IB Physics 1.2: Uncertainties and Errors

12 min readJun 24, 2022

Preface

This is the topic where we start getting a bit more mathematical and scientific with respect to definitions. Most of this article will focus on the tools used by experimental physicists.

These are the formulas from this topic that are available on the data booklet:

All the formulas available for topic 1.2

They might look slightly intimidating and confusing at the moment but by the end of this article, you should be able to understand them.

1.2 Uncertainties and errors

Before we get into the meat of this topic, we need to first understand what errors are, and the different types of errors we can encounter in an experiment.

The first error is the systematic error. Systematic errors are errors that don’t happen randomly, but instead they occur due to an error with the experiment itself. Another way to think of systematic errors is that they are errors that occur in the same direction. What does this mean? Let’s say we were measuring the mass of some objects using a balance. It’s common experimental practice to set the balance to 0 before we place any object. However, if the balance reads 0.1 kg before any object is placed, all the masses we measure will be 0.1 kg higher than the object’s actual mass. In this case, the error occurs in the same direction, ie all the measured masses are 0.1 kg greater than the object’s actual mass.

Some causes of systematic errors include

An incorrect calibration of any instruments used in the experiment (Eg: Not making sure the balance starts at 0)
An improper use of the instrument itself

Digital balance used to measure mass — Science Enteprise

The next type of error is the random error. True to its name, the random error occurs… randomly! This results in the various measurements being spread out. Systematic errors result in errors occurring in the same direction while random errors can result in errors occurring in any direction

Random errors mainly occur due to incorrect experimental technique. For example, when measuring the length of an object using a metre ruler, it is recommended to measure the length perpendicularly to avoid parallax. However, if during the course of the experiment, the researcher measures the lengths at different angles, there will be a random error.

Another example for random errors is while measuring the volume of a liquid in a measuring cylinder. It is common to measure volumes from the lower meniscus. However, if readings are taken from different points of the meniscus, an error can occur.

Lower meniscus of a liquid — Science Notes

Image illustrating the difference between systematic and random errors — Penn State

Accuracy and Precision

Now that we’ve gone over the different errors, we need to go over what the errors can actually affect in an experiment.

Errors can affect either the accuracy, precision, or both in an experiment.

Hold on, what’s the difference between accuracy and precision? It’s very likely you’ve used these words interchangeably, however, they are two completely different things from a scientific point of view.

Measurements are said to be accurate if it is close to an agreed upon value of that measurement (Eg: The mass of an adult is 75 kg)

Measurements are said to be precise if the measurements are close to each other (Eg: The measured masses of three adults are 75.0 kg, 75.2 kg, 75.1 kg)

Here is a common diagram that illustrates the difference:

Hence, along with the image from the errors section, we can see that systematic errors affect accuracy while random errors affect precision. Random errors can also affect accuracy depending on the magnitude of the error.

Experimental practice

In general, we take multiple measurements of the same thing in order to ensure our measurements are reliable. What this is essentially does is it helps combat random error and helps us identify if there is any systematic error in our experiment.

Typically, we tend to take averages of our measurements as well. For example, if we found out that the time it takes for an olympic runner to run 100 metres is 9.8 s, 9.6 s, and 10.2 s, we can average this out to find the approximate time it would take an olympic runner to run 100 metres.

This is done by calculating the mean

This formula looks very complicated, partly because it uses sigma notation. However, this essentially tells us to add up all our measurments and divide by the number of measurements we have. Hence, the average time in the example above would be about 9.87 s, rounding up to 1 s.f we get 10 s.

Another measurement we can use is the standard deviation. As it’s not part of the syllabus, I won’t be going into it.

Representing uncertainties (Propogation of uncertainties)

Whenever we do an experiment, we can truly never be certain of our measurements. This is because instruments will simply never be good enough. We can get really good approximations for our measurements but we can never be certain.

So how do we factor this uncertainty into any calculations we do? The first step would be to actually represent this uncertainty mathematically.

The way we do that is in the form:

a represents the mean of our measurements while Δa represents the absolute uncertainty. Before I go into what that means. Let’s actually understand what this represents.

Let’s say the mass of an object with uncertainty is (7.0 +/- 2.0) kg. (We generally wrap our expression in brackets and put the units outside), This mean that the mass of our object could be anywhere in the range of 5.0 kg (7–2) to 9.0 kg (7+2).

Here is another way of visualising uncertainties

As you can see, it is pretty obvious that the length is greater than 2.7, and less than 2.8. However, it’s not entirely clear how much more than 2.7 it is. This image might make it look like it’s 2.75 but with just the naked eye in an experiment, it’s really hard to distinguish between 2.73, 2.74, and 2.75. This has to be factored into an experiment and is precisely why we need to use uncertainties.

With that being said, a in the equation is easy to calculate, it is simply the mean of our dataset. We can calculate Δa using the equation:

We subtract the smallest value in our dataset from the largest value and divide by 2. Convention generally states that Δa is rounded to 1 s.f. Hence, we round a off to the same number of decimal places as Δa.

Let’s illustrate this with an example:

The following represent the measured lengths of a rod in m.

8.2, 8.5, 8.0, 8.3, 8.4, 8.6

a can be found by calculating the mean. This gives us about 8.3. Δa can be calculated by (8.6–8.0)/2 = 0.3. 0.3 is already rounded to 1 s.f so we have less work. 0.3 has only 1 decimal place, 8.3 also only has 1 decimal place. Hence, our work is basically done.

The uncertainty of measured lengths can thus be given by:

If we have values that are 2 or 3 digits long, we might want to represent it using scientific notation.

For example, if the uncertainty was

We can rewrite this as

Fractional and percentage uncertainties

We found out that Δa represents the absolute uncertainty or the actual uncertainty of the measurement. However, it’s pretty meaningless without context. If I told you the absolute uncertainty of a measurement was 0.8 cm. Without context, you wouldn’t know if it was high or not. If the actual measurement was 100 cm, an uncertainty of 0.8 would almost be ideal. However, if the measurement was 1 cm, an uncertainty of 0.8 is very high.

So how do we get context in the first place? We can do this by representing our absolute uncertainty as fractional and percentage uncertainties.

Fractional uncertainties can be calculated as follows:

All we do is divide our absolute uncertainty by the mean.

The percentage uncertainty can be calculated using:

All we do is multiply our fractional uncertainty by 100%.

Hence using the example of of our measured lengths where the uncertainty was (8.3 +/- 0.3) cm, the fractional uncertainty would be 0.3/8.3 or about 0.04. Hence, the percentage uncertainty would be 4%.

Notice how if I told you the uncertainty in my measurement was 0.3, it wouldn’t really be helpful because there was no context. However, if I told you that my percentage uncertainty was 4%, you’d get a feel for how reliable the measurements are.

Calculations with uncertainties — addition and subtraction

Now that we know how to represent uncertainties, it’s time we learn how to actually incorporate them into calculations. This is where the data booklet will be handy.

If we are doing a calculation which involves addition or subtraction, we simply add or subtract the means as usual. However, we add the absolute uncertainties, even if we’re subtracting the values.

Let’s say we want to add

The result would be (8.2 + 2.4) +/- (0.2 + 0.8) which would give us

What if we wanted to do the following calculation?

The result would be (7.9–1.8) +\- (0.4 + 0.2) which would give us

Remember, we still add the absolute uncertainties even if we are subtracting the values.

Calculations with uncertainties —multiplication and division

If we are multiplying and dividing two values, we carry out the multiplication and division as usual. However, finding the absolute uncertainty is not as simple. Instead, the fractional uncertainty of our measurement is equal to the sum of the fractional uncertainties of our values. We then rearrange the equation to get our absolute uncertainty.

Hold up, that sounds complicated. Let’s look at an example

Let’s say that the following represent the lengths of a rectangle:

Now we want to calculate the area A of the rectangle, along with its uncertainty

The fractional uncertainty of the area is represented by:

We can calculate the value of A easily, we simply multiply 5.2 and 8.4 to give us 43.68.

Hence

We can therefore calculate ΔA by multiplying both sides of the equation by 43.68. Hence ΔA is approximately 2.201. Remember that absolute uncertainties have to rounded to 1 sf. Hence, ΔA is 2. Now 2 has no decimal places, hence round A to 1 sf as well to get 40. Therefore, the area would be

The steps are the same as follows for division, except we divide the two values instead of multiplying them to find the denominator.

Calculations with uncertainties — powers and roots

If we have a power or a root (We must rewrite the root as a power), we simply multiply the absolute value of the power with the fractional uncertainty. Let’s suppose the length of a square was

The area of the square A = l². Therefore

A would just be 2.8² = 7.84. Hence, we can rearrange for ΔA to get 2.24. This has to be rounded to 1 sf which is 2. Hence, 7.84 is rounded to 8. Therefore the area is (8 +/- 2) cm²

Note: We do not take constants like π or e into account for uncertainty calculations.

What if we’re given the uncertainty of an area and we have to solve for the uncertainty for a measurement?

In this case, simply rearrange the equation in terms of the measurement and apply your uncertainty rules as follows. Let’s illustrate with a past-paper question:

The first step is to rearrange for R which gives us

Hence, using our uncertainty rules we get:

Remember, percentage uncertainties are just fractional uncertainties into 100. Hence, if we divide each percentage uncertainty in the question by 100, we get:

This calculation gives us 0.08. This is the fractional uncertainty of R. Hence, the percentage uncertainty is found by multiplying by 100% which gives us 8%. Hence, our answer is D.

Best-fit lines and error bars

We’re done with the meat of this chapter. The last bit deals with how we represent uncertainties on graphs.

This is done via an error bar

Error bars represent the following

When we graph points, we are not able to draw a perfect line between the points. Hence, we have to draw a line of best fit. This essentially allows us to extrapolate our data and make predictions on future measurements based on the trend observed.

To find the uncertainty in the gradient of a line of best fit, we draw “maximum” and “minimum” lines through your maximum and minimum error bars and simply use the formula to calculate absolute uncertainty.

Maximum and minimum lines in a graph with error bars

Linearisation

When we draw graphs, we sometimes don’t get a perfect straight line either. This makes it harder to interpret results. Hence, we often times have to change what we plot on our x and y axis to give us a straight line. For example, if we plotted ln(y) against x, we might get a straight line. Why does this work? Let’s say the graph we get is in the form: