)^{2}+ … + (δz/z)^{2} - δ
*Q*= |1.333| * √(.34/20)^{2}+ (.21/15)^{2} - δ
*Q*=**0.0294**

This gives us a final ratio of **1.333 ± 0.0294** inches.

**Measured Quantity Times Exact Number**

If *A* is known exactly and *Q* = *A*x

Then δ*Q* = |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:

- δ
*Q*= |A|δx - δ
*Q*= |*π*| * 0.3 - δ
*Q*=**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* = *x ^{n}*

Then δ*Q* = |*Q*| * |*n*| * (δ*x/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* = *s*^{3}.

The volume would be calculated as *v* = *s*^{3} = 2^{3} = **8 in. ^{3}**. The uncertainty in this estimate would be calculated as:

- δ
*Q*= |*Q*| * |*n*| * (δ*x/x*) - δ
*Q*= |8| * |3| * (.02/2) - δ
*Q*=**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:

**δ Q = |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.