So as we've said before, we said that with any calculation there lies a level of uncertainty which we call experimental error. Now, here we can talk even more specifically about certain types of uncertainties. We're going to first say that absolute uncertainty represents the plus or minus value associated with any numerical calculation. So, for example, we say a student delivers 25.00 plus or minus 0.02 ml of water to a mixture. In this case, the plus or minus value of 0.02 ml would be the uncertainty. That would represent our absolute uncertainty within this calculation. Now we're going to say, besides our absolute uncertainty, we have our relative uncertainty. The relative uncertainty is the absolute uncertainty divided by the associated measurement. So here in this case, our absolute uncertainty we said again is the plus or minus 0.02 ml so that would go on top divided by our measurement which is the 25 ml. That would give us our relative uncertainty of 0.0008. From there, we could also calculate our percent relative uncertainty. Now, our percent relative uncertainty is just our relative uncertainty multiplied by 100. So going and multiplying our relative uncertainty by 100 gives us 0.08% as our percent relative uncertainty. As we delve deeper and deeper into calculations dealing with uncertainties, it's going to become instrumental that you remember these three different types of uncertainties and their usefulness for different types of calculations we will undergo. Now that we've seen this, we can attempt to do the example question that's left here on the bottom. You could attempt it on your own at first, but if you get stuck, just come back and see how I approach that same question.

# Uncertainty - Online Tutor, Practice Problems & Exam Prep

**Uncertainty** can be thought of as the range (+/-) that is associated with any given value.

## Types of Uncertainty

### Types of Uncertainty

#### Video transcript

### Types of Uncertainty

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So here we have to calculate the relative and percent relative uncertainty from the given problem. Here we have as our question 3.25 ± 0.03. Remember, the ± 0.03 represents our absolute uncertainty. So to figure out our relative uncertainty, remember it's our absolute uncertainty divided by the measurement itself. So when we do that, we're going to get initially 0.009231. Here, it's customary to just round it here to just 1 sig fig. So we'll get 0.009 as our relative uncertainty. Rewriting that, it'd become 3.25 ± 0.009 as our complete relative uncertainty.

Now, if we wanted our percent relative uncertainty, we would take that value that we just got, multiply it by 100, so we'd have 0.9%. So our percent relative uncertainty would be 3.25 ± 0.9%. What these numbers are saying is that we expect our calculation, the correct value, the true value, to be somewhere within this range. So our answer should be 3.25 for the measurement, plus 0.9% of that 3.25. It would last somewhere within that range.

Now that you've seen these two examples, the one that we did above and then just new one here, see if you can attempt to do this practice question. Here, I'm asking for the absolute uncertainty when we're given the percent relative uncertainty from the very beginning. I'll give you guys a hint. Try to work backward. Do the exact opposite of all the steps that we've done and see if you can figure out what the absolute uncertainty is. Once you've done that, come back and take a look at how I approach that same question.

Calculate the absolute uncertainty from the given problem.

6.77 (± 5.6%)

## Propagation of Uncertainty from Random Error

### Propagation of Uncertainty

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So here we're going to talk about the calculations that we have to employ in order to deal with uncertainties when it comes to addition, subtraction, and division. These are all related to random error, of course, when it comes to our calculations. So here we're going to say with addition and subtraction or multiplication and division, we will use certain rules for propagation of our answer. Here, we're going to use what's called the real rule. The real rule says that the first digit of the absolute uncertainty is the last significant digit in the answer. What does that say? So, for example, let's say we had an absolute uncertainty of 0.0005. And we had a number, a measurement of 7.95876. Here, the first digit in the absolute uncertainty that is significant is that 5. Where does it occur? It occurs in the 4th decimal place. That would mean that my measurement has to have 4 decimal places within it. So here are our 4 decimal places here. Because there's a 6 here, I'd have to round this number up. So the measurement here would be 7.9588 plus or minus 0.0005 as our measurement with its absolute uncertainty. We'll do another one. Let's say that our absolute uncertainty here was 0.1. And let's say that we had 13.235. So here, our first digit in the absolute uncertainty, that significant is that 1. It's in the 1st decimal place. That means our measurement has to have 1 decimal place. So here's our decimal place. Because this number here is a 3, we wouldn't do anything to that 2. So this comes out to 13.2 plus or minus 0.1. So when we're doing these propagations of our answers, we have to keep this in mind when we're dealing with addition, subtraction, multiplication, or division. When it comes to the measurement with the absolute uncertainty. Now here, if we take a look at addition and subtraction, we'll learn all the different ins and outs in terms of the on the real rule. Click on the next video to see me go through this particular example when it comes to addition and subtraction.

### Propagation of Uncertainty

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So here for addition and subtraction, the uncertainty in our final answer is determined from each individual absolute uncertainty. So if we take a look here at section 2, we have these three measurements. Now remember, when we're adding or subtracting, our final answer is the one with the least number of decimal places. So here, this has 2 decimal places, 2 decimal we have uncertainty 4 because we already have these 3 initially. Now when it comes to addition and subtraction, so whether our units are adding, the uncertainties are adding or whether the uncertainties are subtracting, it's the same basic method. We're going to say that here, the uncertainty that we're looking for equals the square root of each uncertainty squared added together. For this example, our 3 uncertainties, our absolute uncertainties are 0.04, 0.03, and 0.03. We plug them into this formula, so (0.04)2+0.03)2+0.03)2). Here, right now we'd have the square root of 0.0034 initially. When we plug that in, we get what? We get 0.058. Now this little 8 here means that it is not a significant digit. Remember, when it comes to the absolute uncertainty, we're relying on the real rule. So we're looking at the first digit that's significant within the absolute uncertainty. Here, I'm placing this 8 here to let us know that the first significant digit in my absolute uncertainty is this 5 But because that 8 is there, it's going to be rounded up to 6. We know that our measurement based on the least number of decimal places came out to 6.85. And then here we round up to 6 because this 8 causes this 5 to get rounded up so that's 6. So going back on the real rule, we're going to say here that this first significant digit within my absolute uncertainty happens in the second decimal place. That means that my measurement has to have 2 decimal places within it, which it does. This is my final answer when we've done the propagation of our answer when it's dealing with addition and subtraction. Let's say that that first digit happened in the 3rd decimal place. That would mean that my answer would have to have 3 decimal places. So we'd have to put an additional number here, 6.850. But in this case, it didn't, so we didn't have to worry about it. But just remember, the real rule, we look at the first significant digit in my absolute uncertainty to determine what decimal place it's in so we can determine how many decimal places my measurement will have at the end. So when it comes to addition and subtraction, it's pretty straightforward. We take those uncertainties, we square them, we add them together, take the square root, follow the real rule to get our final answer. With multiplication and division though, it's a bit more strenuous, more work involved in getting our final answer. So come back, take a look at the next video and see how I go step by step to help us get the final answer when it comes to propagation of our answer dealing with multiplication and division.

### Propagation of Uncertainty

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So when it comes to multiplication and division, like I said, the steps are much more in-depth in terms of finding our final answer. So, for multiplication and division, we must first convert the absolute uncertainties into percent relative uncertainties. So here, our values are 3.68 ± 0.05 times 1.15 ± 0.06 divided by 0.92 ± 0.06. Here, for our measurements, we see that the 3.68 has in it 3 significant digits or figures. This one also has 3 significant figures, and this one here has 2. At the moment, our answer for our measurement will be 4.6, but again following the real rule, that could change. So, it would be best to write out the whole answer for now and then round later on. Here we just put it as 4.6 for simplicity's sake right now.

Now, here our uncertainty we do not know. Remember, when it comes to multiplication of our uncertainties or division of our uncertainties, we have to change those absolute uncertainties into percent relative uncertainties. You would square them, add them together, and then take the square root of that answer. So, remember, to find our percent relative uncertainties, we first find our relative uncertainty. That would mean that we will take each absolute uncertainty and divide it by its measurement. So we have 0.05 ÷ 3.68 × 100 to make it a percentage. This absolute uncertainty divided by its measurement times 100, and this uncertainty divided by its measurement times 100. As a result, we have each one of these percent relative uncertainties. Again, here, we're just keeping 1 significant figure in terms of our percent relative uncertainty. So, here we bring those percentages down, we square them, add them together. Doing that gives me 71.25% within this square root. Taking the square root of that gives me 8.4%.

Now, to find our absolute uncertainty, remember that would just be my percent relative uncertainty divided by 100 to get its decimal form, then multiplied by its measurement. Doing that gives me my absolute uncertainty here. Here's my absolute uncertainty. And remember, with the absolute uncertainty, we're going to say that the first significant digit here, this 9, is not significant, but it does cause this 3 to get rounded up to 4. So, remember, following the real rule, the first significant digit in my absolute uncertainty determines the last significant figure within my measurement. My first significant figure in my absolute uncertainty is in the 1st decimal place. My final answer, my measurement has to have 1 decimal place. It's 4.6. Here this would be our measurement with our absolute uncertainty and our measurement with our percent relative uncertainty here.

Again, as before, when it comes to addition and subtraction, it's pretty straightforward. But when it comes to multiplication and division, we have to change our absolute uncertainties into percent relative uncertainties, then back to absolute uncertainties at the end. So, it's quite a bit of work, but these are the methods that we have to use for propagation of our answer depending on which operation we're doing. So as we delve deeper into these calculations, we'll move on from calculations to word problems. We have to keep in mind these different types of rules based on the operation being used.

### Propagation of Uncertainty Calculations

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Determine the absolute and relative uncertainty for the following addition problem. Since we have just simple addition here, or if we had subtraction, we don't have to worry about extra steps needed to get our final answer. First, realize that with measurements, when you're adding and subtracting, it's going to be the least number of decimal places. But following the real rule, the first digit in the absolute uncertainty will determine the last significant figure or digit in my measurement. For now, when I add these three numbers together, it's going to give me 5.028. I make that 8 small because, as of right now, this has 3 decimal places, 2 decimal places, and 3 decimal places. So we're technically supposed to have 2 decimal places at this point. But, again, we don't know that for sure because we haven't calculated the absolute uncertainty yet. So I'm going to mark this little 8 here as not significant.

To do that, we take the square root. We're going to square each absolute uncertainty and add them together. So when I square them all and add them together, it's going to be the square root of 0.0006. Then I take the square root of that number, which will be 0.024495. Following the real rule, we're going to say that the first significant digit within my absolute uncertainty represents the last significant digit for my measurement. Here, it's in the 2nd decimal place, which means my measurement needs to have 2 decimal places. Because of that, I'm going to use this 8 here to round this 2 up to 3. So my measurement with absolute uncertainty at this point is 5.03 ± 0.02.

But we're not done yet. We have to calculate the relative uncertainty now. Remember, your absolute uncertainty divided by your measurement gives you the relative uncertainty. So my absolute uncertainty is 0.02 divided by my measurement of 5.03. This calculation gives me 0.003976. Here, our first significant digit is that 3. It's next to a 9 here, which means we'll round this up to 4. My measurement with its relative uncertainty is therefore 5.03 ± 0.004. So those would be my 2 answers.

Now that we've seen this, attempt to do the one that's here on the bottom. This one here is a mixture of both addition and subtraction, but keep in mind the rules we've followed so far when it comes to addition and subtraction for the propagation of our answer. Once you've figured out what the answers are, come back and take a look and see if your answers match up with mine.

Determine the absolute and relative uncertainty to the following addition and subtraction problem.

8.88 (± 0.03) - 3.29 (± 0.10) + 6.43 (± 0.001)

### Propagation of Uncertainty Calculations

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So here it states, "Determine the absolute and relative uncertainty to the following multiplication and division problem." So here we have a mixture of multiplication and division. Remember, when it comes to determining the uncertainties, we must first figure out what the percent relative uncertainty will be. Before we do that, though, realize here that we have these values multiplying and being divided by 9.17. So, at this point, let's just write down what that would be. When we multiply the two numbers and then divide by 9.17 on the bottom, we get 7.5201. We won't worry about the number of significant figures because, remember, we're going to base our final answer on the real rule when it comes to looking at our absolute uncertainty. All right, so we have at this point 7.5201.

Now, we're going to deal with the absolute uncertainties that we have here. Remember, we're first going to figure out what our percent relative uncertainty will be, so we take each uncertainty and divide them by their measurements, then multiply by a 100. Here, for this one, we're going to get 0.258732%, which we just round to 0.26%. Remember that 6 there is just a placeholder within our calculations to avoid any types of rounding errors.

Next, we deal with the next uncertainty. We divide it by its measurement of 8.921 times 100. Here, that's going to give me 0.22419%, which we will create as 0.22%. And then finally, we have 0.03 uncertainty divided by the measurement of 9.17, multiply that by 100. That's going to give me 0.327154%, which we just decrease down to 0.33%.

So we just figured out the percent relative uncertainty for each one. Now that we have that, we can figure out what the overall percent relative uncertainty will be. So, we square each percentage and add them together. So, then when we do that, we're going to get inside the parenthesis I'm inside the, square root function, square root of 0.2249%. Since we're running out of room, guys, let me take myself out of the image. So, square root of that gives me 0.47236%, which we can just round to 0.47%.

So what we just found here is our percent relative uncertainty. Now, remember, that's our percentage that means that our relative uncertainty, we take that number and divide it by 100. That would give us this value here. This number here would represent our relative uncertainty. And then if you multiply the relative uncertainty by the overall measurement that we got in the very beginning, that'll give us our absolute uncertainty.

So, here, our absolute uncertainty will come out to being 0.035344. Following the real rule, we're going to keep 1 digit here, 0.04. So this will give us our absolute uncertainty. And because our we're going to say here, remember, based on the real rule, we're going to say that the first significant digit within our absolute uncertainty represents the last significant figure, within our measurement. So, basically, because this has 2 decimal places, our measurement at the end will have 2 decimal places. So here, if we wanted the full answer, we'd say 7.52 ± 0.04 represents our measurement with the absolute uncertainty. And then we'd say 7.52 ± 0.005 represents our relative uncertainty.

So those are the steps that we need to take in order to figure out what our answer is anytime we're given uncertainties, and we're dealing with operations of multiplication and division. Now that you've seen this one, attempt the practice one that's left here on the bottom of the page. Attempt it on your own. Don't worry. Just come back, look at the next video, and see how I approach that same exact practice problem. For now, guys, good luck.

Determine the absolute and relative uncertainty to the following multiplication and division problem.

1.12(±0.01) x 0.546 (±0.01) / 3.12(±0.02) x 1.12 (0.03)

### Propagation of Uncertainty Calculations

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Here it states 2 students wish to prepare a stock solution for their lab experiment. Student A uses an uncalibrated pipette that delivers 50.00 ± 0.02 ml to deliver 200 ml to a container. Student B uses a calibrated pipette that delivers 40 ± 0.01 ml to deliver 200 ml to a container. So, we're asked to calculate the absolute uncertainty in each of their deliveries. Alright. So the big thing here is one is using a pipette that is uncalibrated while the other one is using one that is calibrated. This difference means that we're going to have to take different approaches to get to our answer. So, we have Student A here, and then we'll have Student B here. Alright, so if we're looking, it says Student A needs to get to 200 ml's. So that is our goal. And we’re doing it in multiples of 50. So we say to ourselves, okay, I would have to do it 4 times in order to get to 200 ml's. So that's 50 ± 0.02 ml's and you add that 4 times. Okay, so then we add that 4 times. Put this in. So adding that 4 times will get us to the 200 that we want plus last one. Alright. So we know that if we're adding the measurements that comes out to 200, and when it's uncalibrated, we don't do what we normally would to figure out the absolute uncertainty at the end. Because it's uncalibrated, that means my uncertainties are additive. That means I can just add them together. So we're just gonna do ± 0.02 and add it to one another. So at the end, that's gonna give me ± 0.08 ml's. So remember, when it's uncalibrated, we're gonna just add them together. In Student B though, it's a calibrated pipette, so we're going to have to do what we are normally going to do when adding up uncertainties with one another. Here we're doing it in multiples of 40. Again, our goal is to get to 200. So we do 40 each time, that means we'd have to do it a total of 5 times in order to get to 200 mls. So it's 40 ± 0.01, and actually here, for this answer, following the real rule, it’d actually be 200.00 because there are 2 decimal places here. Alright. So then we're adding all of these together. We have to do it 5 times. So adding it 5 times. Okay. So we have that. So we added that 5 times. Now, we know that in terms of the overall volume at the end, we know it's going to add up to 200 ml's, but then we have to determine what our overall absolute uncertainty will be. Remember, when we're adding or subtracting to figure out our new absolute uncertainty, We're going to take the square root, take each one of these absolute uncertainties, square them and then add them all together. Okay. So adding them all up together gives me 5.0×10-4 which just comes down to 0.022361 as my new overall absolute uncertainty. And here, it reduces down to 0.02. So because this has 2 decimal places in it, that means my final volume of 200 also has to have 2 decimal places in it. So here would be 200.00 ± 0.02 ml. So what these two answers are showing us is that by calibrating your pipette, you decrease your absolute uncertainty, which means that you're going to get a final volume that is closer to your desired amount of 200 ml's. When it's uncalibrated, there's a much bigger variation in the final amount that you're going to get. In that case, it would be ± 0.08. So it's 4 times, of a difference in terms of your final volume. So again, out of these 2 students, Student B would be the more correct student because they're using a calibrated pipette, which helps us to get a final answer that's closer to our true value of 200 ml's. Now that we've seen this, attempt to do the practice question that's left on the bottom. We've determined the volume of our 2 students, but now we're asked to figure out the molarity. Remember, molarity is just simply moles over liters. Knowing that is the key to answering this question correctly. Also, one more thing. Since it's liters and these are in milliliters, you'd have to change these values into liters. Remember, that means you divide them by 1000. Both the measurement and the absolute uncertainty will be divided by 1000 so that this becomes liters at the end. Put all those numbers together. Find out what the final answer will be. If you're stuck, don't worry. Just come back and see how I approach this practice question.

Based on the previous example calculate the molarity value for each student if they dissolve 0.300 (± 0.03) moles of analyte.

### Propagation of Uncertainty Calculations 4

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A Class A 50 milliburet is certified by a manufacturer to deliver volumes within a tolerance, i.e., uncertainty, of plus or minus 0.05 milliliters. They tell us the smallest graduations on the burette are 0.1 milliliters. You use the burette to titrate a solution adding 5 successive volumes to the solution. The following volumes were added. So here we have 5 different measurements for the amount of volume added to my total burette by the burette. And it says, what is the total volume added and what is the uncertainty associated with this final volume? What we're going to do here first is, we're going to add up all 5 of these measurements. When we do that, that gives us a number of 30.80 milliliters for my total volume. But now we have to determine what the uncertainty is. Here they're telling us that the manufacturer basically certifies that it's going to do it within this level of uncertainty. But here's the thing, the manufacturer certifying this does not mean that it is a calibrated burette. Once you get the burette in the lab, you yourself have to calibrate it. So at this point, because they're saying it's coming from the manufacturer, we're assuming that this is an uncalibrated pipette or burette. And remember, when it's uncalibrated, that means you're going to have systematic error. And when you have systematic error, we find the uncertainty in a different way. All we do to find the uncertainty is we add up all the uncertainties from each one of the measurements. Each one of these measurements had an uncertainty of plus or minus 0.05 mL. So here are all our uncertainties. And all we're doing here is we're just going to add them all up. So that gives me a total of plus or minus 0.25 milliliters. So remember, them saying that the manufacturer certifies this does that mean that it's calibrated yet? You get it in this form, but then you yourself have to calibrate it later on. Because it's uncalibrated, we have systematic error and so we just add up all the uncertainties. So my final answer would be 30.80±0.25 milliliters. If the burette itself had been calibrated, then we would have random error. And when we have random error, we do the same process that we normally would do for addition and subtraction. We would have taken the square root and squared all of these uncertainties and added them up together. And so on. And then that would have given us our total absolute uncertainty. But again, because the burette is uncalibrated, it is not random error. It is systematic error, which is pretty simple. We just add up all the uncertainties to get the total uncertainty at the end. Again, doing this gives us our final answer of 30.80±0.25 milliliters.

### Propagation of Uncertainty Calculations 4

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Here we're told a Class A 250 milliliter volumetric flask has an uncertainty of plus or minus 0.15 milliliters and a 50 milliliter volumetric pipette has an uncertainty of plus or minus 0.05 milliliters. If I fill a 250 ml volumetric flask to the line and remove 4 50 ml aliquots with a volumetric pipette, I should have 50 ml of solution remaining in the flask. Let's calculate the absolute and relative uncertainty in the remaining volume.

We start with a total volume of Vt+-0.15 ml in the 250 ml flask. We are subtracting 50 milliliters 4 times, which is represented as 50+-0.05 ml each time. This operation involves subtraction, and to calculate the overall absolute uncertainty, we square each uncertainty, sum them, and then take the square root:

- Total squared uncertainty: 0.152 + 0.052 + 0.052 + 0.052 + 0.052
- Square root of total: 0.0325, approximately 0.18

Based on the significant figure rules, we round 0.18 to 0.2. The total absolute uncertainty is thus +-0.2 milliliters, therefore the volume is 50.0+-0.2 milliliters.

To find the relative uncertainty, we divide the absolute uncertainty by the measurement: 0.250.0, giving us 0.004 as the relative uncertainty. In milliliters, the final relative uncertainty is represented as 50.0+-0.004 milliliters.

Remember, the operations within this question involve subtraction. Apply the rules for calculating uncertainty accordingly, differentiating between the rules for addition/subtraction and those for multiplication/division.

I am making a 0.1 M KCl (molar mass 74.551) solution for an experiment. To measure the mass of the KCl, I will use an analytical balance that is only accurate to ± 0.01 g. I place a piece of paper on the balance and set the tare to read 0.00. I then put the KCl on the balance until it reads 6.79 g. What is the uncertainty in this mass?

The volume of the solution I am making is 2.5 L. To measure this volume I will use a large graduated cylinder that can measure volume to ± 10 mL. What is the absolute uncertainty in my concentration?