# Error Propagation

### Introduction

Error propagation is simply the process of determining the uncertainty of an answer obtained from a calculation. Every time data are measured, there is an uncertainty associated with that measurement. (Refer to guide to Measurement and Uncertainty.) If these measurements used in your calculation have some uncertainty associated with them, then the final answer will, of course, have some level of uncertainty. For instance, in lab you might measure an object's position at different times in order to find the object's average velocity. Since both distance and time measurements have uncertainties associated with them, those uncertainties follow the numbers throughout the calculations and eventually affect your final answer for the velocity of that object. How would you determine the uncertainty in your calculated values?

In lab, graphs are often used where LoggerPro software calculates uncertainties in slope and intercept values for you. In other classes, like chemistry, there are particular ways to calculate uncertainties. In fact, since uncertainty calculations are based on statistics, there are as many different ways to determine uncertainties as there are statistical methods. The error propagation methods presented in this guide are a set of general rules that will be consistently used for all levels of physics classes in this department.

In the following examples:

- q is the result of a mathematical operation
- δ is the uncertainty associated with a measurement.
- For example, if you have a measurement that looks like this:
- m = 20.4 kg ±0.2 kg
- Then q = 20.4 kg and δm = 0.2 kg

### First Step: Make sure that your units are consistent

Make sure that you are using SI units and that they are consistent. If you are converting between unit systems, then you are probably multiplying your value by a constant. Please see the following rule on how to use constants.

### Logger Pro

If you are using a curve fit generated by Logger Pro, please use the uncertainty associated with the parameters that Logger Pro give you.

In the above linear fit, m = 0.9000 and δm = 0.05774. The uncertainty should be rounded to 0.06, which means that the slope must be rounded to the hundredths place as well:

m = 0.90 ± 0.06

If the above values have units, don't forget to include them.

### Constants

If an expression contains a constant, B, such that q =Bx, then:

You can see the the constant B only enters the equation in that it is used to determine q.

Example:

F = mg = (20.4 kg)(-9.80 m/s2) = -199.92 kgm/s^{2}

δF/F = δm/m

δF/(-199.92 kgm/s^{2}) = (0.2 kg)/(20.4 kg)

δF = ±1.96 kgm/s^{2}

δF = ±2 kgm/s^{2}

F = -199.92 kgm/s^{2} ±1.96kgm/s^{2}

With the answer rounded to 3 sig figs:

F = -200 kgm/s^{2} ±2kgm/s^{2}

### Addition and Subtraction

Although it may seem intuitive to simply add the uncertainties of numbers that are added together, this can give misleading results. For example, if some number A has a positive uncertainty and some other number B has a negative uncertainty, then simply adding the uncertainties of A and B together could give an answer with an uncertainty of zero. Two numbers with uncertainties can not provide an answer with absolute certainty! To fix this problem we square the uncertainties (which will always give a positive value) before we add them, and then take the square root of the sum. If q is the sum of x, y, and z, then the uncertainty associated with q can be found mathematically as follows:

### Multiplication and Division

Finding the uncertainty in a product obtained by multiplying two numbers together is similar to the method used for addition and subtraction. However, we want to consider the ratio of the uncertainty to the measured number itself. This ratio is called the **fractional error**. This ratio is very important because it relates the uncertainty to the measured value itself. Consider a length-measuring tool that gives an uncertainty of 1 cm. If you measure the length of a pencil, the ratio will be very high. If you're measuring the height of a skyscraper, the ratio will be very low. Mathematically, if q is the product of x, y, and z, then the uncertainty of q can be found using:

Since division is simply multiplication by the inverse of a number, then this multiplication rule can be extended to give:

### Exponents

If a number is raised to a power, then the uncertainty is found to be:

For example:

L = A^{-1/2} where *A* = 44 m^{2} ±2m^{2}

L = (44 m^{2})^{-1/2} = 6.63 m

δL/L =(0.5)δA/A

δL/6.63 m =(0.5)(2m^{2})/(44 m^{2})

δL = 0.15 m

L = 6.63 m ±0.15m

With the answer rounded to 3 sig figs:

L = 6.6 m ±0.2m

### General function of one variable

The uncertainty of a function of one variable can be found with the following:

Example: Measure the constant velocity of a cart rolling on a 50.0 cm track. The time is measured to be 1.32 seconds with an uncertainty of 0.06 seconds.

Since the velocity is the change in distance per time, v = (x-x_{o})/t.

Setting xo to be zero, v= x/t = 50.0 cm / 1.32 s = 37.8787 cm/s.

The derivative, dv/dt = -x/t^{2}.

Using the equations above, delta v is the absolute value of the derivative times the delta time, or:

Uncertainties are often written to one significant figure, however smaller values can allow more significant figures. In this example, the 1.72 cm/s is rounded to 1.7 cm/s. Since uncertainties are used to indicate ranges in your final answer, when in doubt round up and use only one significant figure.

Since the uncertainty has only one decimal place, then the velocity must now be expressed with one decimal place as well. Knowing the uncertainty in the final value is the correct way to officially determine the correct number of decimal places and significant figures in the final calculated result. The final result for velocity would be v = 37.9 + 1.7 cm/s. This tells the reader that the next time the experiment is performed the velocity would most likely be between 36.2 and 39.6 cm/s.

### General function of multivariables

For a function q which depends on variables x, y, and z, the uncertainty can be found by the square root of the squared sums of the variables uncertainty multiplied by the partial derivative with respect to that variable, or:

Example: The previous example measured the constant velocity of a cart rolling down a known track length with an uncertainty in the measured time. If we now have to measure the length of the track, we have a function with two variables. It will be interesting to see how this additional uncertainty will affect the result!

The measured track length is now 50.0 + 0.5 cm, but time is still 1.32 + 0.06 s as before.

As in the previous example, the velocity v= x/t = 50.0 cm / 1.32 s = 37.8787 cm/s.

The derivative with respect to t is dv/dt = -x/t^{2}. The derivative with respect to x is dv/dx = 1/t.

So, rounding this uncertainty up to 1.8 cm/s, the final answer should be 37.9 + 1.8 cm/s. As expected, adding the uncertainty to the length of the track gave a larger uncertainty to the final answer. Now a repeated run of the cart would be expected to give a result between 36.1 and 39.7 cm/s. Also, notice that the units of the uncertainty calculation match the units of the answer.

### Error Propagation in Trig Functions

Rules have been given for addition, subtraction, multiplication, and division. Raising to a power was a special case of multiplication. You will sometimes encounter calculations with trig functions, logarithms, square roots, and other operations, for which these rules are not sufficient.

When propagating error through an operation, the maximum error in a result is found by determining how much change occurs in the result when the maximum errors in the data combine in the worst possible way.

**Example**: An angle is measured to be 30°: ±0.5°. What is the error in the sine of this angle?

Solution: Use your electronic calculator. The sine of 30° is 0.5; the sine of 30.5° is 0.508; the sine of 29.5° is 0.492. So if the angle is one half degree too large the sine becomes 0.008 larger, and if it were half a degree too small the sine becomes 0.008 smaller. (The change happens to be nearly the same size in both cases.) So the error in the sine would be written ±0.008.

The size of the error in trigonometric functions depends not only on the size of the error in the angle, but also on the size of the angle. A one half degree error in an angle of 90° would give an error of only 0.00004 in the sine.