Uncertainties - Physics A-level & GCSE

Here it is, the definitive guide to how to deal with uncertainties. Okay, so let's say that we have a beaker of water and we wanna measure the temperature. We're gonna need a thermometer. Here's my thermometer up in here. And we wanna see how far up the mercury or usually the alcohol has risen up the thermometer. And then we need to check against the scale where it's at. All right, so let's blow this up a little bit. Here's my thermometer. Here's the bit of the scale that the liquid in it has risen to measuring room temperature. And we can see that the liquid has gone past 21 degrees but it hasn't reached 22 degrees. Now, when you take this reading and I use the word reading not measurement for a good reason, you have to make a judgment call as to whether it's closer to 21 or 22 degrees. Now, the resolution of this thermometer is one degree Celsius. In other words, that's what each line on the scale represents. It's the smallest change in number on the scale. Now, in physics, whenever we take readings, we wanna be as accurate as possible. So if our resolution is only to one degree Celsius, that means that we can't say for sure this line is about four tenths of the way up here. Therefore, I'm gonna say the temperature is 21.4 degrees C. We can't do that because we don't have lines in between 21 and 22 degrees. Let's say that we know for sure what the temperature is closer to, 21 or 22 degrees. Temperature equals 21 degrees Celsius. Even though we can clearly see that it's gone past 21 but hasn't gone past halfway. This line can go past 21 all the way up to halfway before we have to say, actually, it's now 22. So it's only when we get to 21.5 that we have to say, actually, no, the number's actually gonna be different. So what can we say for sure that the temperature is 21 degrees plus or minus 0.5 degrees Celsius? And that is half the resolution. With this 0.5 Celsius, we're saying that we can tell when it's closer to one or the other, but we can't say for sure exactly what temperature it is to the nearest, say, 10th or 100th of a Celsius. This is what happens when you take any reading off a piece of equipment. No piece of equipment has an infinitely small resolution. So that means that there's always gonna be an uncertainty in your reading, but that's okay. Now, what we say about a thermometer is that there is no zero error. What do I mean by that? When the piece of equipment was made by the manufacturers, they calibrated it very carefully and they made sure that when the liquid hits zero degrees on here, it absolutely is zero degrees. And we're gonna say that we trust them with that. So there is no uncertainty with what they did when they calibrated it. However, there are other times when there might be zero error. An example of this is when you use a ruler. There's zero millimeters, here's 10 millimeters, then 20, 30, 40, and so on. Now, more often than not, when you have a ruler, the resolution is one millimeter. You can tell just by looking at the scale. Let's blow this up a little bit. What do you have to do when you measure something though? Let's say I wanna measure the length of my SD card here. In order to do that, I'm the one who needs to actually line it up with the zero mark. And this is where there is zero error. And there is a difference between lining something up at a zero point and measuring where it goes to between two points later on, but this is what your exams are going to expect. If there's a zero error here, then we have to say that there is a plus or minus 0.5 millimeter error there. What about my actual reading then? Well, I can see that it comes to here, and that's slightly closer to the three millimeter mark than the two millimeter mark. So therefore, my measurement, we call this kind of thing a measurement because we're lining stuff up to begin with as well. Of course, that's a length. It's gonna be equals to three millimeters, plus or minus, because there's a zero error, and we have to include the uncertainty at this point here as well. Adding those up, we actually have to say that the uncertainty is plus or minus one millimeter. And that's the same as the resolution. Confused yet? I know I am. So let's try and clear things up a little bit and make it simpler. So we had readings, and we have measurements. What's the difference between the two? For a reading, there's no zero error because you're not having to line things up to begin with. You're just looking at a number on the scale, and that's what your reading is. Measurements, there is a zero error because you're lining up with zero or another number, in order that you can then take your measurement. So what does this mean ultimately? And this is the important bit. The uncertainty, we can call this the absolute uncertainty, is half the resolution. With measurements, the uncertainty is equals to the resolution. That's because we have half the resolution at the beginning, half the resolution at the end, so therefore, the uncertainty overall is the resolution. So we set examples of this. We've seen a thermometer already. A measuring cylinder, because you don't have to line anything up with the measuring cylinder. A balance, voltmeters or ameters, pressure gauge, you get the idea. So what about examples with this? We set a ruler, stopwatch, because you're the one who has to press the button to begin with as well. You might argue that there's a greater uncertainty there, but we can still get away with saying that the uncertainty is half a second, then half a second when you stop it, so altogether one second. Micrometers and vernier calipers. Again, I'm not so sure about these because these should technically be calibrated before they get to you, but according to the guidelines, yes, even when it comes to micrometers and vernier calipers, we have to say that the uncertainty is the resolution. Protractor as well, because you have to line up zero degrees and analog meters as well. So ameters and voltmeters that have this gauge here because you calibrate that yourself with a little knob. So technically I should say digital voltmeter and ameter over here. So just be careful when it comes to questions. It all depends on the piece of equipment that you are using. Are you having to line it up or not? What about if you're taking a lot of repeat readings or measurements in an experiment to then find an average to get a more reliable result? In that mean value, the average, there is gonna be an uncertainty. So how do you calculate it? Now, let me just tell you that there are infinitely better ways to calculate an uncertainty in a mean, but it's a nice simple way of doing it and it's good enough for A-level. This is what they want you to do. So let's say that we have lots of readings for speed. What's the mean of these? Add them all up. Don't forget to press equals first because otherwise you'll just be dividing the last one. Divide that by five and we end up with 4.98. And because we need to give means to the same precision as our values, that's gonna be 5.0 meters per second. However, there is an uncertainty in this because our results were all over the place. So how do we find the uncertainty in a mean? All we do is take the range and divide it by two. So let's find our biggest number. It's 5.3. Take away our smallest number, 4.6. Divide that by two. So that gives us 0.7 divided by two. So that gives us 0.35. But as per usual, because we need to give our uncertainty to the same number of decimal places as our value, that makes that 0.4. So average speed is 5.0 meters per second plus or minus 0.4. What if you don't actually know what piece of equipment you're using and you're just given a number? So let's say that we're given a PD of 5.27 volts. We're not told what the resolution of the voltmeter is. However, but we're given the PD to two decimal places. Therefore, we can assume the uncertainty is one of the last significant figure. Therefore, what would this be? Plus or minus 0.01 volts. Let's say that we're given a speed of 28 meters per second. Again, we can assume that whatever's measuring the speed did it to just one meters per second. So we can say that the uncertainty is just one meter per second. That is how you can infer what the uncertainty is without knowing the resolution of the piece of equipment that took the reading or measurement. So what about if we want to combine uncertainties? And this is where it gets complicated. Let's say that we have three displacements, S1, S2 and S3. So Bob goes for a walk and he goes a certain distance this way, certain distance in the same direction and then comes back a little bit as well. So the first displacement that he walks is 10 meters but he's a little bit unsure as to whether he walked 10 exactly. So he's saying that his uncertainty in his displacement is one meter. Forget about how we measured it. We're just saying that that's the uncertainty for now. He actually walks a little bit further and again, he's a little bit unsure as to whether he walked five exactly. So he's saying plus or minus one meter. And then he walks back in the opposite direction. So we're gonna say it's a minus number for this one and that's eight meters, but he really wasn't concentrating. So the uncertainty is plus or minus two meters. So the total displacement is gonna be 10 plus five minus eight. That gives us seven meters. Now you might be thinking that because we're actually taking away a number here, that means that we take away the two meters. And so therefore we're gonna end up with an uncertainty of zero. However, this minus eight is being added. It just happens to be a minus number. So just because we're taking away a number doesn't mean the uncertainty goes down. It actually keeps on increasing. So we add the two and so we end up with an uncertainty of plus or minus four meters. What does that mean? It means that we always add up absolute uncertainties or absinthes for short when adding or subtracting. We never take them away. Bob then wants to find out how fast he is. So he wants to know his speed. And we know speed is measured by taking distance over time. Bob gets his friend Beth to time him and measure the distance. The distance that she measures him running over is 20 meters but she's not too sure about distances either. So there's an uncertainty of plus or minus one meter. She then times him and she times a time of 4.5 seconds. But she knows her uncertainties and she knows that because measuring time is a measurement, that means that there must be an uncertainty of 0.1 seconds in that. So how fast did Bob run in meters per second? 20 divided by 4.5 gives us 4.4. It is 4.4 recurring, so we could say 4.44444 but we know that we have to give values to the same precision as the least precise piece of data that we have. Two sig figs though, we can tell that with the one meter, two sig figs, so this is two sig figs as well. So 4.4 meters per second. However, because there was an uncertainty in the distance and the time, there's gonna be an uncertainty in our speed as well. Question is, what is that? Now you might be thinking that because we divide these numbers, we're gonna be dividing the uncertainties as well. Not so, we never divide uncertainties by each other. As per usual, we add them up. But here's the problem. How can we add up an uncertainty in meters and an uncertainty in seconds? We can only add up absolute uncertainties when they are the same unit. When we times or divide values, we have to add up percentage uncertainties. By converting these into a percentage of the actual value, we're adapting them so they can be added up even though we are dealing with completely different measurements. So what is the percentage uncertainty in D? Well, that's going to be one divided by 20. Let's times that by 100 to turn it into a percentage. That's 5%. So plus or minus one meter is 5% of 20. Let's find out what the percentage uncertainty of this is. 0.1 divided by 4.5, times that by 100 as well, and we end up with 2.2%. When it comes to significant figures for percentages, you don't need to worry about it too much. It's with the absolute uncertainty that that is important. So we have an uncertainty of 5% here, an uncertainty of 2.2% here. Therefore, our total uncertainty is 7.2%. So that means that our speed is 4.4 meters per second. Yep, but that is plus or minus 7.2%. But that's not so useful because we actually wanna know what that is in meters per second. So we have to turn that back into an absolute uncertainty. All we wanna do is find out what 7.2% of 4.4 is. So therefore, we do 4.4 times the percentage uncertainty, but in decimal form, so 0.072. And that gives us this number here, but we need to make sure it's the same as our value, so that's 0.3 meters per second. Let's say that Bob was doing some fancy calculations, so he wanted to know what V squared was. So 4.4 squared, that gives us 19.4. I'm just gonna leave it as three sig figs for now. And yeah, that's gonna be the unit meter squared seconds to the minus two, but we're not so concerned about that. However, what is the uncertainty in this value going to be? Well, we know that in V, there's an uncertainty of 7.2%, and another one, 7.2%. Therefore, in order to find out the percentage uncertainty in a value that's been squared, all we have to do is add the percentage uncertainty to itself. So in other words, we double the percentage uncertainty. That's 14.4%. Hey, let's turn that back into an absolute uncertainty. So that's 19.4, but we wanna know what 14.4% of that is, so we need to put that in in decimal form, and that gives us plus or minus 2.8. There is one more thing. You could come across a situation where you already have the squared number with the uncertainty. Let's say that we have height squared for some reason, and we have a value of 50 plus or minus one. Yeah, that's gonna be meters squared. What's the percentage uncertainty in this? Well, it's one divided by 50 times 100, so that's 2%. So therefore, what would H be? Well, it would be the square root of 50. That gives us 7.1. But what would the percentage uncertainty be in this? Well, in H squared, there was a percentage uncertainty of 2%. Therefore, if we square root it, then just like the opposite of this, the percentage uncertainty should half, so that's gonna be 1% again. And then we can turn that back into a percentage uncertainty, and then we can turn that back into an absolute uncertainty as well. What about if you're doing an experiment, and you got a nice graph going on here, and nine times out of 10, it's gonna be a straight line graph. Let's say that this is V squared, velocity squared against displacement or height. You might recognise this graph from one of the required practicals, determining G via a free-fall method, dropping a piece of card through a photo gate. You should end up with some lovely points that look like that. Nice straight line that maybe goes through the origin. We don't force it through the origin though. Hopefully you've done the hard work of calculating your uncertainties in V squared as well, and you've drawn these as error bars on your graph. Now, my line of best fit has to match up nicely with these, and it has to go through all of the error bars as well. So I can see that that is about there. Lovely. I can tell you that the gradient of this is 20, and according to the equation, V squared equals U squared plus two AS, where U is zero, because we're dropping it. That means that the gradient is V squared divided by S, and that gives us two lots of the acceleration. That means that my acceleration is going to be the gradient 20 divided by two. Anyway, I am a little bit unsure about this number though. The whole point of drawing error bars on a graph is so you can actually find out what the uncertainty in your gradient could be. So what we then do is draw lines of worst fit. Now, lines of worst fit must go through all error bars. If, however, you think that actually one of these points like this one is really not playing ball, then you can just circle it and call it an anomaly, and then leave it out. And we need to go for highest and lowest possible gradients. That does not mean that my highest gradient line of worst fit is gonna go through the bottom of the first error bar and top of the last error bar, or vice versa. That is my highest possible gradient I can achieve while still going through all of the error bars. That's one of my lines of worst fit. That's my highest gradient. And then I'm gonna do the lowest gradient one as well. And I can see that that is there. Now, you only need to choose one of these to do your calculations with. It's good practice to choose the one that's furthest away from the line of best fit. However, in most cases, they are gonna be fairly symmetrical around the line of best fit. So it shouldn't really matter. It's worth drawing just to check, though. I'm gonna choose the bottom one here. Now I can tell you the line of this worst fit is actually 18. So I have the gradient of my line of best fit and the gradient of my line of worst fit that I've chosen to do my analysis with. What comes next? We need to find out the percentage and certainty in the gradient. And that is going to be equals to the difference in your gradients divided by the gradient of your line of best fit. The difference is gonna be 20, take away 18, divided by my original gradients of the line of best fit. So that's gonna give us two divided by 20. That gives me 0.1. Let's times that by 100 to get it into a percentage. That gives us 10%. That is the uncertainty in my gradient. Once you have your percentage and certainty in your gradient, you can then apply that to whatever the gradient is equals to. If there's 10% uncertainty in my gradient, that means there's 10% uncertainty with this here, V squared over S. There's no uncertainty with the number two. That means that there's gonna be 10% uncertainty with my acceleration. Just bear in mind that your gradient might be equals to something involving more than one variable, so you might have to combine this uncertainty with uncertainty from other variables. This is the case with the Young's modulus experiment, for example. So that's how you deal with uncertainties. I hope that that has left you slightly less confused. Uncertainties is one of those things that is gonna feature fairly prominently in your exams, whatever board you're sitting. And the way you get better at it is by having a go yourself. Have a go with some practice questions. If you wanna see some examples, have a look at my past paper solutions playlist.