Lesson video

Hello, I'm Mrs. Lashley, and I'm really looking forward to working with you throughout this lesson.

So in today's lesson, we're gonna be thinking about the mean and what does it actually mean, what is it doing, and how do you apply it to data? Some keywords that you will have met before in your studies.

So just take a moment to read through them, and then when you are ready, we'll move on to a new keyword for this lesson.

Today, we're gonna be working with the mean, and that is a central tendency measure.

So here on the screen is the definition of a central tendency.

Our lesson has got two parts.

The first part, we're gonna think about what does it mean to distribute evenly? And the second part, we're gonna formally calculate the mean.

So we're gonna make a start on that first part.

We're gonna think about what happens when you distribute evenly.

Through the next few slides, we're gonna work with the same concept and that concept is about distributing clay blocks evenly.

We can't add any clay blocks and we can't remove any clay blocks, but all we can do is redistribute them.

So that concept is gonna be running throughout the next few slides.

Here we've got Laura and Andeep.

They've got an uneven amount of blocks each, but can we redistribute the blocks such that they both have the same amount? Well, Andeep had more than Laura, so Andeep could have given the extra block that he had to Laura.

Now, they've both got the same amount of blocks.

So the same concept.

Can we redistribute the blocks such that they end up being distributed evenly? So I'll let you think for a moment.

Izzy has more than Sam, so hopefully, you're thinking about the fact that Izzy needs to give some.

So if she gives one away, they're still not even.

If she gives two away, though, then they now have an even towers.

So if Izzy gives two blocks to Sam, then they've now been distributed evenly.

A quick check for you to do.

So Lucas and Alex have some clay blocks as shown and you need to redistribute the blocks such that they have the same number as each other.

Pause the video whilst you're having a go at that, and then when you're ready to check your answer, just press play.

If Lucas gave one block to Alex, then they have the same amount of blocks.

So now we've increased the amount of people.

So we've got Lucas, Sofia, and Alex.

So at the moment, this is distributed unevenly.

So how do we redistribute the blocks such that they are distributed evenly? Well, we know that Lucas has got more than the other two, so it must be Lucas that's giving some blocks away.

He gives one to Sofia and one to Alex.

So what about Sofia, Jun, and Sam? Can they have clay towers of the same height? Hopefully, you recognise that both Sofia and Sam are gonna have to give away some of their blocks.

And so, if they each give away one block, then they've all got the same amount.

Now, our towers are evenly distributed.

Up to this point, you might have started to think about what was going on.

So what is staying the same in each redistribution? Well, the number of towers is staying the same.

If there was three people, then there were three towers, and that was the same regardless of if it was unevenly distributed or whether it was evenly distributed.

The other thing that was staying the same was the number of blocks in total across all of the pupils.

So what is changing? Well, the number of blocks in each of the towers is changing.

So now we've got four towers, we've got four pupils, we've got Izzy, we've got Jun, Jacob, and Laura.

And the same question, the same concept.

Can we redistribute the blocks such that everybody ends up with the same height of tower? The four of them have decided they're gonna actually all give in their blocks, put all their blocks together, and then they'll redistribute.

All of their blocks have now gone in as a total group and they're gonna start to redistribute them.

And if you was gonna do this, you'd probably do it the same way, which is one to you, one to you, one to you.

So, one's gone to Izzy, one's going to Jun, one's gone to Jacob, one's gone to Laura.

So we've still got some blocks left, so we're gonna repeat that.

We've still got blocks left.

We'll do it again.

There's still blocks so we can do it again.

And we've redistributing the last four and now they've all got the same height.

The colours are all mixed up, but that didn't matter.

It's all about just having an even amount of blocks regardless of the colour.

We've got four different pupils this time.

We've got Alex, Aisha, Jacob, and Sofia, and they want to have towers of the same height as well, again without adding any extra blocks or removing any blocks.

So our total blocks needs to stay the same and we need to end up with four towers.

So remember they were the things, what was staying the same, was the number of blocks in total and the amount of towers.

And what's gonna change is their individual heights.

This method of getting a total number and then sharing them out evenly is what we're gonna see again here.

So we've put them all together.

They all get one block each.

They all get another block, and they all get another block each, but this time, there's one block left.

So here, we've got a problem because we can't just give one block to one person because then they will not be distributed evenly.

So to distribute it evenly, we need to share that one block across all four towers.

And so, that would be a quarter of a block that they all gain because we're dividing by four, we're sharing it into four.

So our towers are now the same height.

We have distributed evenly, but this time, it wasn't whole blocks.

We've had to divide one of the blocks up.

And this happens in real life that sometimes you're gonna have to share things out by splitting it a little bit more.

Okay, so this is what we've done here.

Has it still kept with the same as before? Yes.

The total number is still the same.

The number of towers is still the same and their heights have now changed.

This dashed line is the height where they were evenly distributed.

And this is the mean value.

The mean value for this scenario was 3.

25 or three and a quarter.

So you can see here that Alex lost some blocks, Aisha gained, Jacob gained, and Sofia lost some blocks because we were distributing them evenly.

Aisha has recognised that actually there's the same amount of blocks above the mean line as there is below the mean line.

And you can see that with the empty spaces and the pink ones above the line.

So a quick check.

Can the mean value be positioned here? Pause the video, think about that.

And then when you're ready to check, press play.

So the mean line couldn't be there.

It's always gonna be within the data because some of them will have given away their blocks, and it'll therefore, the mean line will be lower than their starting heights.

And some of them will have gained blocks and so it'll be above their heights.

But this mean value is, or this line is above all of them.

And so, the mean is always within the dataset.

This was where the correct position of the mean would be.

So what we've just been working with is actually the mean.

So the mean is an average.

It's quite common for people to use the word average and are talking about the mean.

The mean is not the only average.

So we should be very careful that we don't just say "average", we say "mean average" or say "the average" and then stipulate that it is the mean or just to say the mean.

The mean as an average uses all of the data values.

So when we looked at those clay blocks, everybody's clay block was being distributed out and it's evenly distributed.

So when they collected them all together, that was our total, and then we shared evenly between the amount of towers.

The value itself, the mean value, doesn't have to be an actual value within the dataset, didn't have to be an original height of one of the towers, but it is a single value that is attempting to describe the centre of the dataset.

And that's a measure of central tendency.

So the mean is a measure of central tendency because it finds a single value that is trying to describe the central part of the dataset.

So three pupils each have four blocks after redistribution.

So in other words, the mean is four.

So how many blocks might each pupil have started with? So now we're thinking about it in the other direction.

If we know the mean, so we know what their heights of their towers end up being, what could their original heights have been? So here's some example.

So here we could have had Lucas with three, Izzy with five, and Andeep with four.

Lucas could have only had two, Izzy have five, and Andeep have five.

Lucas could have had one, Izzy could have had six, Andeep could have had five.

Lucas could have had six, Izzy could have had zero, and Andeep could have had six.

All that mattered was that there was 12 blocks across the three stacks.

The original distribution, the original dataset, doesn't make any difference to what the heights are once they've been evenly distributed.

So here is another example, which is that Lucas has all 12 at the beginning, Izzy has zero, and Andeep has zero.

The mean would still be four even though Lucas started with all 12.

So the mean value of four blocks is because there are 12 in total, evenly distributed, or you could use the word "shared" across the three stacks.

So a check.

Which of these towers would have a mean value of three? Pause the video whilst you're working that one out and then when you're ready to check it, press play.

Hopefully, you figured out it was A.

So the key thing here is if the mean value was three, there needs to be nine blocks in total.

It might have been that you could just see, you could move the one from the middle tower onto the left tower and there would all be three, that would one way you could have worked it out.

But we want to try and start thinking about this idea of a total.

There is nine blocks over three towers.

So that would be a mean value of three.

So now we're gonna do some independent work, some tasks on this idea of distributing evenly.

Question one, you need to match up the diagrams. Pause the video when you're ready to have a go at that question.

And then when you're ready for the next questions, press play and come back.

So question two and three are both on the slide here and these are worded in context questions.

Read them carefully and then when you're ready for the last question, press play.

Okay, and then question four is about a group of friends who have collected some conkers and they're gonna share them evenly.

Pause the video whilst you have a go and then when you're ready to go through the answers, just come back and press play.

Okay, so question one, you needed to match up the diagrams and there was two blanks where you needed to complete them.

So hopefully, you were thinking about totals and sharing evenly, distributing evenly.

So just have a check.

On the top row card, the blank where it was an original distribution, you needed to just have 15 blocks arranged in three rows.

So one example of that is this.

Question two is about the delivery driver trying to even up his crates.

So he had two crates.

One had 18 in it and the other had 12.

In total, there were 30 packages across two crates.

So he needed there to be 15 in each for it to be evenly distributed.

And that would happen by taking three out of the crate of 18 and placing it into the crate of 12.

Question three, batter has been mixed up and then split across three tins.

When it's been done by eye and then weighed, there was one kilogramme in one, 1.

2 kilogrammes in another, and 1.

4 kilogrammes in the third.

And if these were to be baked, there would be different depths because they've got more batter in or less batter in.

So we need to redistribute the batter evenly across the three tins.

So how much batter should there be in each tin? Well, in total, there was 3.

6 kilogrammes of batter, and this needs to be shared evenly between the three tins.

So that would be 1.

2 kilogrammes per tin.

Question four, a group of friends collected some conkers and shared them equally among themselves.

So they've distributed them evenly.

Part A said that there were five friends and they've got six conkers each.

So how many conkers did they collect altogether? So this is saying, that you've got, if we go back to the towers idea, there was five towers and the mean value was six.

So if you've got five towers of six, that has a total of 30.

So 30 conkers would've been collected.

Going on to part B.

Another friend joins them, but they don't find any more conkers.

How many conkers would they each have if they now share them equally? They've not got any more conkers.

So the total of 30 has stayed the same.

What's changed this time is we're not distributing across five because another friend joined.

So you are now distributed across six.

So how would you share 30 conkers across six people? They would have to get five conkers each.

So we've finished the first part and we're now moving on to the second part of the lesson, which is about calculating the mean in a more formal manner.

As the number of towers increases, it becomes more difficult to visualise the even distribution.

It's much more difficult to think about who needs to give who, how many blocks.

As we saw in the first part, the mean is the total and then shared evenly across all of the towers.

So on this image, it's actually 80 blocks in total.

And you could do that by counting them.

There are 10 pupils.

Again, you could count how many pupils there were.

It's a much easier to calculate the mean value than to collect them together physically and then redistribute them or to try and work out who gives who which blocks.

So there are 80 blocks in total.

That's gonna stay the same even after we distribute them.

There are 10 pupils and 10 towers.

And that will stay the same even after the distribution, which would mean that an even distribution of the blocks rather than this uneven distribution would be eight per tower.

And that's the mean value.

And if we've just put the mean value line on there at the moment, again, considering what's happened, the those that have got above the mean, who started with above the mean will have to distribute theirs out.

And those that started with the tower below the mean value gain, all of them would end up with eight blocks in their tower.

The mean is an average and a measure of central tendency, and it's gonna be used for large data sets, not just 10 pieces of data or three pieces of data.

So as the data sets would get larger, physical redistribution, although that's what the mean is doing, the mean is collecting the total and then evenly distributing it among the data points, it becomes an impractical method.

If you've got even more towers, then the idea that you can just move pieces around in order to make them be evenly distributed becomes an impractical method.

So we need to have a better method that works for large data sets because ultimately, statistics is all about data, and data is often very large in terms of how much you are collecting.

So the theoretical even share, which is what we, which is the mean value, is still able to be calculated.

You just need two things.

You need to know the total.

So in all of this concept, it was about the total blocks, but just the total, the grand total.

And then you need to know how many you are sharing it between.

So in this concept, it was about how many pupils, how many towers, but ultimately, how many pieces of data have you just totaled? If this was a large data set, we've not got the images anymore because our large data set has got 3,500 values.

So just to link that back to the towers idea, that's saying that there was 3,500 towers.

We can't possibly imagine showing that on the screen.

And the total of the data set is 15,750.

So again, with the blocks idea, that's how many blocks, and the 3,500 is how many towers.

So if we want the mean value, we need to evenly distribute the total by how many data points there are.

So it's gonna be the total of the values divided because we're sharing evenly, so division is the operation we're gonna use by the number of values.

15,750 divided by 3,500 is 4.

5.

So if this was a tower situation, they'd all end up with four and a half blocks per tower.

So one for you to have a go.

A large dataset has 4,800 values and the total of the dataset is 35,520.

Pause the video, have a try at that.

Think about which one's the amount of blocks, which one's the amount of towers, and then you are sharing, so that's the division evenly.

When you are ready to check your answer, just press play.

So you hopefully, did 35,520 'cause that's the total, that's what you need to share out.

And you are sharing it between 4,800 pieces of that data.

So 7.

4 was your mean value.

So we've got a list of data.

So it's not a large data set per se, but there's a few data points on the screen and you can represent data using what's called a dot plot.

So each dot represents a piece of data and the dot appears above their value.

So here we've got 20, 21, 22, 23, all the way up to 40.

What the dot plot is showing is that there is one data point for each of those values because there is only one dot above it.

So just to check your understanding of a dot plot, how many data points are represented on this dot plot? Pause the video and then come back when you're ready to check your answer.

So there are 10 data values and we know that because there are 10 dots.

So the amount of dots is the amount of data values.

Another check.

Write out as a list the data set represented on this dot plot.

Pause the video whilst you're writing those down and then when you're ready to check your list, press play.

So hopefully, your list of data reads 23 because there is a dot above 23, 25, 28, 30, you should have written down two 31's and that's because there are two dots above the 31, 32, 35, 36 and 40.

So this is the same dot plot that we have seen previously.

So there is one of each of those points.

Each of those would be the amount of blocks per tower.

If we added them up and then redistributed them evenly, the mean value would be 30.

Each tower would end up with 30 blocks.

Some of them would've gained blocks, some of them would've lost blocks.

And because the data is completely symmetrical, there's no missing numbers in between, there is a point, they're integer values, they're consecutive, it's completely symmetrical, the mean is the central value.

And remember that the mean is a measure of central tendency.

In this case, because of the symmetry, the mean is the central value.

So it is trying to describe the centre of the data and in this case, it gets it completely accurate because it's symmetrical.

This dot plot represents the dataset 28, 28, 32, and 32.

There are two dots above the 28.

There are two dots above the 32, and that's why there are two of each of them.

The mean is still 30.

So if you was to add it up and divide by four, 'cause there are four data points, you'd still get 30.

The data is symmetrical about the mean value there.

There are two points, two away from the mean and two points above the mean.

It is symmetrical and the mean is therefore 30.

This set on the dot plot is less symmetrical in its distribution.

And Andeep and Jacob are discussing what the mean value would be.

Andeep said the mean value for these data will be central to the two extreme points, and Jacob said, but is it not matter that there's quite a lot of data bunch towards the right side? So does that data not have an impact on the mean value? So the mean value is a measure of central tendency and it calculates it.

I remember, what do we do? We total everything and then we share it evenly.

We distribute evenly across the data points.

So where will the mean be? Well, the mean is 33.

3.

If you calculate it, you add them all up and you divide by how many points there are, you come out with 33.

3.

So that isn't central to 21 and 40.

That would be 30.

5 if it was central to those two.

Jacob recognised that there was a grouping of data towards the right hand side, and that's why the mean has sort of being pushed up because all of those have contributed to the total.

So the total is larger because of those larger values.

So when it's being evenly distributed, it's towards the majority rather than right in the middle.

The mean is a measure of central tendency.

It's trying to find a single value that describes the data.

And 33.

3 is more descriptive of the dataset than 30.

5.

When we are looking at data, and especially, if we've got it on a dot plot, where we can sort of see shape and distribution, we can can get a vague idea of where we think the mean will be.

So here, we've got a dot plot showing a data set.

So without calculating, without adding up all of the values and dividing by how many there are on that dot plot, what will the mean be? So the mean's gonna be close to 27 as that's fairly central to the majority of the points, but that large data point of 38 has been included in the total.

So it'll increase the mean slightly and the mean is at 27.

8.

Here is a check.

Laura and Sophia are discussing the mean of a dataset shown on the dot plot.

I would like you to read both of their suggestions and then think about who you agree with.

Pause the video whilst you are reading them and making a decision.

And then when you're ready to check your answer, press play.

Sophia would be correct.

She's identified that the data is fairly symmetrical and we are trying to find a measure of central tendency with the mean, and therefore, it's roughly 34.

5, which would be central to 33 and 36.

It's slightly higher because of the total being increased by those larger data points.

Sophia has done some recording of the outside temperature every two hours over the course of the day.

And here is the findings of her temperature collection.

Work out the mean temperature for that 24-hour period.

We need to add up all of those temperatures that she collected and divide by how many there are.

There are 12 pieces of data because she did 12 recordings.

So we'd do the total and then we're gonna divide by 12, and that comes out as 4.

75 degrees Celsius.

So this is the mean temperature over a 24-hour period.

If we then decided that we wanted to only find the mean of certain points, so the middle sort of, of the day from 10 till 2:00 PM, then we've got only three points.

So the difference here is we are still gonna add up and we're still gonna divide, but we're only gonna divide by three because we are only totaling three recordings.

And this comes out with a mean temperature of 10 degrees.

So that middle section of our sort of waking day was 10 degrees Celsius, double figures.

And you can see how that's very different to the whole 24-hour period.

We can do the same thing again, but looking at the very final three recordings of the day, so from 6:00 PM until 10:00 PM.

Looking at the data that we've got there, we've got two positives and we've got one negative.

So what's that gonna do for the mean value? When we total them and divide by three, it comes out with a mean value of three degrees.

And that's because the data is symmetrical about that three, the seven is four degrees above and the negative one is four degrees below.

So there is the symmetry of the distribution, the mean is the central value.

Here's a check.

And here, you're not working out the mean, you're just considering what the mean will be based on the data that you would use.

The mean temperature for the first three recordings, so that's from midnight, two o'clock in the morning, four o'clock in the morning to the first three data points, would the mean temperature be positive, negative, or would it be zero? Pause the video whilst you think about that and then press play to check your answer.

Well, negative, and that is because the total will be negative.

Negative four plus negative one plus two is still a negative answer.

So regardless of what you have to divide it by, in this case, it's dividing by three, then the result would still be negative.

We're now gonna think about means and comparing means.

Izzy has calculated that the mean age of students in her class is 12.

5 years or 12 and a half years old.

Lucas has calculated that his family have a mean age of 12.

5 years.

So these two data samples, one is the class and one is Lucas's family have got the same mean age.

So does this imply and does this mean that the class and Lucas' family are all the same age? I'm hoping, logically you thought, well, that cannot be the case.

Lucas's family cannot have the same ages as is his classmates.

And so, the classmates will all be of a similar age.

The way that sort of our system of education works, that you are all of the same age in one class.

So there'll be a mixture of 12-year-olds and 13-year-olds, those that have had their birthdays and those that are still waiting to have their birthdays.

Whereas Lucas's family must have at least one adult, and therefore, the adult age is gonna be much higher than 12.

5 years, the mean age.

So to counter that, to make that adult age drop down to the mean age of 12.

5, there must be younger members of Lucas's family as well.

The dot plot here shows a suggestion of how they could both be 12.

5.

So above the scale line is the classmates.

We've got the same amount of students that are 12, the same amount of students that are 13.

It's therefore, very symmetrical.

The mean is in the centre of the two, which is 12.

5.

For Lucas's family, which is below the the scale line, we've got Lucas as the age of 12, then we've got an adult, and then we've got two younger siblings.

Their mean age of the data is the same, but it doesn't mean they are all the same data points.

Two data sets are shown on dot plots.

So what is the same and what is different? Well, the mean value's the same.

You've got that marked onto both of them.

The mean value is 30.

8 in both of the data sets.

It happens to be here that the number of data points is also the same on the dot plot at the top and the dot plot at the bottom, there are the same number of points.

If we were to add all of those data points up, the total would be the same.

Is there anything different between the two? The data set itself is different.

For example, the top one has two 27's, whereas the bottom one doesn't have any 27's.

So the data set that is being represented here is different and the distribution and the shape of the data, the bottom one is more spread out, goes from 20 to 40, the top one only goes from 25 to 37.

So this is another example of what we just saw with the ages, that the mean values can be the same, but actually the data set themselves can be quite different from each other.

If they've got the same amount of points, then their total will be the same.

If you've got different amount, like in Lucas's family and the class, their totals were different.

Jun plays basketball, and each week at practise, he attempts 10 free throws.

And what he does is he collects the data about the number that he successfully gets in out of 10 attempts.

So he is gonna do it 10 times and he is gonna record how many of those he actually gets through the basket.

The data for six weeks is that he gets seven every week.

He's got seven out of 10 of his practise throws into the basket.

What's the mean average number of baskets that he scores out of 10 attempts each week? The data is already evenly distributed.

So remembering that the mean evenly distributes the data, it collects it all together by finding the total and then it shares it between how many points there are.

Here, the data is already shared evenly because he got the same amount each week.

So therefore, the mean value is seven.

Continuing with that thought process, here is a check.

Pause the video whilst you read through the options, make your decisions, and then when you're ready to check them, just press play.

So it is true.

It doesn't matter how many data points there are.

If all of the data points have the same value like we just saw in Jun's example, then the mean value will be that same value.

Justification is B.

And you can see there the algebraic explanation of why this is always true, that if A is the value, so that was the seven engines example, and N is the number of points, which was six, then A multiplied by N, divided by N will just be A, and it will always be A for any number.

And so, that's why it doesn't, regardless of how many points, if they're all the same, the mean will be that value.

Now you're gonna do some practise for calculating the mean.

So question one, calculate the mean for the following sets of data which are being represented on a dot plot.

Pause the video whilst you're trying that question and then when you're ready for the next one, press play.

Question two is about large data sets, where you don't need to do the totaling yourself.

You've been given that, instead, you just need to calculate the mean.

So question two, calculate the mean given the following information.

Pause the video whilst you do that, and then when you're ready, press play.

The last question of this practise, question three, you need to match up the totals on the left hand side with the values and the mean value on the right hand side.

There are two blanks for you also to fill in by yourself.

Pause the video whilst you complete that question three, and then when you press play, we're gonna go through the answers to all the questions.

Question one, you needed to find the total.

So that was adding up all of the values of the dots and then dividing by how many there were.

There were 10 on each of the dot plots.

The mean for A was 35.

4 and the mean for B was 30.

Question two, this was the large data set where you needed to calculate the mean, given the total and given the amount of values.

So if part A was 3.

25, part B was 6.

5, and part C was 16.

8.

On part C, the wording was the other way round.

So you were given the total before the amount of values.

Hopefully, you picked that up and you did the division the correct way round.

Question three, here are the matches and the missing values was the mean of five and a total of 143.

To summarise this lesson, which was check and understanding of the mean, the mean is a measure of central tendency.

It's not the only measure of central tendency, but the mean is one of the measures of central tendency.

The mean is the value at which each data point would take if the total value has been evenly distributed.

And to calculate them mean, you sum all of the values and divide by the number of values in the set.

Well done today.

I look forward to working with you again in the future.