Estimating uncertainties in raw data

According to the subject guide  "an attempt should always be made to quantify uncertainties" in raw data, this implies that the student does  not necessarily have to get it right but that they have made a reasonable attempt. There are some simple rules that students can apply but I prefer them to use judgement, provided that they justify their value then it is acceptable, anyway here are two rules that form a starting point.

 

Two simple rules

Digital devices

The uncertainty in a digital readout is given by the least significant digit. The example shown here is the time taken for a cart to pass between two photogates. The least significant digit in the display is .0002 so the uncertainty is ±0.0001s

Analogue devices

The uncertainty in a measurement made with an analogue device such as a ruler or an analogue voltmeter is 1/2 the smallest division of the scale. So for a typical meter rule with mm divisions the uncertainty would be ±0.5mm.

Applying these rules is perfectly acceptable but sometimes the value obtained is unreasonable since the uncertainty depends on the way the device is used not just the size of the scale. If, for example, a meter rule is used to measure the length of a piece of paper then the uncertainty could be less than 0.5mm, especially if the edges of the paper line up with the lines on the ruler. However if we use the same rule to measure the maximum height of a bouncing ball then the uncertainty is going to be much bigger. So the method will influence the uncertainty and that where judgement comes in.

 

Using Judgement

I prefer students to use their judgement rather than the simple rules above but whatever method they use they must state how they arrived at the value given. Here are some factors to consider:

Using a ruler

• Is the edge straight, curved or uneven?
• Is the object to be measured stationary or moving?
• Would it help to use a magnifying glass to view the scale?
• Is there a parallax error?
• If measuring the position of an image or the node in a string how well can you judge its position?
• Is there an error at both ends of the ruler?

Using a voltmeter or ammeter

• Does the reading fluctuate?
• Will changing the range reduce the uncertainties?
• What is the manufactures uncertainty of the instrument?
• If holding the contacts by hand does the pressure applied change the reading?

Using a sensor connected to a datalogger

• Does the position of the sensor change the reading?
• If reading a point from a graph, how well can you judge the point from the display?
• If using the gradient of a line does the part of the line you choose change the gradient?
• If reading from a digital display, how many decimal places is reasonable?

 

Why do measurements vary?

There are two reasons why a measurement might vary, one is because the instrument does not give the same reading every time and the other is because the quantity itself is varying.

Device

If we measure the mass of a steel ball on a top pan balance that measures to the nearest 0.1g then we may find that the reading varies between 52.3g and 52.4g, this could be because the actual value is somewhere between the two and the electronics in the balance is causing the value to flicker between one and the other. This is an instrument uncertainty.

Method

If we measure the time taken for a ball to drop 0.4m we might get measurements ranging from 0.2781s to 0.2828s. This is not due to some random fluctuation in the clock but it's because the ball does not drop the same way every time, this is an uncertainty in the method.

If we were to measure the width of a piece of an A4 paper with a ruler we would get a value of 21.0 ±0.5 cm (actually i'd say ±0.1cm since the edge of the paper lines up very closely to the line division on the ruler). if we repeat this measurement we get the same every time. This is because any change in the paper is so small that it is not detected by the instrument used to measure it (the ruler is not sensitive enough)

 

Repeating measurement to find uncertainty

One reason for repeating the measurement of a quantity is so we can take an average to reduce the uncertainty, but to reduce the uncertainty significantly would require a lot of measurement and we rarely have time for more than 5 repeats. it also helps the students to understand that measurements are not exact. A simple (and acceptable) way of finding the uncertainty is to take the half difference between the maximum and minimum values (½ the range) and use this as the uncertainty in the average. The problem is that this does not take into account the fact that taking an average actually reduces the error so students often get exaggerated error bars. Awareness of this problem is enough, detailed analysis is beyond the scope of the course.

There has been a lot of debate about uncertainties on the OCC and there are many different ways to do error analysis, in my experience taking repeated measurements gives the students an appreciation that there are uncertainties in measurement and that's all that we are after at this level.

Example

A common experiment is to measure the acceleration of gravity by measuring the time taken for a ball to drop from varying heights. If you have a custom built set up then repeated measurements take little time so it is realistic to expect students to repeat each of 10 different heights 5 times (this may not always be possible).if we look at these results we can see that the times vary quite a lot from one run to the next. Sometimes there might be a rogue result due to a failure to reset the clock or something. Students should know the difference between a mistake and an uncertainty.

In this case the student did not repeatedly measure the height. If they had it wouldn't have changed but this doesn't mean the uncertainty is zero it is the uncertainty in the scale (±1mm).

In excel it is a simple matter to calculate the uncertainties in each value in an extra column. This is best done by using the Excel formulae Max() and Min() these select the max and min values from the range selected. For example if you type =Max( C2:C6) in a cell then press return, the largest value in the range will then appear in the cell. You don't actualy have to type C2:C6 just highlight the range you are interested in with your mouse and close the bracket. The formula to calculate the uncertainty would therefore be =(Max(C2:C6)-Min(C2:C6))/2

 

presenting uncertainties in the raw data

As can be seen in the table above each header has an uncertainty, this is essential for a complete score according to IB criteria aspect 1. Notice that these values are simply the estimates arrived at from considering the apparatus not the values calculated from the range of data, this counts as processed data so comes into aspect 2

 

 

Processing uncertainties

If you are simply going to graph the raw data (e.g. s against t) then you simply use the estimated uncertainties  for the error bars. However if you are going to process the data (e.g. find t²) then you will need to propagate the uncertainties. To explain why this is the case it is best to use an example:

Here is some made up data representing times measured to an uncertainty of ±1s

The columns should be self explanatory. The idea is to find the error in t² by finding half the difference between is maximum and minimum possible values. Notice that if you do this then the % uncertainty is doubled, this is an alternative way to deal with uncertainties but I find that simply taking ½(max-min) is easier to understand and can always be applied.

If you square a number that is less than 1 it gets smaller, the uncertainty also decreases as shown in the next table.

This sometimes confuses students since they expect the uncertainty to be bigger. The % uncertainty is bigger but the absolute value is less.

 

Practice makes perfect

The best way to learn about uncertainties and how to deal with then is by doing experiments. The first 4 practicals that I use with my students contain detailed instructions on how to estimate and process uncertainties. I they do these then they should have a good idea what to do.

Inclined Plane

Measuring g

Hooke's law

Newton's second law