Understanding Measurement Uncertainty in Labs

Hello, this is a lecture topic video over uncertainty in measurements. So you guys are going to be in the laboratory a lot this semester, and you're going to be making lots of different measurements. And there's a lot of different types of instruments that we can use to make these scientific measurements. So in this case, this is a balance, which allows us to gauge the mass of whatever our sample is. And it has a digital... readout, meaning we can read the value of the mass of that beaker plus the solution, right, directly from that display, okay? And notice that there are several digits. There are two before and four after the decimal place. We would want to record all of these values, okay? And there are other pieces of glassware that you guys will be using that we can also use to make measurements. So in that case, in this case, excuse me, we don't have an interface that allows us to read off what the measurement is. They typically have, in the case of these two pieces of glassware, these kind of etched or these marks in the glassware. So if we're looking at the meniscus of our solution, remember the meniscus is kind of that dip, right? We want to measure from this. point, okay, when we measure from the meniscus, all right, if that meniscus is right on that line, okay, that tells us that these pieces of glassware, these volumetric pieces of glassware, will deliver a particular volume that they were created to deliver. So in the case here, this is a pipette that's been designed and engineered to deliver 25 mil of volume. versus this volumetric flask, if we have that meniscus at that line, okay, then we will be delivering precisely 250 mils of volume. Now, we also have other pieces of glassware, all right, that have what we call graduations. You can see here, this is a graduated cylinder, okay, and you can see that it has these big lines, and it also has these little lines, okay. The big lines are measuring off milliliters and tens, so 10, 20, 30, so on and so forth, milliliters, versus these little lines, in this case, each represent two milliliters, okay? So those are graduations, all right? And then we have a 50 milliliter burette, okay? This one's set up a little bit differently. It is set up to be able to read a particular volume delivered. Okay, so you can see that we start with zero up here and 50 mils here. So if we start all the way with our solution with the meniscus line at zero, and we essentially end up with the meniscus down here on the 50 mil mark, we will know that we have delivered 50 mils of volume. Okay, so these are some of the things that you will use in the laboratory this semester. However, there is degrees of uncertainty in all of these measurements. even within the measurement on this balance. Okay, so what you'll notice when you use a balance in the lab, this last value is likely to bounce around, okay? And that is because of electrostatics, vibrations, so on and so forth. But there is uncertainty even in this measurement. That doesn't mean that we don't record that value, all right? It just means that that is a point of uncertainty. because we would still want to record all six values here. So let's do a particular example here using a burette. Imagine this is our burette and it's filled all the way at the zero line. So remember this guy would keep going and eventually somewhere up here, we would have zero. Let's imagine this kind of orangish solution. The meniscus line was right on the zero, okay? And what we do is we start delivering volume. And the question is how much? volume have we actually delivered using this burette? Okay, so our volume's going down, and then finally we stop, and you can see we kind of focus in here on the position where we stop, all right? And what we can say for certain, all right, is that we are past the 20 milliliter mark, okay? And if we zoom in and we actually look at these graduations, because again, look, these graduations are measuring off Milliliters, whole milliliters, but these These smaller graduations are a tenth of a milliliter, okay? So if we zoom in, we can actually see that our meniscus is past the first tenth, but it hasn't quite made it to that second tenth, okay? So what we can say for certainty, these are our certain digits, is that the volume delivered is between 20.1 mils and 20.2 mils. However, what we can do is we can estimate a value that is between 20.1 and 20.2. And that would be somewhere lying between 20.10 and 20.19. Okay. This estimate, whatever it is for you, okay, that estimated value, that is our uncertain digit. Okay. It is the measurement. Okay, that's estimated, that's between the calibrated lines of this particular instrument, which happens to be a burette. Okay, so for me, all right, I would say that this has approximately 20, we know that for certain, 0.1, we also know that for certain because that meniscus is beyond the 0.1 milliliter mark, right, and then I would say four, okay. 20.14 milliliters. So this is what I would record in my notebook, okay? Because I can estimate this value. This is my uncertain digit. In other words, this measurement, as precise as it is, still has a degree of uncertainty. So how do we start to deal with these uncertainties that we get in our measurements, okay? Well, we got some things, a couple of terms that we can use to help us kind of wrap our brains around this. And that is precision and accuracy. Precision is the degree of agreement among several measurements of the same quantity. So if we go back here to this example, usually if we're doing a titration, that's what we'd usually be using a curette for. We're not only going to do it once. OK, we're going to do it multiple times. OK, and so we're going to get multiple measurements. for the same quantity. Well, precision tells us something about the reproducibility of those measurements. What is the spread of those measurements versus what we want to call accuracy. Accuracy is how close is our particular measurement to the true value. So imagine I give you a 24 ounce sample of gold. All right. Well, if we go all the way back to that balance, probably hope that that balance reads after we do a conversion from grams into ounces that we have 24 ounces of gold. OK, that would tell us that that balance is giving us an accurate reading. OK, so again, precision, something about the reproducibility of a measurement. Accuracy is how close our measurement is to the true value. So what we can do is we can kind of focus our attention down here to the dartboards, okay, where the arrows, okay, or the darts, excuse me, are meant to kind of represent our measurements in our experiment, okay? So if we're looking at this first one to the bottom left here, okay, what we see is that our measurements are kind of spread throughout, okay? So they're not clustered. They're definitely not hitting our bullseye here, okay? And so... And so these kind of results of our experiment are neither precise nor accurate. OK, so our reproducibility and our measurements wasn't very good. And kind of the average of all of these measurements is nowhere near accurate, which would be the center of this bullseye. OK, so what we can do is we can connect yet another term to this. This would be called random. error, okay? Random error. That's an error that has an equal probability of being high or low, okay? In other words, if we keep throwing darts at this thing, we don't really know where they're going to end up. There's just going to be a kind of an even spread over an infinite number of measurements, okay? This is called an indeterminate error, okay? So there's really no way to determine the cause of... this type of error. That's what indeterminate kind of is suggesting. But what we can see from random error is that, again, over an infinite set of measurements, our average would end up being close to the true value. All right, so now let's focus our attention on to the second bullseye. Okay, in this case, all of our measurements are clustered very closely to one another, meaning that they're precise, but unfortunately they still are not accurate. Okay, so this on the other, this also has another term associated with it as well. This would be systematic error, an error that always occurs in the same direction. Okay. So in this case, it's all high, but you can also imagine that another set would be low if you, say, change the experiment and use a different piece of equipment. This is a determinant error. Most of the time, we can get rid of this type of error by calibrating our instrument towards a known quantity or value. And typically, we do this using standards, so things of a known quantity or value. value. We kind of calibrate our instrument towards those standards. And hopefully, if we do it correctly, we can get rid of the systematic error. So that what we can do is we can shift, all right, these kind of precise but inaccurate measurements and shift them towards a scenario where now we are both precise and accurate. All right, so what we have to do is we have to think about how to deal with these uncertain digits in our measurements, okay? And to do this, we're going to start working with significant figures, okay? So there are several rules for this, okay? And with a little bit of practice, I think everyone can master these, okay? So the first rule is that all non-zero integers Okay, so what is a non-zero integer? That's 1, 2, 3, 4, 5, 6, 7, 8, and 9. When any of these numbers show up in a measurement, they are significant, okay? They are significant. Now, zeros are where it kind of gets tricky, okay? But again, with practice, I know we can all get through this, all right? So there are three types of zeros. The first one are leading zeros. These are zeros that precede all the non-zero digits. These are more or less placeholders. They are not significant. In other words, if we look at an example, the 0.0025, there are three zeros in front of two and five. Two and five, those are non-zero integers, so therefore they are significant. But those zeros, all they're doing is kind of acting as a placeholder. In other words, they're making sure that that 2, in this case, is in the thousandth place. And the 5 is in the ten thousandth place. So they're simply indicating the position of the decimal point. So therefore, these zeros are non-significant, and this number only has two significant figures. All right, so leading zeros are not significant. Captive zeros are zeros between non-zero digits, okay? They are always significant. So for example, this 1.008, it has four significant zeros. Those two captive zeros, all right, are significant. And then we have something called trailing zeros. These are zeros at the right end of a number. They are significant. only if the number contains a decimal point. So the number 100 is kind of ambiguous. It only, what we can for sure say, okay, or concretely say is that it has one significant figure, okay? And, but we don't, we're not really sure about those other two zeros, okay? So what we can do to kind of figure this out, if we intended those two zeros to be significant, significant, what we could do is put it in scientific notation, all right, so we could report it as 1.00 times 10 to the 2, okay, because if there are zeros after the decimal place, okay, they are significant, so this would tell us that the 100, okay, as reported by 1.00 times, the 2 has three significant, three significant numbers, okay, or figures, excuse me. The number 100, okay, again, while it's kind of ambiguous, all right, if we put a decimal place right here, as it's shown right here, okay, that tells us that, again, that we have three significant figures, okay. So, trailing zeros are at the right end of the number and they are significant only if that number contains a decimal point. Alright, so that takes us to what we call exact numbers. So, a lot of times, as we can see here, many calculations involve numbers that are not obtained using measuring devices, but were determined by counting. So, imagine I have... Okay, three apples, right? That's what the example is here in this question. Three apples is three apples, okay? These are exact numbers, okay? Because they're considered to be exact numbers, they are assumed to have an infinite number of significant figures, okay? If we look at some formulas like two pi r, which is the circumference of a circle, or four thirds pi r cubed, which is the volume of our sphere, the two, okay, in their circumference calculation, is considered an exact number. Okay. This four thirds is considered an exact number. Okay. And we can also get exact numbers showing up in kind of definitions or these conversion factors that we've already kind of talked about. Okay. So for example, one inch is defined as exactly 2.54 centimeters. So if we're going from the English unit, of distance into a metric unit of distance, all right, this is a exact definition, okay? So therefore, these guys are assumed to have an infinite number of significant figures, okay? So these numbers will not determine the significant, the final significant figures when we do calculations, all right? So if you'd like to take rules and put them into flowcharts, I would recommend you looking through this flowchart that we've made for you. OK, so I'm not going to go through it, but it's here because I know some people would rather look at a flowchart relative to something like this. All right. So let's look at a sample problem. OK, so the first thing we can do is just look at all the non-zero integers. And we know that. they are significant by definition. So the three and the five. All right. So in terms of significant figures, there's at least two. Okay. And then we can go back to our rules and we can see that leading zeros that precede all non-zero digits. Okay. Are not significant because again, they're just essentially indicating a position or they're acting as a placeholder. Okay. So that would tell us. all right, that these two values are not significant, okay? They are leading zeros. They are essentially acting as placeholders for the 30500, okay? And then what we can do is look at the trailing zeros, okay? So that's these two guys here. So this is not significant. That is not significant. The captive zero is, okay? So that's our second rule back here. Remember captive zeros, zeros between those non-zero digits. All right. And then finally, these trailing zeros. Okay. These two guys are following a non-zero value and they are after a decimal place. Therefore, according to rule three back here, okay, right here, or C, I guess I should say. These both are also significant. So how many significant figures does this value have? Well, it has five significant figures. Okay. All right. So now we want to start thinking about how do we handle significant figures when we're talking about mathematical operations. All right. We got two sets of equal math operators. So we have multiplication and division. and addition, subtraction, okay? And so when we're talking about multiplication division, the number of significant figures in the result is the same as the number in the least precise measurement used in the calculation. Okay, so let's take this example right here. Okay, so if we're following all of our rules, we got three non-zero integers. That's three significant figures. Here are two, so we got two here. So what we can do, all right, is we can do a multiplication. And if we're looking at our calculator, we get 6.384. which for a math class is perfectly acceptable answer, but we got to start thinking about significant figures, okay? Because we're going to be in the laboratory where we're taking measurements and those measurements have degrees of uncertainty, all right? And so we have to actually look at what two values went into giving us this final answer, if you will, okay? Well, we got one that's got three, okay? Three significant figures and we also have a value that only has two, okay? So we have one of these values, the one with two significant figures, that is less precise than the other. All right. Therefore, it dictates how many significant figures we can have in our final answer. OK. And so because we can only have two, the answer, instead of being reported as six point three eight four, is actually reported to six point four. All right. So now for addition and subtraction. All right. The result has the same number of decimal places as the least precise measurement used in the calculation. Again, we're starting to see this theme that the least precise measurement determines the precision of the overall answer, of the final answer. OK, and so what you can see here is we have three values and what we're going to do is add them together. And I like the way this is set up. This is set up in a column format. So I highly recommend. You set these kind of questions up like this. And so what we can do, all right, is we can add all these values up. 12.11, all right, 18.0, and 1.013. And adding all those up, we get 31.123, all right? Well, what we can see here is that this guy has two decimal places. This has a single decimal place. And this has... three decimal places. Well, the one with the fewest numbers of decimal places actually dictates the overall precision of our answer. So what we can do is we can start to think about kind of drawing a line down here where everything on the left-hand side of that line is our final value. Okay, so the correct response here, so 31.123 is actually 31.1. All right. And again, that's because 18.0 only has one decimal place. Now, there are some rules for rounding. Okay, I expect that most of you know these rules. Okay, but the first thing I would say, all right, is I would do all my calculations and keep all those digits until the final result and then I would round. Okay, you're gonna run into fewer errors that way. all right and it's actually the kind of the best practice if you will all right but if you have to remove um or do some rounding okay if the digit to removed all right is less than five okay that preceding digit stays the same for example 1.33 okay if we want to get rid of that last three there that means our our that's less than five therefore that never should round to 1.3 okay that's i i suspect that everyone knows that rule. And I also suspect that everyone knows rule B. So if it's greater than five, that preceding digit is increased by one. So 1.36, in this case, since this is greater than five, it would round to 1.4. Okay. This last one is kind of tricky. All right. This is one that I suspect you may or may not know. Okay. So if that digit to be dropped is equal to five, then Okay, you want to round up if there are more non-zero digits following that five. So if this is the number we're going to drop here, all right, what you can see is there's a one after it. Okay, therefore, what we can do is we round this up. Okay, so instead of 1.37, we round it up to 1.38. However, if there are no digits following that five, we keep it the same. Okay. So, for example, again, this is the guy that we want to drop. You see there's no numbers after it. Therefore, this rounds to 1.37. Again, with some practice, I'm sure that these are not going to give you any problems. But again, it's going to take practice, okay, because we all know that this is a struggle for many students. All right, so here's a sample problem. Okay, so what we have to do first off is remember our order of operations. Okay, in other words, do we add first or do we divide first? So on and so on. And then the next thing we want to do is keep track of the significant figures throughout the problem. All right, so first thing we want to do is think about order of operations. Which mathematical operation do we do first? Okay. And there's a little thing that can help you remember that. P-E-M-D-A-S. All right. And that stands for please excuse my dear Aunt Sally. Okay. Please excuse my dear Aunt Sally. All right. Where the P. kind of stands for things in parentheses. They're done first, okay? And then what do we do? We do exponents next, all right? And then M in D stands for multiplication division, all right? Those are equal operators, so you can do either of those, okay? And then addition and subtraction, they are also equal operators, right? So that is the order of operations, okay? And then again, Our number two thing we wanted to do here is to keep track, if I could spell, keep track of our sig figs throughout. Okay. All right. So following the order of operations, what we want to do is multiplication division first. Okay. So. But before we start in, let's go ahead and count up our sig figs. So all non-zero integers, and we're talking about this guy. So the first value here, so there's 4 divided by 3.1, which again, all non-zero integers are 2. Then we got 0.470. So we have a leading zero, which is not significant. We have a non, or we have a... tailing zero after a non-zero integer. So that is significant because it's after a decimal place. Okay. So this one would have three significant figures. Then we got dividing by 0.623. Again, the leading zero, all right, is non-significant. So 623 is, all right. And then we have 80.705. All of these are considered. significant because those two zeros are both captive zeros. We got five and then we got 0.4326. That is a leading zero. It is not significant. All right. But the 4326 is. Okay. So therefore, that is how many significant figures each of these values have. All right. So now we have to do order of operations. Okay. So let's do all the divisions first. And we don't, again, we don't want to round off until the very end. Okay, so if we do 2.5 to 2.526 divided by 3.1, our calculator gives us 0.8148387. Okay, 8387. And if we do 0.470 divided by 0.623, we get 0.7544141. Okay. And if we do 80.705 divided by 0.4326, we get 186.55. 8, 0, 2, 1, 3. All right, so now let's think about the significance of each of these numbers. All right, so we got something. Our numerator's got four sig figs divided by a number that's got two. So overall, if we're following our rules, we should only have two sig figs in our answer. So our answer would be 0.81. We're going to keep everything. We're not going to round until the very end. If we're looking at the second division, we have an equal number. So we got three significant figures here. And then here we have something with five being divided by something with four. And so we want to keep all those, but we should expect four significant figures. All right, so now if we line these up in a column format, 0.81483871. Let's get rid of that actually at one, make it easier. And then the next one is 0.7544141. and then 186.5580213. And this is an addition, so we wanna add all these guys together. And we can again underline our significance. This guy is right here. We can add these up and we get 180. 8.1272741. Okay. That is our value. Okay. And now what we want to do is we see, okay, that the 186.5 is the guy that should be dictating this operation, this addition. Okay. So our answer should only have. four significant values equal 188.1. And this would be the value that I would report in my lab notebook. Okay, with that, this is a participation question. All right, the mass of a watch glass was measured on four different balances. The masses were 99.98 grams, 100.008 grams, 102.1 grams, and 100. 0.005 grams. The question is, what is the average mass of the watch glass using the correct amount of significant figures? Okay. So remember that what we need to do, okay, is we need to strategize. We need to set the question up. Okay. We need to get a solution and then does it make sense? All right. So again, I want to keep stressing that because it's going to help you as you proceed through the course this semester. All right. So we need an average. All right. And so remember what the average is. All right. That is the sum of our measurements divided by the number of measurements made. OK. All right. So that's your campus participation question. Good luck and again, reach out if you need some assistance. I am here and happy to help. Thanks.

readout, meaning we can read the value of the mass of that beaker plus the solution, right, directly from that display, okay? And notice that there are several digits. There are two before and four after the decimal place. We would want to record all of these values, okay?

And there are other pieces of glassware that you guys will be using that we can also use to make measurements. So in that case, in this case, excuse me, we don't have an interface that allows us to read off what the measurement is. They typically have, in the case of these two pieces of glassware, these kind of etched or these marks in the glassware.

So if we're looking at the meniscus of our solution, remember the meniscus is kind of that dip, right? We want to measure from this. point, okay, when we measure from the meniscus, all right, if that meniscus is right on that line, okay, that tells us that these pieces of glassware, these volumetric pieces of glassware, will deliver a particular volume that they were created to deliver. So in the case here, this is a pipette that's been designed and engineered to deliver 25 mil of volume. versus this volumetric flask, if we have that meniscus at that line, okay, then we will be delivering precisely 250 mils of volume.

Now, we also have other pieces of glassware, all right, that have what we call graduations. You can see here, this is a graduated cylinder, okay, and you can see that it has these big lines, and it also has these little lines, okay. The big lines are measuring off milliliters and tens, so 10, 20, 30, so on and so forth, milliliters, versus these little lines, in this case, each represent two milliliters, okay?

So those are graduations, all right? And then we have a 50 milliliter burette, okay? This one's set up a little bit differently. It is set up to be able to read a particular volume delivered. Okay, so you can see that we start with zero up here and 50 mils here.

So if we start all the way with our solution with the meniscus line at zero, and we essentially end up with the meniscus down here on the 50 mil mark, we will know that we have delivered 50 mils of volume. Okay, so these are some of the things that you will use in the laboratory this semester. However, there is degrees of uncertainty in all of these measurements.

even within the measurement on this balance. Okay, so what you'll notice when you use a balance in the lab, this last value is likely to bounce around, okay? And that is because of electrostatics, vibrations, so on and so forth. But there is uncertainty even in this measurement. That doesn't mean that we don't record that value, all right?

It just means that that is a point of uncertainty. because we would still want to record all six values here. So let's do a particular example here using a burette. Imagine this is our burette and it's filled all the way at the zero line. So remember this guy would keep going and eventually somewhere up here, we would have zero.

Let's imagine this kind of orangish solution. The meniscus line was right on the zero, okay? And what we do is we start delivering volume. And the question is how much?

volume have we actually delivered using this burette? Okay, so our volume's going down, and then finally we stop, and you can see we kind of focus in here on the position where we stop, all right? And what we can say for certain, all right, is that we are past the 20 milliliter mark, okay? And if we zoom in and we actually look at these graduations, because again, look, these graduations are measuring off Milliliters, whole milliliters, but these These smaller graduations are a tenth of a milliliter, okay? So if we zoom in, we can actually see that our meniscus is past the first tenth, but it hasn't quite made it to that second tenth, okay?

So what we can say for certainty, these are our certain digits, is that the volume delivered is between 20.1 mils and 20.2 mils. However, what we can do is we can estimate a value that is between 20.1 and 20.2. And that would be somewhere lying between 20.10 and 20.19.

Okay. This estimate, whatever it is for you, okay, that estimated value, that is our uncertain digit. Okay. It is the measurement.

Okay, that's estimated, that's between the calibrated lines of this particular instrument, which happens to be a burette. Okay, so for me, all right, I would say that this has approximately 20, we know that for certain, 0.1, we also know that for certain because that meniscus is beyond the 0.1 milliliter mark, right, and then I would say four, okay. 20.14 milliliters. So this is what I would record in my notebook, okay?

Because I can estimate this value. This is my uncertain digit. In other words, this measurement, as precise as it is, still has a degree of uncertainty. So how do we start to deal with these uncertainties that we get in our measurements, okay? Well, we got some things, a couple of terms that we can use to help us kind of wrap our brains around this.

And that is precision and accuracy. Precision is the degree of agreement among several measurements of the same quantity. So if we go back here to this example, usually if we're doing a titration, that's what we'd usually be using a curette for.

We're not only going to do it once. OK, we're going to do it multiple times. OK, and so we're going to get multiple measurements. for the same quantity. Well, precision tells us something about the reproducibility of those measurements.

What is the spread of those measurements versus what we want to call accuracy. Accuracy is how close is our particular measurement to the true value. So imagine I give you a 24 ounce sample of gold.

All right. Well, if we go all the way back to that balance, probably hope that that balance reads after we do a conversion from grams into ounces that we have 24 ounces of gold. OK, that would tell us that that balance is giving us an accurate reading.

OK, so again, precision, something about the reproducibility of a measurement. Accuracy is how close our measurement is to the true value. So what we can do is we can kind of focus our attention down here to the dartboards, okay, where the arrows, okay, or the darts, excuse me, are meant to kind of represent our measurements in our experiment, okay?

So if we're looking at this first one to the bottom left here, okay, what we see is that our measurements are kind of spread throughout, okay? So they're not clustered. They're definitely not hitting our bullseye here, okay?

And so... And so these kind of results of our experiment are neither precise nor accurate. OK, so our reproducibility and our measurements wasn't very good.

And kind of the average of all of these measurements is nowhere near accurate, which would be the center of this bullseye. OK, so what we can do is we can connect yet another term to this. This would be called random. error, okay? Random error.

That's an error that has an equal probability of being high or low, okay? In other words, if we keep throwing darts at this thing, we don't really know where they're going to end up. There's just going to be a kind of an even spread over an infinite number of measurements, okay? This is called an indeterminate error, okay? So there's really no way to determine the cause of...

this type of error. That's what indeterminate kind of is suggesting. But what we can see from random error is that, again, over an infinite set of measurements, our average would end up being close to the true value.

All right, so now let's focus our attention on to the second bullseye. Okay, in this case, all of our measurements are clustered very closely to one another, meaning that they're precise, but unfortunately they still are not accurate. Okay, so this on the other, this also has another term associated with it as well. This would be systematic error, an error that always occurs in the same direction. Okay.

So in this case, it's all high, but you can also imagine that another set would be low if you, say, change the experiment and use a different piece of equipment. This is a determinant error. Most of the time, we can get rid of this type of error by calibrating our instrument towards a known quantity or value. And typically, we do this using standards, so things of a known quantity or value. value.

We kind of calibrate our instrument towards those standards. And hopefully, if we do it correctly, we can get rid of the systematic error. So that what we can do is we can shift, all right, these kind of precise but inaccurate measurements and shift them towards a scenario where now we are both precise and accurate. All right, so what we have to do is we have to think about how to deal with these uncertain digits in our measurements, okay? And to do this, we're going to start working with significant figures, okay?

So there are several rules for this, okay? And with a little bit of practice, I think everyone can master these, okay? So the first rule is that all non-zero integers Okay, so what is a non-zero integer? That's 1, 2, 3, 4, 5, 6, 7, 8, and 9. When any of these numbers show up in a measurement, they are significant, okay?

They are significant. Now, zeros are where it kind of gets tricky, okay? But again, with practice, I know we can all get through this, all right?

So there are three types of zeros. The first one are leading zeros. These are zeros that precede all the non-zero digits.

These are more or less placeholders. They are not significant. In other words, if we look at an example, the 0.0025, there are three zeros in front of two and five. Two and five, those are non-zero integers, so therefore they are significant.

But those zeros, all they're doing is kind of acting as a placeholder. In other words, they're making sure that that 2, in this case, is in the thousandth place. And the 5 is in the ten thousandth place.

So they're simply indicating the position of the decimal point. So therefore, these zeros are non-significant, and this number only has two significant figures. All right, so leading zeros are not significant.

Captive zeros are zeros between non-zero digits, okay? They are always significant. So for example, this 1.008, it has four significant zeros.

Those two captive zeros, all right, are significant. And then we have something called trailing zeros. These are zeros at the right end of a number.

They are significant. only if the number contains a decimal point. So the number 100 is kind of ambiguous. It only, what we can for sure say, okay, or concretely say is that it has one significant figure, okay?

And, but we don't, we're not really sure about those other two zeros, okay? So what we can do to kind of figure this out, if we intended those two zeros to be significant, significant, what we could do is put it in scientific notation, all right, so we could report it as 1.00 times 10 to the 2, okay, because if there are zeros after the decimal place, okay, they are significant, so this would tell us that the 100, okay, as reported by 1.00 times, the 2 has three significant, three significant numbers, okay, or figures, excuse me. The number 100, okay, again, while it's kind of ambiguous, all right, if we put a decimal place right here, as it's shown right here, okay, that tells us that, again, that we have three significant figures, okay.

So, trailing zeros are at the right end of the number and they are significant only if that number contains a decimal point. Alright, so that takes us to what we call exact numbers. So, a lot of times, as we can see here, many calculations involve numbers that are not obtained using measuring devices, but were determined by counting.

So, imagine I have... Okay, three apples, right? That's what the example is here in this question.

Three apples is three apples, okay? These are exact numbers, okay? Because they're considered to be exact numbers, they are assumed to have an infinite number of significant figures, okay? If we look at some formulas like two pi r, which is the circumference of a circle, or four thirds pi r cubed, which is the volume of our sphere, the two, okay, in their circumference calculation, is considered an exact number. Okay.

This four thirds is considered an exact number. Okay. And we can also get exact numbers showing up in kind of definitions or these conversion factors that we've already kind of talked about.

Okay. So for example, one inch is defined as exactly 2.54 centimeters. So if we're going from the English unit, of distance into a metric unit of distance, all right, this is a exact definition, okay?

So therefore, these guys are assumed to have an infinite number of significant figures, okay? So these numbers will not determine the significant, the final significant figures when we do calculations, all right? So if you'd like to take rules and put them into flowcharts, I would recommend you looking through this flowchart that we've made for you. OK, so I'm not going to go through it, but it's here because I know some people would rather look at a flowchart relative to something like this.

All right. So let's look at a sample problem. OK, so the first thing we can do is just look at all the non-zero integers. And we know that. they are significant by definition.

So the three and the five. All right. So in terms of significant figures, there's at least two.

Okay. And then we can go back to our rules and we can see that leading zeros that precede all non-zero digits. Okay.

Are not significant because again, they're just essentially indicating a position or they're acting as a placeholder. Okay. So that would tell us.

all right, that these two values are not significant, okay? They are leading zeros. They are essentially acting as placeholders for the 30500, okay?

And then what we can do is look at the trailing zeros, okay? So that's these two guys here. So this is not significant. That is not significant. The captive zero is, okay?

So that's our second rule back here. Remember captive zeros, zeros between those non-zero digits. All right.

And then finally, these trailing zeros. Okay. These two guys are following a non-zero value and they are after a decimal place.

Therefore, according to rule three back here, okay, right here, or C, I guess I should say. These both are also significant. So how many significant figures does this value have?

Well, it has five significant figures. Okay. All right.

So now we want to start thinking about how do we handle significant figures when we're talking about mathematical operations. All right. We got two sets of equal math operators. So we have multiplication and division.

and addition, subtraction, okay? And so when we're talking about multiplication division, the number of significant figures in the result is the same as the number in the least precise measurement used in the calculation. Okay, so let's take this example right here. Okay, so if we're following all of our rules, we got three non-zero integers.

That's three significant figures. Here are two, so we got two here. So what we can do, all right, is we can do a multiplication.

And if we're looking at our calculator, we get 6.384. which for a math class is perfectly acceptable answer, but we got to start thinking about significant figures, okay? Because we're going to be in the laboratory where we're taking measurements and those measurements have degrees of uncertainty, all right?

And so we have to actually look at what two values went into giving us this final answer, if you will, okay? Well, we got one that's got three, okay? Three significant figures and we also have a value that only has two, okay? So we have one of these values, the one with two significant figures, that is less precise than the other.

All right. Therefore, it dictates how many significant figures we can have in our final answer. OK. And so because we can only have two, the answer, instead of being reported as six point three eight four, is actually reported to six point four.

All right. So now for addition and subtraction. All right.

The result has the same number of decimal places as the least precise measurement used in the calculation. Again, we're starting to see this theme that the least precise measurement determines the precision of the overall answer, of the final answer. OK, and so what you can see here is we have three values and what we're going to do is add them together. And I like the way this is set up.

This is set up in a column format. So I highly recommend. You set these kind of questions up like this. And so what we can do, all right, is we can add all these values up.

12.11, all right, 18.0, and 1.013. And adding all those up, we get 31.123, all right? Well, what we can see here is that this guy has two decimal places.

This has a single decimal place. And this has... three decimal places. Well, the one with the fewest numbers of decimal places actually dictates the overall precision of our answer.

So what we can do is we can start to think about kind of drawing a line down here where everything on the left-hand side of that line is our final value. Okay, so the correct response here, so 31.123 is actually 31.1. All right. And again, that's because 18.0 only has one decimal place. Now, there are some rules for rounding.

Okay, I expect that most of you know these rules. Okay, but the first thing I would say, all right, is I would do all my calculations and keep all those digits until the final result and then I would round. Okay, you're gonna run into fewer errors that way.

all right and it's actually the kind of the best practice if you will all right but if you have to remove um or do some rounding okay if the digit to removed all right is less than five okay that preceding digit stays the same for example 1.33 okay if we want to get rid of that last three there that means our our that's less than five therefore that never should round to 1.3 okay that's i i suspect that everyone knows that rule. And I also suspect that everyone knows rule B. So if it's greater than five, that preceding digit is increased by one.

So 1.36, in this case, since this is greater than five, it would round to 1.4. Okay. This last one is kind of tricky.

All right. This is one that I suspect you may or may not know. Okay.

So if that digit to be dropped is equal to five, then Okay, you want to round up if there are more non-zero digits following that five. So if this is the number we're going to drop here, all right, what you can see is there's a one after it. Okay, therefore, what we can do is we round this up. Okay, so instead of 1.37, we round it up to 1.38. However, if there are no digits following that five, we keep it the same.

Okay. So, for example, again, this is the guy that we want to drop. You see there's no numbers after it. Therefore, this rounds to 1.37.

Again, with some practice, I'm sure that these are not going to give you any problems. But again, it's going to take practice, okay, because we all know that this is a struggle for many students. All right, so here's a sample problem.

Okay, so what we have to do first off is remember our order of operations. Okay, in other words, do we add first or do we divide first? So on and so on.

And then the next thing we want to do is keep track of the significant figures throughout the problem. All right, so first thing we want to do is think about order of operations. Which mathematical operation do we do first?

Okay. And there's a little thing that can help you remember that. P-E-M-D-A-S.

All right. And that stands for please excuse my dear Aunt Sally. Okay. Please excuse my dear Aunt Sally. All right.

Where the P. kind of stands for things in parentheses. They're done first, okay? And then what do we do?

We do exponents next, all right? And then M in D stands for multiplication division, all right? Those are equal operators, so you can do either of those, okay? And then addition and subtraction, they are also equal operators, right? So that is the order of operations, okay?

And then again, Our number two thing we wanted to do here is to keep track, if I could spell, keep track of our sig figs throughout. Okay. All right. So following the order of operations, what we want to do is multiplication division first.

Okay. So. But before we start in, let's go ahead and count up our sig figs. So all non-zero integers, and we're talking about this guy.

So the first value here, so there's 4 divided by 3.1, which again, all non-zero integers are 2. Then we got 0.470. So we have a leading zero, which is not significant. We have a non, or we have a... tailing zero after a non-zero integer. So that is significant because it's after a decimal place.

Okay. So this one would have three significant figures. Then we got dividing by 0.623.

Again, the leading zero, all right, is non-significant. So 623 is, all right. And then we have 80.705.

All of these are considered. significant because those two zeros are both captive zeros. We got five and then we got 0.4326. That is a leading zero. It is not significant.

All right. But the 4326 is. Okay. So therefore, that is how many significant figures each of these values have.

All right. So now we have to do order of operations. Okay. So let's do all the divisions first. And we don't, again, we don't want to round off until the very end.

Okay, so if we do 2.5 to 2.526 divided by 3.1, our calculator gives us 0.8148387. Okay, 8387. And if we do 0.470 divided by 0.623, we get 0.7544141. Okay.

And if we do 80.705 divided by 0.4326, we get 186.55. 8, 0, 2, 1, 3. All right, so now let's think about the significance of each of these numbers. All right, so we got something. Our numerator's got four sig figs divided by a number that's got two. So overall, if we're following our rules, we should only have two sig figs in our answer.

So our answer would be 0.81. We're going to keep everything. We're not going to round until the very end. If we're looking at the second division, we have an equal number.

So we got three significant figures here. And then here we have something with five being divided by something with four. And so we want to keep all those, but we should expect four significant figures.

All right, so now if we line these up in a column format, 0.81483871. Let's get rid of that actually at one, make it easier. And then the next one is 0.7544141.

and then 186.5580213. And this is an addition, so we wanna add all these guys together. And we can again underline our significance.

This guy is right here. We can add these up and we get 180. 8.1272741. Okay. That is our value. Okay.

And now what we want to do is we see, okay, that the 186.5 is the guy that should be dictating this operation, this addition. Okay. So our answer should only have.

four significant values equal 188.1. And this would be the value that I would report in my lab notebook. Okay, with that, this is a participation question.

All right, the mass of a watch glass was measured on four different balances. The masses were 99.98 grams, 100.008 grams, 102.1 grams, and 100. 0.005 grams. The question is, what is the average mass of the watch glass using the correct amount of significant figures? Okay. So remember that what we need to do, okay, is we need to strategize.

We need to set the question up. Okay. We need to get a solution and then does it make sense?

All right. So again, I want to keep stressing that because it's going to help you as you proceed through the course this semester. All right. So we need an average. All right.

And so remember what the average is. All right. That is the sum of our measurements divided by the number of measurements made. OK.

All right. So that's your campus participation question. Good luck and again, reach out if you need some assistance. I am here and happy to help. Thanks.

Transcript for:Understanding Measurement Uncertainty in Labs

Transcript for:
Understanding Measurement Uncertainty in Labs