4. Error analysis

Cécile Fradin

4. Error analysis

Performing error analysis is one of the more important skills needed when analyzing and interpreting the results of experiments. Below is a reminder of the basics of error estimation and propagation.

4.1 Estimating errors on measured quantities

If you are measuring a quantity during an experiment, there are at least two ways you can estimate the error associated with that measurement:

1. You can consider that the error on your measurement is equal to the precision of the instrument you are using to perform that measurement.

For example, if you measure a length $L$ with a ruler that has 1 mm markings, then the precision of your measurement cannot be better than about 0.5 mm, and you can consider that the error on the value of $L$ you measured is $\dd L$ = 0.5 mm.

As another example, if you measure the length of a line $L$ in a digital image with pixel size of 5 μm, the precision on that length cannot be better than the size of the pixels, so you can estimate that $\dd L$ = 5 μm.

It should be noted that using the precision of an instrument as the error means that all other sources of errors are neglected. For instance in the examples above, we are neglecting possible errors having to do with the placement of the ruler or the drawing of the line on the digital image.

2. A very different way of determining error which does not require us to explicitly state the source of this error is to repeat our measurement a number of times, then to calculate the average and standard deviation of this set of measurements (for example using Excel). We can then equate the error on the measurement to the standard deviation.

For example, you could measure the same length five times, and find values of 53, 52, 54, 53 and 49 mm. The average of these five measurements is 52.2 mm, and the standard deviation is 1.9 mm. The error on that measurement can be considered to be 2 mm.

You can in fact use both ways to estimate errors (make an estimate based on instrument precision, and also repeat your measurement and look at the standard deviation). In that case, you should use the largest of your two error estimates.

For example, if you measure a length using a ruler with 0.5 mm precision, but also find that the standard deviation of repeated measurements is 2 mm, then report the error as 2 mm.

4.2 Propagating errors

Often, quantities we are interested in are calculated from measured quantities. We then need to propagate the error estimated for the measured quantities to the calculated quantities, and there are simple rules we can use to do that.

A good introductory reference for understanding uncertainties and error propagation is J.R. Taylor’s Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements. Below, we summarize some simple error propagation rules.

1. If the calculated quantity is the sum or the difference of other quantities, then we can just add up the absolute errors of these quantities.

For example, if we measure two lengths, $L_1$ = (39 ± 2) mm and $L_2$ = (7 ± 2) mm, then their sum would be reported as $L = L_1 + L_2$ = (46 ± 4) mm, where the error on $L_1$ and $L_2$ have just been added.

If we wanted to calculate the difference between these two lengths, it would be reported as $L' = L_1 - L_2$ = (32 ± 4) mm, where again the error on $L_1$ and $L_2$ have been added (not subtracted!).

This concept can be extended to more complex calculations, as long as only addition and subtraction are involved. For example if $L = L_1 - L_2 - L_3 + L_4$ , then $\dd L = \dd L_1 + \dd L_2 + \dd L_3 + \dd L_4$

Note that the absolute error ( $\dd L$ ) on a quantity ( $L$ ) has the same dimension (unit) as the quantity itself. If $L$ is a length, then $\dd L$ is a length too, and therefore when we add absolute errors we are adding the same type of quantities (quantities with the same unit).

2. If the calculated quantity is the product or the ratio of other quantities, then we can just add up the relative (or percent) errors of these quantities.

For example, if we measure two lengths, $L_1$ = (39 ± 2) mm and $L_2$ = (7 ± 2) mm, and then calculate their product $A = L_1 \times L_2$ = 273 mm², we need to consider relative errors to find $\dd A$ . We have:

$\dd L_1 / L_1$ = 2/39 = 0.051 = 5.1%, and $\dd L_2/L_2$ = 2/7 = 0.286 = 28.6% .

Thus the relative error on $A$ is:

$\dd A/A= \dd L_1 /L_1+ \dd L_2/L_2$ = 0.051 + 0.286 = 0.337 = 33.7% .

The absolute error on $A$ is then \dd A $= 0.337 \times A$ = 92 mm². The result would be reported as $A$ = (270 ± 90) mm².

Similarly if we were calculating the ratio between these two lengths, we would have:

$r = L_1/L_2 =$ 5.57 and $\dd r/r = \dd L_1/L_1 + \dd L_2/L_2$ = 0.337.

Thus, $\dd r$ = 5.57 x 0.337 = 1.88. This result would be reported as $r$ = 5.6 ± 1.9.

This concept can be extended to more complex expressions involving only multiplications and divisions. For example if: $N = 4\pi R2S$ then $\dd N/N = \dd (4)/4 + \dd \pi/\pi + \dd R/ R + \dd R/R + \dd S/S.$ Since “4″ and “ $\pi$ ” are constant, $\dd 4 = \dd \pi = 0$ , so in the end: $\dd N/N = 2 \times \dd R/R + \dd S/S.$ And the absolute error on $N$ is $\dd N = N \times (2 \times \dd R/R + \dd S/S).$  Note that relative errors are dimensionless, so it is ok to add the relative errors of quantities that themselves have very different units.

3. If you have a mix of addition, subtraction, multiplication and/or division, you might have to propagate the error step by step.

For example, imagine you are calculating the quantity $y = (a+b)/c$ . You should first calculate the (absolute) error on $(a+b): \dd (a+b) = \dd a + \dd b$ . Then you can calculate the relative error on $y: \dd y/y = \dd (a+b)/(a+b) + \dd c/c = (\dd a + \dd b)/(a+b) + \dd c/c.$ In the end, the (absolute) error on $y$ is given by: $\dd y = y [(\dd a + \dd b)/(a+b) + \dd c/c].$

Note: A more precise way of adding errors can be used if the added errors are independent from each other, called adding errors in quadrature. If you are familiar with that way of adding up error, you are welcome to use it (See the Physics 1CC3 Lab Manual for reference).

4.3 Reporting quantity and errors

When reporting quantities and associated errors remember to pay attention to significant digits! The two rules you need to remember are:

1. Keep only 1 or 2 significant digits for the error.

For example, if you calculate your error to be $\dd L$ = 0.23492334001 mm, just report it as being $\dd L$ = 0.3 mm or $\dd L$ = 0.24 mm (it is a good idea to round your errors up, not down, in order to avoid underestimating it).

2. Adjust the number of significant digits such that the last digit used is the same as the last digit in your error.

For example, if you measure a length of $L$ = 341.2842523524 mm with an error of $\dd L$ = 0.3 mm, then report it as $L$ = (341.3 ± 0.3) mm. If you kept your error as $\dd L$ = 0.24 mm then you should write $L$ = (341.28 ± 0.24) mm.

If you measure a length of $L$ = 0.7984834 mm with an error of $\dd L$ = 0.3 mm, then report it as $L$ = (0.8 ± 0.3) mm.

4. Error analysis

4.1 Estimating errors on measured quantities

4.2 Propagating errors

4.3 Reporting quantity and errors

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