**Error propagation** (or *propagation of uncertainty*) is what happens to measurement errors when you use those uncertain measurements to calculate something else. For example, you might use velocity to calculate kinetic energy, or you might use length to calculate area. **When you use uncertain measurements to calculate something else, they propagate **(grow much more quickly than the sum of the individual errors). To take this propagation into account, use one of the following formulas in your experiments.

These formulas assume your errors are random and not correlated (e.g. if you have systematic errors, you can’t use them).

**Error Propagation Contents:**

- Addition or Subtraction Formula
- Multiplication or Division formula
- Measured Quantity Times Exact Number formula
- General formula
- Power formula
- Error Propagation in Calculus

## 1. Addition or Subtraction

Where:

- a,b,c are positive measurements
- x,y,z are negative measurements
- δ is the error associated with each measurement (the absolute error). δa is the uncertainty associated with measurement a, δb is the uncertainty associated with measurement b, and so on.

## Example of Worked Formula

Let’s say you measured your height (a) as 2.00 ± 0.03 m. Your waistband (b) is 0.88 ± 0.04 m from the top of your head, which means your pant length P would be p = H – w = 2.00 m – 0.88 m = 1.12 m.

The uncertainty, using the addition formula, is:

Giving a final measurement of 1.12 m ± 0.05 m.

## 2. Multiplication or Division formula

**When calculating errors, there is no difference between multiplication and division.**

## 3. Power formula

If n is an exact number and Q = x^{n}, then

## 4. Measured Quantity Times Exact Number formula

If A is exact measurement (e.g. A = 9 or A = π) and Q = Ax, then:

**δQ = |A| δx**

## 5. General formula for Error Propagation

You might wonder why you can’t just add (or multiply, or divide) the errors and be done with it. Why do we have to use formulas? Very basically, one small measurement error on an independent variable, when applied to a function (say, a formula for area, kinetic energy, or velocity) is going to result in a much larger error on the dependent variable.

## Error Propagation in Calculus

*Why *the formulas work requires an understanding of calculus, and particularly derivatives; They are derived from the Gaussian equation for normally-distributed errors. If you have some error in your measurement (x), then the resulting error in the function output (y) is based on the slope of the line (i.e. the derivative).

The general formula (using derivatives) for error propagation (from which all of the other formulas are derived) is:

Where Q = Q(x) is any function of x.

Error propagation formulas are based on taking partial derivatives of a function with respect to the variable with the uncertainty. Let’s say you had a function with three variables (x, u, v) and two of those (u, v) have uncertainty. The variance of x can be approximated by [1]:

**Example question**: The volume of gasoline delivered from a pump is the difference between the initial (I) and final (F) readings. If each reading has an uncertainty of ±0.02mL, what is the error in the volume delivered?

**Solution**:

V = F – I; σ^{2}(V)

= σ^{2}(I) + σ^{2}(F)

= (0.02mL)^{2} + (0.02mL)^{2}

= 0.0008mL^{2}

= 0.028mL

The error in the volume delivered is 0.028mL.

This calculation will only work when the partial derivatives are 1. The formula changes slightly when x is a product (x = uv) or quotient (x = u/v):

**Example question #2**: A shipping container is 12 x 10 x 8 feet with an uncertainty of 0.1ft. What is the uncertainty in the volume?

Step 1: Calculate the volume:

V = 12 x 10 x 8 = 960 ft^{3}.

Step 1: Work the formula:

(0.1 ft)^{2} / (12 ft^{2}) +

(0.1 ft)^{2} / (10 ft^{2}) +

(0.1 ft)^{2} / (8 ft^{2}) =

0.0.00145149572

Step 3: Find the volume variance (Step 1 * Step 2):

(0.00145149572)(960 ft^{3})^{2} = 1337.698 ft^{6}

Step 4: Take the square root of Step 3 to find the uncertainty in the volume:

√(1337.698) = 36.57 ft^{3}.

## References

10. Error Propagation tutorial.doc. Retrieved April 14, 2021 from: https://foothill.edu/psme/daley/tutorials_files/10.%20Error%20Propagation.pdf