Lesson video

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

So during the lesson today we're gonna look at calculating the mean, but not just from a list but instead from different representations.

So for example, frequency tables and different graphs.

Some keywords that you've met before and we'll be using during the lesson are on the screen now.

So pause the video and just re-familiarize yourself with those and then when you're ready, press play.

Our lesson has got three learning cycles within it.

The first one is calculating the mean from a line graph.

The second one is calculating the mean from a frequency table.

And the third one is calculating the mean from a bar chart.

Our outcome is all about calculating means from different representations.

So we're gonna start with the line graph.

So data has been collected about the number of days of frost in February over nine different years, and we can see that in the table.

So the mean number of days of frost has been calculated by Aisha and Jacob.

Aisha's done 67 divided by eight and got 8.

375 days of frost.

Jacob has done 67 divided by nine and that's approximately 7.

4.

So there's a little bit of rounding with his value at the end there, which is why it's approximate.

Who is correct in their calculation? So Jacob is correct.

The zero, there was no days of frost recorded in February, 1961, but that is still a piece of our dataset.

That is still a recorded value.

So there are nine pieces of data.

So they both got the same total when they added up their data.

But Aisha divided by eight.

She ignored the zero, but the zero is a piece of data.

That same data could be represented as a line graph or a time series graph.

So each point is one of the data.

So we can see that zero in 1961 where the line graph has gone back to the horizontal axis.

So to find the mean, we would need to read each point, add them up, and divide by how many there are.

This orange line that's just appeared is the mean value.

So the mean value of frost, which is about 7.

4 days of frost.

And we can see that we've got data above the line and data below the line 'cause the mean is a measure of central tendency.

How many data points are there? Pause the video whilst you're working that out.

You should have just counted how many points there were and there were 23 points.

So another check for you, it's the same line graph.

So pause the video, work out how much rain there was in April, 2014 and then press play to check your answer.

40 millimetres.

So it was about going to the year 2014 and reading off from the vertical scale.

The line graph here shows the time taken for someone to travel to work over the course of one week.

So it's a working week, Monday, Tuesday, Wednesday, Thursday, Friday.

To find the mean time taken, we need the total time and then we're gonna share it evenly across the five days.

But as a check, I'd like you to take the data off of that line graph so you can complete the table on the right.

You've got Monday to Friday, and how long did it take on each of those days? Pause the video and then come back when we can move on.

Okay, so Monday was 22 minutes to get to work.

Tuesday 26, Wednesday 24, Thursday 23, Friday 18.

The table has now just been transposed.

It now runs horizontally but it's the same data.

So we've got Monday to Friday and then the minutes taken per day.

So to find the mean we need to total the travel time.

So just by adding them up that's 113 minutes.

And so the mean time taken per day, we're gonna evenly distribute that time across all five days.

Gives you 22.

6 minutes a day.

So now you're onto a bit of practise for line graphs.

So here you've got to use the line graph to complete the table of data.

This line graph is representing the maximum temperature on the first of every month in 2022 rounded to the nearest degree Celsius.

Pause the video whilst you're writing down that data and then when you're ready for the next question, press play.

This question, you need to calculate the mean from the line graph.

This line graph has got two sets of data represented.

So this is about minimum temperature on the first of each month in 2022.

We've got data from Aberporth and data from Lowestoft.

Pause the video then when you're ready, head back.

So from question one, you needed to pull off the data.

Question two, you needed to calculate the mean for two different sets of data, but both were showing the same context, which is the minimum temperature on the first of each month.

So Aberporth, if you pulled the data off, it was the blue line, then the total temperature was 101 degrees Celsius and there were 12 recordings.

12 points because it was once each month for a year.

And so the mean average would be 8.

42 degrees Celsius and for Lowestoft, the total temperature was 96 degrees and their mean temperature was eight degrees Celsius.

So slightly lower than Aberporth.

So we're now moving on to the second part of the lesson, which is still about calculating the mean.

But instead of from a line graph, we're now gonna think about a frequency table.

Jun is asked the question, can the mean be calculated for all types of data? And Izzy said, while the mean is calculated by finding the total and dividing by the number of data points.

So data such as 3, 5, 5.

5, 6, 6.

5, 10 has a mean of six.

We can add them up, we can find the total, and we can share them evenly between how many pieces of data there were or data such as this.

So here is data from a spreadsheet.

The mean can still be calculated by adding up all of those numbers.

There are lots of numbers.

So we'd classify this as a large data set.

All the numbers can be added up and then you could divide by how many there are.

And digital technology is very useful to do this for us.

And so the mean if you add up and the sum code gives you how to find a total and then divide by how many pieces of data there are, which is 814, we get a mean value of 9.

24.

So when the data sets get larger, technology can be really useful to do the adding up for us and the division.

Data is red, blue, red, pink, orange, red.

So Izzy said, well the mean is calculated by finding the total, but I can't find the total for this.

So where the data points are not numbers, where they don't have a numerical value, it's not possible to find a total.

So you cannot find a mean.

So Jun's question is, can the mean be found for all types of data? Well the answer is no.

If it's non-numerical data, then the mean is not an average that you're gonna be able to use to do any analysis on it.

So a check.

For which of the following data can the mean be calculated? Pause the video, think about that and consider that.

And then when you've got what you think is the answer, press play to check it.

A and C were the data sets that you could find them in 'cause you can total them.

So here we've got a plan view.

So looking from above of a restaurant and its tables.

The brown is the table, a circle, the blue disc is representing where a person is seated.

And so therefore if you've got a brown with two circles on it, that shows you that there are two people sat at that table.

So how many tables are there at this restaurant? It's a plan view so we're looking from above.

That's just a case of counting how many brown shapes there are.

And there are 11 tables, different sizes, but there are 11 tables in the restaurant.

How many people are dining at this restaurant? So again, remember that the blue circles are representing a person is sat there because that's their dinner plate.

So we need to just count up the dinner plates and there are 33 of those on our plan view.

So we've got 33 people sat at the restaurant over 11 tables.

The same data can be shown on a dot plot.

So we've got a brown table there and what that is showing, we've got one 'cause there is one table above the zero.

We've got one table where there is nobody sat.

So a restaurant, you're a diner when you're sat there.

It's the same dot plot, we've just got teal colour now here.

There are 11 dots and that's because there are 11 tables in the restaurant and and we knew that from counting them on our plan view.

What we also knew was that there was 33 diners in the restaurant and we got that from our counting of the blue circles.

But how do we get that off of this dot plot? Well this dot represents zero diners.

There is one table, but the number of diners per table is zero.

So one table of nobody still means that there are no diners.

So our total is zero.

This dot above the one means that there is one table of one person.

There is one person on that table.

So that adds a one to our total.

So now we've got a total of one.

These two dots represent four diners because there are two tables with two people on them.

So each of those dots represents two people.

So there are four diners.

So our total have gone up by four.

The next two dots represents six diners and that's because each table has three diners at it.

So that's an extra six.

These three dots represent 12 diners.

There are three dots above the four.

The four represents four diners at the table.

Three lots of four gives us 12.

We've got another 12 diners in the restaurant.

And then lastly, we've got these last, the larger groups of people, which is five per table.

And there are two dots which represents two tables.

So we have 10 diners sat in groups of five.

And so if we add them up we get our total of 33.

So the amount of dots shows us the 11 tables, but the 33 is less obvious and that's because each of those dots were representing a different amount of people depending on where they were located.

So the same information could be shown as a frequency table.

The zero group size is basically asking you the question, how many empty tables are there? If the group size is zero, then the table has no diners and we've got one and we can see where that is in the restaurant.

How many tables have one diner? And there's only one of them as well.

So we can see that there on our plan view.

How many groups of two are there? How many couples are there in the restaurant? Well there are two of them, and here is the completed frequency table.

So what does the three in that frequency column represent there? Well, it represents the three tables that have four people sat at them.

So the three is the frequency and the three means the amount of, and that's the amount in this case, the amount of tables with four people at them.

So we know that there are 11 tables in the restaurant because we could count them from our plan view.

But if we only have the frequency table, how would you get 11? Well this would be adding up the frequencies.

It would be the total frequency.

How many people are sat in groups of five? So again, if you didn't have the plan view, if you couldn't just look for that on the plan view, if you only had the frequency table, where would you find the answer to that question? Well, it's the two in the frequency.

So how many people are sat in groups of five means we're gonna look at group size five.

We're on that row and the frequency tells us the how many, the amount.

So there are two groups of five, which means that there are 10 people sat in that configuration.

So we also know that there are 33 diners in the restaurant.

But how again could you get that from the frequency table? Well here's a frequency table.

We've got that three in the frequency column and this represents three tables each with four diners.

So that means that there are 12 diners in that row.

So if we've got three tables of four, then in total, that's 12 people.

And the two here, we've already used this one.

The two here represents two tables each with five.

And so that contributes another 10 diners in the restaurant.

We've got 12 diners that are sat in groups of four and we've got 10 diners that are sat in groups of five.

And so we can continue this thought process to find our total diners of 33.

And so we are multiplying the group size by the amount of tables sat like that.

So there was one group of zero, so that contributes zero to our data.

There was one table of one, so that contributes one diner.

There was two tables of two.

So that gives us four diners.

There were two tables of three.

That gives us six and then we've got our 12 and our 10.

If we now add them up, so if you think of them like subtotals, the amounts as zeros, the amounts as ones, the amounts as twos.

Then when we add it up, we get our 33 in total.

So a quick check to do with frequency table.

This data is about the restaurant tables, but it's another day.

So how many people were eating in groups of four? Pause the video whilst you consider the frequency table and the answer to that question and press play, once you want to check it.

You may have said three, and that means you've gone to the correct part of the frequency table.

But the answer to how many people would be 12, so there were three tables of four.

That's what the three in the frequency table means.

So there was three tables of four, which means that there was 12 people eating as groups of four.

Okay, another check.

How many people were eating at the restaurant in total? So pause the video whilst you calculate this one.

Remember of what you've just worked out on that last one, that 12 people were eating in groups of four.

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

So there were 32 diners in total, and this came from doing quite a lot of calculation.

So from the first row there were no empty tables on this day, so that's still no diners.

There was one table of one, so that would be one diner.

There were four tables of two, so that gives you eight more people.

Then there were two tables of three and you've got your three tables of four and then you've got one table of five 'cause that's the 32.

So going back to the day where there was 11 tables and 33 diners in total, what would the mean number of diners per table be at this restaurant on this day? We want the mean of diners per table.

So our number of tables is 11, so we are trying to share the 33 diners across the 11 tables and that gives us three diners per table.

So here is our original plan view of the restaurant and this is what it might look like if it was a mean instead.

So the mean tells us the number of diners at each table once it's evenly distributed.

So in the context of this, what are we working out if we add up the frequency column? Well it's the number of games played by all the teams in that one football season, which comes out to be a hundred.

Alex has said if I add up the goals scored column, this tells me that over the course of the season, the club scored a total of 10 goals.

So Alex has done zero, add one, add two, add three, add four, and has got 10.

Is Alex right? Alex isn't correct.

And that's because we already know from the frequency table that there were 34 times that a team scored one goal in their match.

So if there was 34 times where they scored one goal, that's already 34 goals scored regardless of any of the other matches.

So adding that first column on a frequency table is meaningless.

It never tells you anything important or gives you any information regarding your data.

So when we are looking for finding the total goals scored over the season, you're not gonna find that number by adding the first column, the sort of context column of a frequency table.

So instead to try and work out the number of goals scored across the season, we need to bring back this idea of the diners per table and getting these sort of subtotals.

So the 42, there were 21 games where two goals were scored.

So in each of those games, there were two goals.

So 21 times two, means that there was 42 goals scored in those 21 games.

If we do this for each amount of goals scored, so over 19 games, the frequency is 19.

There were 19 games where no goals were scored, which means it contributes zero to our total of goals scored.

Then there were 34 games where one goal was scored.

So that means that there was 34 goals going towards our total.

There was 42 goals scored in 21 matches.

There was 45 goals scored in 15 matches.

And that's because in each of those 15 matches, three goals were scored.

So three times 15 is 45.

And then lastly there was 11 games in that season where four goals were scored.

So 11 times four is 44.

So now we can add them up to see that there was 165 goals in total scored.

So the mean number of goals scored per match, we know that there is a total number of goals of 165 and we know that there was a hundred matches played, means that there was an average, mean average, of 1.

65 goals per match.

You might be questioning how.

How can there be 1.

65 goals per match? Well because of how the mean is calculated, it's a theoretical number that gives us sort of this idea of a measure of central tendency.

It's more often than not, it'll gonna give us a number that isn't actually part of our data.

So this data is about goals scored.

You either score one goal or you score two, but that mean is 1.

65.

So if the number of goals was evenly distributed, it would be 1.

65 goals per match.

And it's important that we don't round that because then that loses the sort of shape and distribution of the data.

So a check here.

So which of these mean values seems reasonable for these data? Pause the video whilst you think about that.

You shouldn't have to calculate the mean for this one.

So just think about reasonable.

Come back when you're ready to check your answer.

As I say, you shouldn't have had to calculate it.

B is the reasonable mean value and that's because our data is about star ratings and the star ratings go between one and five.

And so the mean value does not have to be a piece of data within the set.

So here again we've got 3.

5 and 3.

5 was not an option but it does need to lie within the data.

So our data runs from a star rating of one up to a star rating of five.

So our mean needs to be in between one and five inclusive.

So 10.

7 is just too large, and 112 is also too large.

So we need to be careful and sense check in.

Every time you do a calculation, does it seem reasonable for the data that you've been working with? So now you're gonna do some practise for the second learning cycle and calculating the mean from an ungrouped frequency table.

Question one, there's three questions all regarding that frequency table on the screen.

Pause the video and then when you're ready for the next question, just press play.

Question two, the frequency table shows the number of bedrooms in different houses within an area of a town.

And so which calculation correctly finds the mean for this frequency table? For each of the incorrect answers, why is it wrong or why is it unreasonable? Pause the video and then when you're ready for the answers, press play.

Question one, we've got this frequency table which is about a quiz taken by a class and the quiz was marked out of six and we've got our results for the class.

So how many pupils are in the class? Well, that's 33 pupils.

You should have added up the total frequency.

How many marks did the class get as a group? Here we needed to do the sort of subtotals as you went along.

There were three pupils that scored zero.

So how much did they contribute to the total marks? They contributed zero.

Then there were four students that scored one each.

So they as a group contribute four marks.

At the other end, we've got, you know, four pupils, that's called five out of six.

So they contribute 20 marks to the total.

And so there was 99 marks as a class awarded.

Hence what is the mean marks per pupil? So we need the marks, the total marks, which we've already calculated is 99, and we need to share that evenly across the 33 pupils.

So the mean mark was three marks per pupil.

Question two, so there were six calculations for the mean and only one of them was correct, and actually A was the correct one.

So this was about number of bedrooms in different houses and the mean number of bedrooms is 2.

5.

So again, it doesn't have to be a value within the data.

You can't have two and a half bedrooms, but it does need to be within.

It needs to be a number that fits within.

Our data was from one to four.

So now we needed to go through why were the other five wrong? Well B, D, and F are all wrong for the same reason.

You may have come up with different reasons.

You may have just said about them being unreasonable answers, but what made them an unreasonable answer is what I'm gonna go through.

So all of these are connected by the 10 and the 10 is where they've added up the bedrooms column and that is meaningless.

So remember that adding up the context column of a frequency table.

If it's horizontal, the top row is never given you any information that's useful.

And then C and E, the division by four.

And again this happens quite a lot and it gives you a really unrealistic, unreasonable answer.

So if you do forget and do this sometimes, if you're checking your answer, doing a sense check, then you should recognise that your answer just cannot be a mean value for the data and then hopefully you'll figure out where you went wrong.

So what they've done here is the four is not a value, is they've not used four bedrooms. It's not the four from there.

This is because there are four categories, four rows in the frequency table.

And so when we say add up all the data and divide by how many there are, that "how many there are" is where that misconception can come in.

So when we're calculating the mean from a frequency table, we've just gotta be mindful of there are some parts of it that can mislead us.

So adding the first column, meaningless.

Dividing by how many rows, meaningless.

But if you check your answer, is it a reasonable mean value? Does it show us that sort of central value, that central tendency? Is it within the data? Then you're more likely to be going the right way with your calculations.

We're now gonna move on to the third and final part of the lesson, which is calculating the mean from bar charts.

So bar charts can show data in lots of different ways and here we've got a bar chart representing the data of five pupils and their attendance marks since September.

So each bar is one pupil and what this is actually showing us is just five pieces of data.

The height of the bar is their attendance marks in September.

So if we wanted to calculate the mean from this style of bar chart, then we've only got five data points.

So we're just gonna read the data points off, add them up, so get their total attendance marks, and then divide by how many pupils there are, how many bars there are, which is five.

And that will give us our mean attendance mark for the five pupils, which in this case is 24.

2.

Whereas, this type of bar chart is showing us data with frequency as the vertical axis.

So this isn't just showing us four pieces of data because there are four bars.

This is showing us many more pieces of data and it's actually 200 and we'd know that there are 200 pieces of data by adding up the heights because the first bar has its own frequency.

The second bar has its own frequency.

So the total frequency comes from adding up the heights and in this case it's 200.

For the rest of this cycle, we are gonna be working with data represented on frequency bar charts.

So we've got a bar chart here showing the number of each table type in a restaurant and we've worked with this style of question already this lesson.

So to find the number of tables in the restaurant, we're gonna add up the frequency.

So we've got two with nobody sat at them, one with one person sat, three with two people sat at them.

And so that gives us a total of 13 tables.

So we know that in this restaurant, there are 13 tables.

In your check here, this is a bar chart for the reviews of a restaurant.

So how many reviews has this restaurant received? Pause the video whilst you're working out the answer to that and then when you're ready to check it, just press play.

Hopefully, you've got 40 reviews and that should have come by adding up the heights or the frequency for each bar.

So if we head back to the bar chart regarding the number of diners per table, there were six diners that were sat as pairs, and where is that represented on this bar chart? Well that's represented by this bar because the pairs part means sat in twos, two diners per table.

That's on our horizontal scale and there are three.

The frequency is three.

So if we've got three tables each with two diners, then that gives us six diners in total.

If we overlay a dot plot on top of the bar chart.

We've seen dot plots before.

Then this can help us see how we get the total diners within the restaurant.

We know there are six sat in pairs.

We saw that on the last slide.

The first bar has got two dots because it's got a frequency of two, and that bar represents empty tables because it's a zero diners per table that contributes no diners to our total.

The next bar has got one dot because it has a frequency of one.

And this is the bar that represents the tables where there is one diner per table.

So we've got one table of one person.

So that gives us one diner into the restaurant.

We've seen this bar, we spoke about this bar already.

So this one contributes six diners and that's because there are three tables of two.

The next one will contribute three and that's because there is one table of three diners.

The next bar, we've got three tables and at each of those tables, there are four diners.

So three tables of four gives us 12 diners at the restaurant.

The next one we've got two tables of five, so that's 10.

And the last one we've got one table of six.

So that is another six diners.

So these are like subtotals.

These are the total of diners in each group size.

So if we add all of those subtotals up, we get our total diners, which is 38.

So just to check that sort of follows on from that concept.

Going back to your bar chart about reviews from the restaurant, which star rating contributed 36 stars to the total? Pause the video whilst you ponder it, and then when you're ready to check, press play.

Four star rating contributed 36 stars because they had nine reviews who gave them four stars.

How many stars were given in total? So this one, you're gonna need to do a bit of calculation.

You might need to write something down.

So pause the video, work out the number of stars given in total across all of the reviews and then when you're ready, press play to check it.

So in total, that's 134 stars that the restaurant has been awarded by their reviews.

Going back to the table idea, the table example and our dot plot, the mean number of diners per table.

We need the total diners, which we've already calculated and we need the number of type tables, which we've already calculated.

The total diners was 38.

We did that when we did our multiplication for each category and then we totaled it up.

The number of tables is 13, which we can do by adding up our frequencies or with a dot plot, we could just count the dots.

And the mean number of diners per table is therefore 38 shared evenly across 13, which is approximately 2.

92 diners per table.

You've been working on this same bar chart throughout in your check.

So far, you've worked out that there were 40 reviews given to the restaurant and you have worked out how many stars were given across those 40 reviews.

So now what was the mean star rating of this restaurant? Pauses video whilst you calculate that and then press play to check your answer.

The total was 134 stars.

There were 40 reviews.

So if we share those stars evenly across the 40 reviews, if everybody gave the same star rating, the star rating would be 3.

35 stars.

In the first learning cycle of this lesson, we looked at line graphs or what might be called time series graphs, and to find the mean, we added up the data points divided by how many there were.

On a frequency bar chart, that's not what you do.

You do not add up the frequency and then divide by how many bars.

We have seen that there are bar charts that that is the method for, but they're not frequency bar charts.

So we're only focusing here on frequency bar chart.

So why does adding the frequency and dividing by how many bars on a frequency bar chart not calculate the mean for the data? These states have five data points.

Each of the points have a frequency of one as there is no repeating value.

So it's time taken to travel to work and we've got the data in a sort of list there.

We know Monday's time, Tuesday's times, Wednesday's time.

None of the data points were repeated.

There was different discrete data every day.

So if we were to convert this onto a frequency bar chart, then the time travelled would be our horizontal axes and our frequency would be one because there's only one of each of them.

So if you add up the frequencies and divide by the number of bars, you're gonna get one.

And that's not the mean time taken to travel to work, but instead, that's just the mean frequency per bar.

So if you were to add up the frequency and divide by how many bars there are, that's all you're working out, is the mean frequency per bar.

So if we return to the bar chart about the star rating of the restaurants, we'd already calculated in your check that the mean would be 3.

35, but if we did it the wrong way, if we added up the frequencies and divided by how many bars there are, the total frequency is 40.

That's how many reviews that they got.

And there's five bars because there was five different ratings.

That would come out as eight and sensibly or reasonably, that can't be the mean rating because eight stars is not even an available choice.

So again, if we consider whether we get an answer that seems sensible for the data, that can help us recognise if we've maybe made a mistake.

So a check on this.

So to find the mean from a frequency bar chart, you add up the frequencies and divide by the numbers of bars.

True or false, and then justify your answer.

Pause the video whilst you're reading that and considering your answer.

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

Hopefully, you went for false.

So you do not add up the frequencies and divide by the number of bars when it is a frequency bar chart.

And to justify this, it was B, the total frequencies is the number of items within the data and that's the value that you should be dividing by when you calculate the mean.

So you're onto your last practise part of this lesson.

So question one, use the bar chart, it's a frequency bar chart.

It's about number of cars per household to answer those three questions.

Pause the video whilst you do that and then press play for question two.

Question two, the bar chart shows the length of time customers spent waiting at self-service till to the nearest minute.

So what is the mean wait time? Pause the video whilst you calculate that.

And when you're ready for the answers, head back and press play.

Question one, using the bar chart to answer some questions and then calculating the mean.

So how many households are there? There was 175.

And you got that by taking the frequencies of each bar and adding them up.

How many cars are there in total across all households? So this one needed you to work out how many cars each bar represented and then find that total.

What is the mean number of cars per household? So we know that there are 350 cars across 175 households.

So when we evenly distribute it, we share them, which is why we do division.

It comes out as two cars per household.

Question one, you sort of did it in three parts.

Question two just needed you to do all three parts again for this different data.

So first of all, you needed to work out how long all the customers waited for.

So the values from each bar is 0, 1, 10, 18, 12, 10, and 24.

The 18 comes from six lots of three minutes.

So the total time waited was 75 minutes.

Then how many customers are there? Well that's just adding up the frequencies.

And so that was 25 customers.

And so the mean wait time comes out as three minutes.

To summarise today's lesson, which was about calculating the mean, but specifically from different representations, you need to consider how the data is represented and where the data points are within each representation.

So to find the mean from a line graph, you're gonna add up the data points and divide by how many points there are.

You need to take the data back off of the graph.

To find the mean from a frequency table and a bar chart you need to find the overall total and divide by the total frequency.

But that's a bar chart.

That is a frequency bar chart.

Well done today.

I look forward to working with you again in a future lesson.