 Right, let's have a look at this electrolytes practical. So let's just check we've got, yeah, you can see it in Excel there. I'm not gonna cover the exact answers or show you any sample data. I'm just gonna show the method here because you might want to generalize it at some point. So we've got, as we add in this reactant, the conductivity decreases steadily and then it reaches an endpoint and the reaction changes slightly. So suddenly the conductivity begins increasing again. And this is a little bit like a hydration. There is an endpoint here where everything's equal. And we wanna figure out where's that point because it's clearly not at four and it's clearly not at five. It's somewhere in between and we could maybe eyeball it as four and a quarter. But how do we get that precise and how accurate do we know it? So let's do that all kind of from scratch. So what I'm gonna do is right click and add a trend line in Excel here and show that the equation and the R squared and all of that. I'm gonna get rid of these grid lines because it'll get in the way. So we've got this trend line going through them but that's not the trend line we want. It's actually it's linear coming down here. So we need to exclude some of the data. Now I've probably shown a few people have to do this before but if you want to really quickly add a second data point or a second data series, highlight the whole graph in Excel and then you see this purple and blue box on left. Click the tag in the bottom and drag it to a new column. That adds a second data series automatically. If you right click and go to select data, you see that there's a series one and series two. And what we need to do is drag some of those data points onto the other series. You can just highlight them all and then drag across. And if there's some ambiguity about which is on what series, you can drag it back and forward and see what looks best or perhaps even delete them. You could delete the ones closest to your end point if they're ambiguous. And certainly with real data, you can run into that sometimes. But I'm just gonna leave them like this for now. I'm also going to add in trend line on this side, add my R squared in equation. Now I've got two simultaneous equations basically. Now I want to find out what is here? Where do these two cross? Where are they equal? And so for that, I'm gonna do a little bit of work by hand just to show you where this comes from. So you can actually take those two equations, type them into all from alpha and it will actually solve simultaneous equations for you automatically. You just have to type them in. But instead I'm gonna show you how to do this by hand. So we've got two equations. We have a Y equals, let's say, M plus, we've got two equations like that. So I'm just gonna label them M1 and C1. And I've actually got a second equation like that as well. Y equals M2 plus C2. So those are the two best fit lines we've got here. And how do we equate these together? Well, we just set them equal to each other. We know that this X is the same in the crossing point. We know this Y is the same in the crossing point. So we actually just take this and make it equal to that. The Y doesn't really come into it or at least we can get rid of it. So we're gonna type that out. Again, keeping some kind of color involved. And actually I'll tell you what, I've missed out the X, so that's actually, this is one method of doing simultaneous equations. You can just set if they're in this form Y equals something, Y equals something. You just set the two Ys to be equal to each other. And then you've got an equation with just the one variable. And we're then going to begin collecting like terms and working with some brackets and so on. So all of these M's here have an X associated with them. So I'm going to put those on the same side. M1, X, bring this over, it becomes minus M2, X. And that's going to be equal to C2. And I'll bring this side over to become minus C1. Okay, so fairly straightforward, so far. We'll rearrange that stuff. Next, I'm gonna pull this X out of the bracket because that's what we want to calculate. So if we bring that all out, we have X and M1 minus M2. And for completion's sake, I'll write C2 minus C1 here. Great, that's fine so far. And now we just want to keep X on one side. We do X. And well, we've got to bring this whole thing across. If you're not comfortable with the algebra at this point, this is not like one object that we can. C2 minus C1 minus M2. Now, it doesn't matter which one M1 and which one M2, it doesn't matter. We've got two sets of numbers. As long as we're consistent and we remember which rear end they go. So if we have this from one equation and that from the other, we can't swap them around at the bottom, it won't work. So I've got this here. We can return back to our spreadsheet and have a look at what we've got. In fact, I'm gonna get the data out of this first. I'm going to type in equals line ST. The known Ys, the known X values, true, true. If you're working on the newer versions of Excel, you don't need to do the control shift enter. It will do something called a spill and automatically fill in everything for you line ST. So that's our first one. First line, otherwise it's identical to what I've taught before. And the second line, line ST, known Ys here, known Xs, true, true. And the reason I've done this is because we're later, we're gonna go on to propagate the error of it as well as in space. And X, the crossing point is going to be equal to C1. So C2 minus, in fact I'm going to bracket that off, divide that by M1 minus M2. They've got to be swapped the other way around. And this comes out to 4.38, which is roughly where we expected it to be. If we do a line down here, kind of draw that in straight down there. Oops, it's about four and a half, so it works. Now, if you type that straight into all form, I'll give you that number exactly as well. So now we're gonna go on to part two of this. We're gonna ask what's the propagation of error for this? For this, we are going to break this formula down into a couple of pieces. I've written this derivation on the calendar's question, but I'll go into it anyway. So this is a quotient. It's one thing divided by another, which is the same as a multiplication. So if what we're gonna do is write this down as a new variable. And I just tend to write just delta C, because delta in uppercase there means a difference. And then delta M, just a difference there. And in principle, it doesn't really matter which way you get them around now for this, for the errors, but it will for the actual value. So in a quotient, the uncertainty in X here, divided by X, so it's a relative error, is equal to the square root. Of two things squared added to each other. And it's specifically the uncertainty in delta C, divided by delta C itself, that's it. It's a little bit scrunched up here, isn't it? There's a lot of numbers to fit in, but the same thing there. The uncertainty in the value divided by the value itself. So this is like a relative error. We could call that a relative error, we want that. But how do we get the error in this? Well, it comes from these two things, being added together, right? So if we want to get the error in this, we need to do another propagation of errors. So the uncertainty in say delta C there, well that's an addition for me, or it's slightly different. We don't put the relative errors here, we put the absolute errors in. So that's just gonna be the uncertainty in C one, squared, and the uncertainty C two, squared, there you go. And may as well write it out for the other one as well. So the uncertainty in this number here, call it what if you like, but again is equal to the square root, two other uncertainties added together, right? Now, this is the uncertainty that we want, but these are the numbers that we actually have already because these are the ones in Excel that we've got from that line stats formula. So let's have a look at that now, let's see. So these are gradients and slopes, then the numbers just underneath are those standard errors. So I'm gonna set underneath this X here, actually what I was thinking about. So I'm gonna look at building in these different values. So all I need is that difference in the two intercepts. I'm just gonna call that delta C because I can't be bothered to put in the delta sign. And that is C1, sorry, it's C2, look at that, look at my notes, C2 minus C1. So that is gonna be that number minus 14. Now, what's the del is gonna be M1 minus. So again, if we did that divided by this, we get the same number as before. Right, 4.38, that's our X value. Now we're gonna talk about the uncertainty in each of those. So uncertainty in delta, well, that's our big propagation of errors thing. So if we do the sum squares of the two C's, which are the numbers underneath that one and that one, and we also then need to square root it. That's the uncertainty in that top number. We've done some squares of those two standard errors of the intercept and then square root it afterwards. That's what sum squares mean there. And maybe we will do the same for M here. Now, press control D under there or duplicate the exact formula where I will need to move these to pick up the standard errors of the slope instead. Do those two uncertainties. Now, how do we then get the uncertainty in X? Well, I'm gonna start with the uncertainty of X divided by X, that's the relative error. And that is, we're gonna do the same thing. We're gonna do a square root in sum squared, but the inputs here need to change. We go back to the off formulas here. We've got to have an error divided by the absolute number, the error divided by the absolute number together all together. So let's go back to here. So the sum squared is the uncertainty in delta C divided by C itself, right, there we go. And the uncertainty in delta M divided by, the uncertainty delta M that way around divided by the actual value of delta M itself, there we go. So that's actually a relative error now, it's a percentage. So we're actually within seven percent. And we wanna get that back. Well, look at how the formula would be. We've got that divided by X, we need to multiply by X. So we take that times 4.3, not 0.3. So whatever the uncertainty is in this crossing point, it's 0.3, which you can imagine, these numbers are not particularly good towards the end. So maybe if this was a bit higher, let's go back and we measure that point, for instance. Our error goes down, because this is a bit more of a straight line. The R squared has gone up to 0.997, I have where it's previous, it was 991. So that is kind of the propagation of errors for this practical. It is a little bit involved, there's quite a few things going on in it. And basically, let's go back to here for a second. This is kind of how you would go about building propagation of errors through a more complicated formula, because we've rearranged something to get a formula for what we want, right? And if we know that, we actually start putting propagation of error formulas together, you literally pipe these two into there, because we've broken it up into smaller bits. We've done a quotient, so we need a quotient. But these values that go into the quotient comes from these linear additions, which is this formula here. So we need two versions of that, they pipe into there, and that's our uncertainty, remembering that we've just got to rearrange and put that on the other side, at least, to get the overall value. So that's hopefully the end of it. Hopefully you got that right, and this helps you understand how to go about it.