Uncertainty: Study with Video Lessons, Practice Problems & Examples

Understanding uncertainties in measurements is crucial in scientific calculations. Absolute uncertainty indicates the range of error in a measurement, while relative uncertainty is the absolute uncertainty divided by the measurement itself. For addition and subtraction, the total uncertainty is found by taking the square root of the sum of the squares of individual uncertainties. The real rule helps determine significant figures based on the first significant digit of the absolute uncertainty. Mastering these concepts enhances accuracy in experimental results and calculations.

Types of Uncertainty

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±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 25 ml. That would give us our relative uncertainty of 0.001.

From there, we could also calculate our percent relative uncertainty. Now, our percent relative uncertainty is just our relative uncertainty multiplied by a 100. So going and multiplying our relative uncertainty by 100 gives us 0.1% 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'll 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.

Types of Uncertainty

So here we have to calculate the relative and percent relative uncertainty from the given problem. Here we have our question, 3.25 plus or minus 0.03. Remember, the plus or minus 0.03 represents our absolute uncertainty. To figure out our relative uncertainty, remember it's our absolute uncertainty divided by the measurement itself. When we do that, we initially get 0.00923076923076923.

Here, it's customary to just round it here to just one significant figure. So, we'll get 0.009 as our relative uncertainty. Rewriting that, it becomes 3.25 plus or minus 0.009 as our complete relative uncertainty. Now, if we wanted our percent relative uncertainty, we would take that value we just calculated, multiply it by 100, so we'd have 0.9%. So our percent relative uncertainty would be 3.25 plus or minus 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 or minus 0.9% of that measurement. So our actual value would be in the range from -0.9% of this 3.25 to +0.9% of that 3.25. It should be found somewhere within that range. Now that you've seen these two examples, the one that we did above and this 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 backwards. 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%) 

• A. 0.38
• B. 0.056
• C. 0.83
• D. 0.038

Propagation of Uncertainty

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, multiplication, and division. Now, 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. So it'd 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 1st decimal place, that means our measurement has to have one 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 calculations necessary to get our correct answer. But remember, always rely 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

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 places, 2 decimal places, which is why our answer here has 2 decimal places. But now we have to determine what the absolute uncertainty is right here which will be our 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 0.042+0.032+0.032. 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. So 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. 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 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

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. Here, our values are 3.68 ± 0.05 times 1.15 ± 0.06 divided by 0.92 ± 0.06. In our measurements, we see that 3.68 has 3 significant digits. This one and the 0.92 also have 3 and 2 significant figures, respectively.

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 take the square root of that answer. 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 0.053.68 × 100 to make it a percentage.

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 one significant figure in terms of our percent relative uncertainty. But here, we put these little numbers here to say that they are not significant. But based on what their values are, it could cause these numbers to round up or round down. 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%.

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. So 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.

So my first significant figure in my absolute uncertainty is in the first decimal place. My final answer, my measurement has to have one decimal place, so it's 4.6. Here, this would be our measurement with our absolute uncertainty and our measurement with our percent 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 this is the method that we have to use for propagation of our answer depending on which operation we're doing. As we delve deeper and 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

Determine the absolute and relative uncertainty to 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 answers. So what we're going to do here first is realize with our 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, 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 don't know the absolute uncertainty yet. So I'm going to put this little 8 here as being not significant as of yet. Now at this point, we have to figure out what our new absolute uncertainty will be. So 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, add them together, it's going to be 0.00062. Then I take the square root of that number, so I'm going to get 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 second 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 the absolute uncertainty at this point is 5.03 ± 0.02. But we're not done yet. We have to do it for the relative uncertainty now. Remember, your relative uncertainty equals your absolute uncertainty divided by your measurement.

So my absolute uncertainty is 0.02 divided by my measurement of 5.03. So that's going to give 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. So here, my measurement with its relative uncertainty is 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 that we've done 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 my answers.

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) 

• A. 0.1 absolute; 0.012 relative
• B. 0.1 absolute; 0.010 relative
• C. 0.1 absolute; 0.008 relative
• D. 0.01 absolute; 0.0012 relative

Propagation of Uncertainty Calculations

So here it states, determine the absolute and relative uncertainty for 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.

So when we multiply the two numbers and then divide by 9.17 at 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.

So remember, we're first going to figure out what our percent relative uncertainty will be. So we take each of those uncertainties and divide them by their measurements, then multiply by 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. Now, next, we do 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 square root, we're going to square each percentage and add them together. So then when we do that, we're going to get inside the parenthesis, I mean, 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, the square root of that gives me 0.474236%, which we can just round to 0.47%. So what we just found here is our percent relative uncertainty. Now, remember if 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 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 plus or minus 0.04 represents our measurement with the absolute uncertainty. And then we'd say 7.52 plus or minus 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 to do 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)

• A. 0.003 absolute; 0.01 relative
• B. 0.03 absolute; 0.006 relative
• C. 0.006 absolute; 0.03 relative
• D. 0.006 absolute; 0.01 relative

Propagation of Uncertainty Calculations

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. 

• A. 1.5 ± 0.2 M for both
• B. 1.5 ± 0.2 M for student A and 1.5 ± 0.1 M for student B
• C. 1.5 ± 0.1 M for student A and 1.5 ± 0.2 M for student B
• D. 1.5 ± 0.04 M for student A and 1.5 ± 0.01 M for student B

Propagation of Uncertainty Calculations 4

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. 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 certifies that it's going to deliver 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. At this point, because they're saying it's coming from the manufacturer, we're assuming that this is an uncalibrated pipette or burette.

Remember, when you're 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 of 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 not 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. Uncertainties.

My final answer would be 30.80 plus or minus 0.25 milliliters. If the burette itself had been calibrated, then we would have had 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 plus or minus 0.25 milliliters.

Propagation of Uncertainty Calculations 4

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. Here we're told if I fill a 250 ml volumetric flask to the line and remove 4 50 milliliter aliquots with a volumetric pipette, I should have 50 ml of solution remaining in the flask. What is the absolute and relative uncertainty in the remaining volume? All right. So, we're starting out with a total of 250 ± 0.15 ml.

And we're subtracting 50 milliliters four times. So that means I'm just subtracting out 50 ± 0.05 milliliters. And I'm just going to do that over and over again four times. Minus 50, plus or minus 0.05 milliliters, minus 50, plus or minus 0.05 milliliters. So, in this operation here, we have subtraction that's being undertaken.

And remember, when it's subtraction or addition, in order to find our overall absolute uncertainty, we're going to take each one of these uncertainties, square them, and then add them all together. So, we're adding them all together. Our total at this point would be 0.0325. We take the square root of that, which gives me 0.180278. So here, we're going to use our first significant figure, which is this one here. And because there's an 8 next to it, it rounds up to 2. So, my absolute uncertainty, my total absolute uncertainty, is ± 0.2. So here, this would be 50.0 ± 0.2 milliliters.

So that's my absolute uncertainty. Now, to find my relative uncertainty, I would take my absolute uncertainty and divide it by my measurement. So that gives me 0.004 for my relative uncertainty. So at the end, it would be 50.0 ± 0.004 milliliters as my relative uncertainty. So just remember what operations are being undertaken within the question and then apply the rules that we've learned previously. Addition and subtraction follow one set of rules, whereas multiplication and division follow another set.

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? 

• A. 0.010 g
• B. 0.014 g
• C. 0.0002 g
• D. 0.020 g

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? 

• A. 0.0001 M
• B. 0.0100 M
• C. 0.4499 M
• D. 0.0002 M

Here’s what students ask on this topic:

What is absolute uncertainty and how is it calculated?

Absolute uncertainty represents the range of error associated with a measurement. It is expressed as a ± value. For example, if a student measures 25.00 ± 0.02 mL of water, the ± 0.02 mL is the absolute uncertainty. This value indicates the possible deviation from the measured value. Absolute uncertainty is crucial for understanding the precision of measurements and is typically determined by the limitations of the measuring instrument or the method used.

How do you calculate relative uncertainty?

Relative uncertainty is calculated by dividing the absolute uncertainty by the measured value. It provides a dimensionless ratio that indicates the uncertainty relative to the size of the measurement. The formula is:

$\frac{\mathrm{Absolute} \mathrm{Uncertainty}}{\mathrm{Measured} \mathrm{Value}}$

For example, if the absolute uncertainty is 0.02 mL and the measured value is 25.00 mL, the relative uncertainty is:

$\frac{0.02}{25.00}=\; 0.0008$

What is the real rule in uncertainty calculations?

The real rule states that the first digit of the absolute uncertainty is the last significant digit in the measurement. For example, if the absolute uncertainty is 0.005 and the measurement is 7.95876, the first significant digit in the uncertainty is 5, which is in the fourth decimal place. Therefore, the measurement should be rounded to four decimal places, resulting in 7.9588 ± 0.005. This rule ensures that the precision of the measurement matches the precision of the uncertainty.

How do you propagate uncertainty in addition and subtraction?

When propagating uncertainty in addition and subtraction, the total uncertainty is found by taking the square root of the sum of the squares of the individual uncertainties. The formula is:

$\sqrt{{\mathrm{\Delta x}1}^2+{\mathrm{\Delta x}2}^2+...+{\mathrm{\Delta x}n}^2}$

For example, if the uncertainties are 0.04, 0.03, and 0.03, the total uncertainty is:

$\sqrt{{0.04}^2+{0.03}^2+{0.03}^2}=\; 0.058$

Following the real rule, the final uncertainty is rounded appropriately.

How do you propagate uncertainty in multiplication and division?

In multiplication and division, the relative uncertainties of the individual measurements are added together to find the total relative uncertainty. The formula is:

$\Delta Q=Q\times \surd \left(\frac{\Delta A}{A}\right)^2+\left(\frac{\Delta B}{B}\right)^2+...+\left(\frac{\Delta N}{N}\right)^2)$

For example, if you have measurements A and B with relative uncertainties of 0.02 and 0.03, respectively, the total relative uncertainty is:

$\sqrt{{\frac{\Delta A}{A}}^2+{\frac{\Delta B}{B}}^2}=\; 0.036$

Then, multiply this relative uncertainty by the calculated value to find the absolute uncertainty.