Lesson video

Hello, my name's Mrs. Navin, and today we're going to be talking about uncertainty in measurements as part of our unit on calculations involving masses.

Now you may have come across this idea of uncertainty from your previous learning, but what we do in today's lesson will help us to better appreciate its impact on the choice of appropriate equipment that is used for measurements in an investigation, and also its impact on answering that big question of what are substances made of? So by the end of today's lesson, you should feel more comfortable explaining why there is some uncertainty whenever a measurement is made, and how we can minimise that uncertainty.

You'll also be able to calculate uncertainties from an investigation.

Now throughout this lesson, I'll be referring to some keywords and these include uncertainty, measurement, resolution, apparatus, and mean.

Now the definitions for these keywords are given in sentence form on the next slide, and you may wish to pause the video here so that you can make a quick note of what each represents so that you can refer back to them later on in the lesson or later on in your learning.

So today's lesson is broken into two parts.

First, we'll look at what we mean by what an uncertainty is in a measurement, and then we'll move on to look at calculating mean uncertainty.

So let's get started by looking at uncertainties that arise when making a measurement.

Now you may recall that there are three fundamental concepts in chemistry, and they are the macroscopic or the practical work, the things we can actually see and do within the laboratory setting.

Then there's the symbolic.

That's the way we represent our practical work.

So things like chemical equations, and then there's this idea of the submicroscopic.

That's when we talk about particles, interpreting essentially those chemical equations that are representing that practical work.

So when we are talking about atoms, the molecules, or those formula units.

Now measurements that are collected during investigations, actually provide a lot of information upon which future discussions are actually based and they create a very key link between those three fundamental concepts in chemistry.

So the investigations, it's clear, that's our practical work, but the information then that we collect from practical work represents those chemical equations, okay? So what we're doing in the lab is linked up to the information that we collect from it, and that leads into those future discussions, which tends to be based around the particles that were actually reacting or formed within that investigation.

And that then could lead into future practical work.

So the link between these arrows moving from the macroscopic to the symbolic to the submicroscopic, are those measurements that are collected in an investigation.

Now, an individual measurement provides a scientist with three really important pieces of information.

They tell us the size of the measurement, they tell us the comparison standard, and they also give us an indication of the uncertainty of that measurement.

Now the size and comparison standard of a measurement that's taken is pretty explicitly indicated when it's recorded.

The size is simply the number, and the comparison standard then is the unit.

So if we look at these examples here, the size of the measurement there is the numbers, and the comparison standard, then is the unit.

And that's one of the reasons why it's so important that we're not just writing a number down when we're recording a measurement, we're also including that unit, that comparison standard somewhere.

Now that could be either next to the number or it could be in a heading in our table of results.

Now the uncertainty of a measurement, though, that third piece of essential information, isn't explicitly added here, it's implied by the size of that measurement.

So it's implied by the number of our measurement and the uncertainty of that measurement then, depends on the resolution of the apparatus that's been used to make that measurement.

Now the resolution of an apparatus is simply the smallest measurement change that it can indicate, and it's simply shown by the markings on that apparatus.

So if I have a ruler here and when I zoom in, I can see very clearly that between those numbered markings of 0 and 1, there are 10 other individual markings between those values.

So what I can do then is look at the numbered value that I have, subtract those, so 1 minus 0, and then because there are 10 markings between them, I divide by 10.

And that gives me a value of not 0.

1.

That means that this particular ruler has a resolution of 0.

1 centimetres.

The smallest measurement it can make, its resolution, is 0.

1 centimetres.

Let's stop here for a quick check.

What is the resolution of this measuring cylinder? You may wish to pause the video here and come back when you're ready to check your answer.

Well done if you said B, 0.

2 centimetres cubed.

We can see here in the zoomed-in section that we have the numbered markings of 10 and 8, so I'll subtract that, and then between them, there are 10 markings.

So when I take 10 minus 8 divide by 10 for the markings, I get a value of 0.

2.

So the resolution of this measuring cylinder is 0.

2 centimetres.

Very well done if you've got that correct, guys, great start! Now, similar measuring devices can actually have different resolutions.

So if I look at my two rulers here, the one on the left, when I zoom in and calculate the number of markings between the values of 0 and 1, it gives me a value of 0.

1.

That means the resolution of this ruler on the left is 0.

1 centimetres.

Now when I look at the ruler on the right, I can see that between the same values of 0 and 1, there are actually 20 markings and that gives me a value of 0.

05, meaning the resolution on this particular ruler, that does the exact same thing, it measures the length of something, the resolution on this ruler is 0.

05 centimetres.

So I have two apparatus, two measuring devices that are used for the exact same thing, but they have different resolutions.

Now, I said earlier that the uncertainty of a measurement is related to the resolution of the apparatus that was used to make that measurement.

Now the uncertainty that we're talking about here is essentially, a measure of the doubt that we have in the accuracy of that measurement we're taking, is what we are measuring on that resolution marking or somewhere between them? Now, because of that, we can actually make a calculation here.

The apparatus uncertainty is going to then be equal to the resolution of that apparatus divided by 2, and that value then, that measure of uncertainty will be provided as a plus or minus to that measurement, it gives us a range within which we could find the accurate measurement using that particular device.

So if we go back to those rulers we were talking about a moment ago, the resolution that we said for the ruler on the left was 0.

1, its uncertainty then can be calculated as 0.

05.

So we're just taking that resolution and dividing by 2, we can do the same thing then for our ruler on the right, its resolution was 0.

05 centimetres.

Its uncertainty then is simply the resolution divided by 2, giving us an uncertainty value of plus or minus 0.

025 centimetres.

So there's a very quick and easy way that you can determine the uncertainty of any apparatus that is given to you, and I'm gonna go through an example here using this thermometer.

So the first thing I need to do is to determine the resolution.

So for this, I'm going to find two labelled markings and then divide by the number of markings between those labels.

So for this one, I'm gonna stick with the 0 and 10 labels, and I can see that there are 10 markings between them, so the resolution for this thermometer is 1.

Then I can use that to calculate the uncertainty 'cause it's simply the resolution divided by 2.

So for this thermometer, the uncertainty is plus or minus 0.

5 degrees Celsius.

So what does this mean? It means then that this particular thermometer can accurately measure the temperature to within a not 0.

5 degrees Celsius of the actual temperature.

What I'd like you to do now then is to calculate the uncertainty of this apparatus.

You may wish to pause the video and come back when you're ready to check your answer.

Okay, let's see how you got on.

So the first thing we needed to do, was to determine the resolution, and then for this particular apparatus, it should be 0.

25.

You then need to take that value and divide by 2 to find its uncertainty, which should then be plus or minus 0.

125 centimetres.

So very well done if you manage to get that final answer, give yourself two marks because it was two sets of calculations here.

You first needed to find the resolution, that's one mark, and then you needed to find the uncertainty, which was the second calculation, and that's your second mark.

So don't worry if you didn't get the final uncertainty, did you at least manage to calculate the resolution? Very well done, guys, you're doing brilliantly! Let's have another quick check.

What is the uncertainty of a measurement using this apparatus? Pause the video and come back when you're ready to check your answer.

Well done if you said, D, plus or minus 0.

05 centimetres cubed.

Now if you didn't get that answer, I've shown the work here on the side so you can double-check just in case you've gone wrong, and if you have identified that error so that we can fix it going forward.

But very well done if you manage to choose D.

Great job, guys! Now sometimes scientists have to analyse measurements without having any access to the apparatus that was used to make that measurement.

For instance, maybe they're peer evaluating somebody else's data, they can get an understanding of the apparatus and the uncertainty in those measurements, simply by looking at the precision of the measurement that is recorded.

So what we can say is that a more precise measurement, that's one with more decimal places, indicates that the apparatus that was used to make that measurement had a higher resolution, had more markings on that apparatus.

And apparatus that has a higher resolution, I.

e.

, more markings on it, tends to also have lower uncertainty.

So let's go back to those rulers from before.

This time, we're going to measure this purple box and it's the same length on both of our rulers.

If we look at the measurement that's been recorded, we can say that the measurement on the right has more precision because it has more decimal places.

Now, that decimal place, because we have a second decimal place on the one on the right, it indicates that there were more markings on it, that the resolution is a little bit higher because that resolution is to 0.

05 centimetres, whereas the ruler on the left, only goes to one decal place and has a resolution of 0.

1.

Now we know then that the uncertainty of a measurement is simply half the size of the resolution.

So if the resolution is to more decimal places, the uncertainty is going to be slightly lower, and that gives us an indication purely from that measurement, what kind of apparatus was used to take that measurement and the amount of uncertainty or doubt we might have in the accuracy of that measurement.

Now when scientists are planning an investigation, they will aim to use the most appropriate apparatus for their measurements, that is going to minimise the uncertainty of those measurements.

And they do this by ensuring that the apparatus is both the appropriate size for the measurements they expect to be taking, and the resolution for that measurement, and I say appropriate size because the highest resolution isn't always the most appropriate, it might give you the most precision and the lowest uncertainty, but it's not always the most necessary.

I don't need to use a four-decimal place balance when I'm only going to be measuring out 10 grammes of a substance, a one-decimal place balance will suffice.

I don't need to have the highest resolution for this particular measurement.

So it's about balancing out the measurement device that you're using, the resolution of it, and the size of the measurement you expect to take.

Let's look at an example.

Now, if I need to be measuring the length of a room that's about 4.

7 metres long, I have got a lot of different apparatus that I can use to measure the length of something.

I could use a ruler that has a maximum measurement of 30 centimetres.

I could maybe use a tailors ruler, that has a maximum measurement of 150 centimetres, or I could use one of those extending measuring tapes, that could go as much as 500 centimetres.

Now if I calculate the resolution and uncertainty for all three of this apparatus, I can see, they all have the same resolution and the same uncertainty.

So how do I make a choice about which of these three apparatus is most appropriate for measuring out this room because they all have the same resolution and the same uncertainty? Well, the thing we need to think about is the number of times you need to use that apparatus to make your entire, overall measurement.

So if I look at the ruler, I'd have to use it 16 times in order to measure out a length of room that was 4.

7 metres long.

That means I'm actually multiplying the uncertainty 16 times because I'm having to take 16 different measurements.

So actually, the overall uncertainty using that 30-centimeter ruler will be 0.

8 centimetres plus or minus.

If I used the measuring tape or the tailor's tape, I'd have to use that particular one four times, to fully measure out a room that was 4.

7 metres long.

So I'd have to multiply the uncertainty using that device by 4.

So its overall uncertainty is plus or minus 0.

2 centimetres.

If I use the extender measuring tape that has a maximum measurement of 500 centimetres, I'd only have to use it once, and therefore, the overall uncertainty of that particular device will be just the one uncertainty, plus or minus 0.

05 centimetres.

So you can see that the uncertainty changes depending on how many times you need to use that particular device.

So the best choice here would be the extender measuring tape so that I'm only using it once, and I'm minimising the amount of uncertainty in that overall measurement that I'm taking.

Let's stop here for another quick check.

Which apparatus will provide measurements with the lowest uncertainty? And I'd like you to explain your choice.

So I'm looking for a because clause, you may wish to discuss your ideas with the people nearest you, so I'm gonna recommend you pause the video here and come back when you're ready to check your answer.

Well done if you chose A.

So give yourself a mark if you chose A, but really what we are looking for was that because clause.

So what I was looking for here, was the most precise measurement.

So I'm looking for a value on my balances that had the most decimal places because that indicates that I am using a piece of apparatus with the highest resolution.

And remember, the higher the resolution, the lower the uncertainty.

So very well done if you manage to choose A, you guys are doing supremely well, keep it up! Okay, time for the first task of today's lesson.

What I'd like you to do, first of all, is to match up each key term to the best description.

So you may wish to pause the video and come back when you're ready to check your answer.

Okay, let's see how you got on.

So the resolution then is the smallest change that is measured by the markings on the apparatus.

So it was markings that gave that one away from me.

The measurement then is the quantitative information about an object or a substance.

So when we think of quantitative, we need to be thinking about the word number that we're getting about it.

So remember a measurement always starts with a number.

Apparatus then, is a piece of equipment that's been designed for a very particular use, and then the uncertainty IS equal to half the value of the device's resolution, okay? And it's always shown with that plus or minus, so it was that plus or minus that gave it away that that should be linked up to the uncertainty.

So very well done if you managed to match those up correctly, guys, great start! Okay, for the next part of this task, what I'd like you to do is to determine both the resolution and the uncertainty for each apparatus shown.

So pause the video here, and come back when you're ready to check your answers.

Okay, let's see how you got on.

So for device A, some of you might've recognised is an egg timer you might have used in a kitchen, and the resolution for this device was 1, and its uncertainty was plus or minus 0.

5 minutes.

And don't worry if you didn't get the units for this, did you get the values for resolution and uncertainty correct? For B, again, another device we might use in the kitchen, this was a scale.

Now the resolution here was 10, meaning its uncertainty was plus or minus 5 grammes.

Now the units here were shown on the picture, but don't worry again, if you didn't get that, did you get the values correctly calculated? And for device C then, this was actually a Newton metre and you may or may not have recognised that, don't worry if you didn't, but the resolution for this device was 0.

2, and its uncertainty then was plus or minus 0.

1 Newton's.

And again, did you get the values correct? Don't worry about the units at this point.

Very well done, though, if you've got all of those correct, give yourself at least two marks for each of those, one for the resolution, and one for the uncertainty.

Well done, guys! Okay, for the last part of this task, I want you to help Sam.

Sam is writing a method during which 50 centimetres cubed of acid is going to be measured out and they're wondering which measuring cylinder to include in their equipment list.

So what I'd like you to do is read through the advice of some of other classmates, and then I want you to decide who do you agree with and why? So here we have a picture of four common measuring cylinders that you can find within the laboratory, and these are the options Sam has.

Now, Andeep suggests that, "We should use the 100-centimeter cubed measuring cylinder because it's wider, so it's gonna be easier to make out that 50-centimeter cubed mark that we need." Izzy recommends, "Using the 50-centimeter cubed measuring cylinder because you'll only have to measure 50-centimeter cubed once." Laura reckons that, "We should use the 10-centimeter cubed one because it's the smallest, and therefore, it would have the lowest uncertainty." And Lucas recommends using, "The 25-centimeter cubed measuring cylinder because it has a low uncertainty of 0.

25 centimetres cubed." So of these four suggestions, who do you agree the most regarding which measuring cylinder is best to measure out 50 centimetres cubed of acid for Sam's investigation? Pause the video here, and come back when you're ready to check your answers.

Okay, let's see how you got on.

Now, the first thing I would always recommend before making a choice about which apparatus is most appropriate for a measurement is to calculate the overall apparatus uncertainty for that measurement.

So we're gonna go through each of these different apparatus to decide its overall uncertainty.

So for the 100-centimeter cubed measuring cylinder, it has an overall uncertainty of plus or minus 0.

5 centimetres cubed.

And it's the same for the 50-centimeter cubed.

Both of these devices will only be used once for that measurement of 50-centimeter cubes of acid.

When we move to the 10-centimeter cubed measuring cylinder, it also has an overall uncertainty of 0.

5 centimetres cubed because it's being used five times.

So we have to multiply its individual uncertainty by 5 because it will be used five times.

The same is true then for our 25-centimeter cubed, we'd have to take its uncertainty and multiply by 2 to give its overall uncertainty of 0.

5 centimetres cubed.

So what this shows is that depending on the measurement, multiple apparatus can actually yield the same uncertainty for that particular measurement.

And because of that, it's actually really important that we are always calculating the uncertainty for each apparatus option before making a choice to decide, confirm really, narrow down your options, is one more appropriate than the other? And when we look at our options that were given to us for the measuring cylinders, every single one of them has the same overall uncertainty.

So if the overall uncertainty for all of our apparatus choices are the same, we're gonna have to look at another reason why one piece of apparatus might be more appropriate, and this time, we're gonna look at how many times do you need to use it to get that measurement? Because if you have to use it multiple times, you are multiplying that uncertainty and increasing actually the possibility for error.

So really, we should be choosing apparatus that is reducing the number of measurements that we need, reducing the number of times we need to use it to get that measurement.

So going back to those calculations we made before, we know that the overall uncertainty for all of our apparatus is exactly the same, so we're gonna focus in now on the number of times it needs to be used.

And we can see that the 150-centimeter cubed measuring cylinders are used once, the 10-centimeter cubed is used five times and the 25-centimeter cubed measuring cylinder is used twice.

Now all of that information then helps us to decide which of the suggestions of Sam's classmates is the best regarding which piece of equipment they should be including in the equipment list for their investigation.

Now Laura and Lucas both suggested the smallest of the measuring cylinders, and they both discussed uncertainty, which makes me think, "Ooh, maybe those are the ones that are correct." Hmm, the problem is whilst they're correct about the uncertainty for one measurement using those devices, they didn't take into account the need to make more than one measurement using those measuring cylinders.

And what that means is that it multiplies that uncertainty and increases the possibility of error.

So I'm gonna say that Laura and Lucas maybe weren't the best options for Sam to follow.

Now if we move on to Andeep and Izzy's suggestions then, because the uncertainty of both of the measuring cylinders, they've suggested, the 50 and 100-centimeter cubed measuring cylinders are exactly the same as the others, and they only need to be used once to get to that 50-centimeter cubed measurement, both Andeep and Izzy's suggested measuring cylinders would work, but Izzy's reasoning is a lot stronger than Andeep's, okay? She's talking about how many times you'd need to use that measuring cylinder, whereas Andeep is simply suggesting it's gonna be easier to see that 50-centimeter mark, centimetre cubed mark, sorry.

But you also might remember from your previous learning that the thinner the measuring device is, particularly for our liquids, it would create a more prominent meniscus or curve of the liquid within that measuring device, and it actually makes it easier to make that measurement more accurate.

So the thinner the measuring device for a liquid, the more prominent the meniscus, and the more accurate your measurement is going to be.

So taking all of that into consideration, I would say that Izzy's argument is the strongest for suggesting a 50-centimeter cubed measuring cylinder is the best apparatus to use in Sam's investigation to measure out 50-centimeters cubed of acid.

So very well done if you chose Izzy, and excellent work if you manage to include a strong because clause in your reasoning as well.

Fantastic work, guys, keep it up! Now that we're feeling a little more comfortable talking about the uncertainty involved in taking individual measurements, let's look at how we can calculate the mean uncertainties.

Now, we said earlier in today's lesson, that the precision of a measurement, so the number of decimal places it might have, implies the uncertainty or the measure of doubt we might have in that single measurement.

And at best, any measurement that we take is actually an estimate that's based on the resolution of the apparatus that we used to take that measurement.

And what we can see then is when we compare devices that do the same type of measuring, so here we have a scale that is measuring the mass of a material, that as the precision of our measurement increases, so the number of decimal places increases, the resolution of our apparatus is increasing as we move from left to right, the uncertainty, which is listed below with that plus or minus value is decreasing.

So the increasing resolution and the increasing precision of our measurement includes an implied decrease in the uncertainty of that measurement.

So the uncertainty of any measurement, really, indicates the range within which we can find the actual value for that measurement.

So when we're looking at this particular example with a thermometer, we could say the uncertainty, that plus or minus 0.

5 degrees Celsius of our measurement, which we would say here is probably around about 39.

0 degrees Celsius, could actually be read as the actual temperature is somewhere in the region of 38.

5 and 39.

5 degrees Celsius, okay? Why do we care about this? Well, because the uncertainty also gives us an indication of the range within which we'd expect to find any repeat measurements.

Now, the great thing about taking repeat measurements is you could then go on to calculate a mean value.

Now remember, when you are calculating a mean value, it's really important to first compare the results within a dataset and remove any anomalous results.

So those are results that don't seem to fit an obvious trend before you then go on to calculate that mean value.

So let's look at an example.

I have here a set of measurements that were taken for the extension of a spring with different masses that were hung on that spring.

And as a reminder, to calculate a mean value, you take the sum of the results for that set of data, divided by the number of results in that set of data.

So if I look at this first row, I can see that all the values are within about 0.

1 of each other.

So I don't seem to have any anomalous results, so I'm gonna add all of them together, divide by 3 'cause I have three different data results, and I get a mean value then of 5.

7.

If I move on to the next row, for the 200-gram mass, I've got a value of 11.

4, 11.

3, and then 16.

9, that stands out as not fitting the obvious trend.

So when I calculate the value ,the mean here, I'm going to exclude that 16.

9 value, and that will give me then a mean of 11.

4.

For the 300-gram mass, then there are no anomalous results, and I would carry all of them through to calculate a mean of 22.

7.

Let's stop here for a quick check.

Which of these values do you think should be excluded as being anomalous when calculating the mean value? Well done if you said D.

When I compare these different values, I can see that the vast majority of them are in the low 40s, except for one, 49 seconds seems quite high when you compare them against the rest, and therefore, it's anomalous because it doesn't fit the trend of the other data points.

So well done if you chose D, good job, guys! Now we learned earlier in today's lesson that we can calculate the uncertainty for an individual measurement.

However, the uncertainty that we could calculate from a calculated mean, so that's repeated measurements, actually provides a far better indication of the range within which we can find an actual correct value for our measurement.

And so the mean uncertainty is going to be equal to the range of repeated values divided by 2.

So let's go through an example.

I have here the values of our repeated measurements from that previous question, and we already said that the 49 seconds was anomalous, so I'm gonna discount it from my dataset.

To find the range of repeat values, I'm going to take the highest value and subtract it from the lowest value.

So I need to find the highest and lowest value within my set of data and subtract them.

And for this set of data, the answer is 2.

I then take that range of repeat values and divide it by 2, and here I get a value of 1.

That means then, that my mean uncertainty for this set of data is plus or minus 1 second.

Let's go through another example.

I have here the repeat measurements for the melting point of a specific substance.

And I would like to know the mean uncertainty for this melting point.

What is the range within which I should be able to find the actual melting point for this substance? So the first thing I need to do is look at the data that's been collected and decide if there are any anomalous data points.

And when I compare these, I can see that the temperature of 167.

0 degrees Celsius does not match the trend of the other melting points, and therefore, I'm going to exclude it as I go forward.

The second thing I need to do then is I need to find the range within which my data points have been collected.

So I find my highest and lowest values, subtract them, and my range here then is 2.

5.

To determine the mean uncertainty, then, I'm going to take the range and divide by 2.

So the melting point uncertainty for the mean is plus or minus 1.

2 degrees Celsius.

What I like you to do now then, is to determine the mean uncertainty for the repeat values that were taken for the stopping distance of a car that was travelling 50 miles per hour.

And you've got then the outline of the example that was given to you on the left to help you in your calculations.

So pause the video here, and come back when you're ready to check your answer.

So the first thing we needed to do was to identify the anomalous data if there was one in this dataset, and we did find one, it was 49.

4 metres.

Once that's been removed from our dataset, we find the range within the data to then find the uncertainty, and when you've done that, you should have had a final answer of plus or minus 0.

9 metres was the mean uncertainty for this set of data.

So very well done if you managed to calculate that correctly.

Remember, each step in here is usually worth at least one mark.

So be kind to yourself as you're marking your work.

Did you manage to find that anomalous measurement? Don't worry if you didn't, it's better if you have, but if you have forgotten it, did you at least do the next two steps correctly showing out your working, okay? Because if you do make a mistake and kept that anomalous result in, if you show your working going forward, you can get what's called an error carried forward mark, so it's always important that you're showing your work.

If you did get that correct answer, though, at the end, very, very well done! Keep up the great work, guys! Okay, now for the last task of today's lesson, what I'd like you to do is to find the mean value and its uncertainty for each set of repeat measurements below.

This is gonna take a little bit of time, you may wish to discuss whether or not there's an anomalous result within this data with the people nearest you, so definitely pause this video and come back when you're ready to check your answers.

Okay, let's see how you got on.

So the first thing I would do whenever I'm being asked to do multiple calculations is just to remind myself of the processing that's involved.

So to remind myself the mean is equal to the sum of the results, divided by the number of those results.

And then the mean uncertainty was equal to the range of my results divided by 2.

Now, before I can even calculate the mean, I need to double-check, are there any anomalous results? And in this set of data there was, it was 150 degrees Celsius.

So you needed to remove that from your dataset, and then calculate the mean, and the uncertainty of that mean, and when you do that, you should have had these answers, 1 29 0.

5 degrees Celsius, and that's plus or minus 3.

5 degrees Celsius.

So very well done if you managed to get those correct! Now, if we carry that process through for the rest of our questions, remember the first thing we need to do, is see are there any anomalous results? And in this set of data there was, and then to calculate the mean and the uncertainty, you would've carried out this processing, which gives you a mean value of 12.

7 centimetres cubed.

And the uncertainty then, was plus or minus 0.

6 centimetres cubed.

Now if you didn't get those answers, maybe pause the video, double-check the working out, and you can see where you may have gone wrong, but very well done if you've got those correct, guys! So for part C, then again, looking for any anomalies.

And when I compare them, I see that there aren't any.

And it's really important to remember that sometimes, datasets don't have any anomalies.

So when I am calculating the mean and uncertainty for this dataset, I need to include all of the values.

And when I do that, these are the answers I get, a mean of 624.

1 grammes with an uncertainty of plus or minus 0.

3 grammes.

Moving on to the last question in this task, then, again, looking for those anomalous results if there are any.

And when I compare these results, I find two anomalous results.

And that does happen sometimes, so it's important that you're looking at all of them, it's not always that there is one and there's not the only, that there's only one, you can have multiple anomalous results in your dataset.

So by removing those two values, I get then a mean volume of 24.

17 centimetres cubed, with an uncertainty of plus or minus 0.

025 centimetres cubed.

So incredibly well done if you manage to get these correct, guys.

Now don't forget that you should be giving yourself at least as a minimum two marks per question here, one for your mean, and one for the uncertainty, and I would have it a guess to give you another one for that anomalous result if you manage to identify that as well.

So really, really well done! Now, we have done a lot in today's lesson that can be very easily confused.

So let's summarise what we've gone through today.

Well, we've learned that there's always some level of uncertainty, a measure of doubt whenever a measurement is made because we're never quite sure if that measurement we're taking is on the line of our measuring device or between those lines.

But the uncertainty of an individual measurement can actually be calculated and it's equal to half the resolution, the markings of the apparatus that was actually used to take that measurement.

And that the higher the resolution, so the more markings on that apparatus, the lower the uncertainty is in an individual measurement.

Now, the range of your repeated measurements can actually be used then to calculate an uncertainty for a mean value.

And that's really useful because the uncertainty of our measurement, of a mean then, provides a range within which the actual value can be found.

And then all of it links back to that choice of equipment.

Why did you choose that particular apparatus for your measurement? So some really, really important aspects here for practical work going forward.

I hope you've had a good time learning with me today, I certainly had a good time learning with you, and I hope to see you again soon.

Bye, for now!.