Lesson video

Hi there, my name's Ms. Lambo.

You've made such a fantastic choice deciding to join me today to do some maths.

Come on, let's get going.

Welcome to today's lesson.

The title of today's lesson is constructing a cumulative frequency graph and that's within the unit graphical representations of data and cumulative frequency and histograms. By the end of today's lesson, you'll be able to construct a cumulative frequency graph.

A keyword that we'll be using in today's lesson, which you may not already be familiar with, is cumulative frequency.

Cumulative frequency is the total of all frequencies up to a defined class.

It is basically the running totals of the frequencies.

Like I said, this may be a new word to you today, so don't worry about it.

I know by the end of this lesson you'll totally understand what we mean by cumulative frequency and why we use it.

Today's lesson then, I've split into two separate learning cycles.

In the first one, we will just concentrate on being able to calculate that cumulative frequency.

And in the second learning cycle, we will look at constructing cumulative frequency graphs.

Let's get going with that first one.

So we're just gonna concentrate on calculating cumulative frequencies.

Let's go.

A class of 30 pupils were asked to time how long they could hold their breath.

This table below shows the results.

We can see in the column on the left, we've got their time in seconds.

And on the column on the right we've got the frequency.

From this table, we can see that four people could hold their breath between zero and 10 seconds.

If we look at the second group, we can see that three people could hold their breath for between 10 and 20 seconds, but not include in the 10.

Why is it not including 10? That's right, because we've got that greater than symbol.

T is greater than 10, not greater than or equal to 10.

We are now gonna calculate the cumulative frequency.

This means we will accumulate the frequencies as we move through the table.

So there's that link there between those two words, cumulative and accumulate.

And like I said, when I talked about the keyword at beginning of this lesson, you might like to think of it as a running total.

I'm gonna add a third column to my table and this is going to be where I'm going to record my cumulative frequency.

We're going to be looking at the frequencies and we're gonna find the cumulative frequencies.

We're gonna accumulate the frequencies.

Let's see what that looks like.

First question we need to ask ourself is, how many of our pieces of data are less than or equal to 10 seconds? And we can see less than or equal to 10 seconds, there are four pieces of data.

Four pieces of data, so that is going to be my first cumulative frequency.

We now need to consider what happens as we move onto the next group.

How many of our pieces of data are less than or equal to 20 seconds? This time I'm gonna pause so that you can work it out.

Now I'm going to change the time column because here, I now want it to be anything up to and include in 20 seconds.

What did you decide was the cumulative frequency? It's the total of the first two groups, four, up three, hopefully that's what you said, and then you would've calculated that a seven.

So the second cumulative frequency is seven.

And then we'll move on to the next group.

How many pieces of our data are less than or equal to 30 seconds? Again, I'm going to change that first column now because I want it to be anything up to and including 30 seconds.

And that will be the total of the first three groups, four and three and eight, is that what you said? Of course you did, well done.

15, cumulative frequency is 15.

And the next group, how many of our pieces of data are less than or equal to 40 seconds? Changing the group, and it's a total of four, three, and eight.

So the sum of those four separate frequencies, which gives us 23.

And let's finish off then, the finally the last group.

How many of our pieces of data are less than or equal to 50 seconds? I'll change the group, so it's anything up to and including 50 seconds, which is a total of all of the frequencies.

So it's a sum of four, three, eight, eight, and seven, which is 30.

It's always worth checking that your final cumulative frequency is the same as the total if it's given in the question.

And here we've got a total frequency of 30 and in the question we can see that we were told that 30 pupils were asked to time how long they can hold their breath.

So if you are told the total in the question, it's always worth doing that little double check because sometimes it's easy to make an error when we're doing our addition.

Here is exactly the same table.

The only difference is, instead of putting in the total cumulative frequency, I've given the calculations that we did to find them.

Sam says, "I can see a quicker way to work out the cumulative frequencies after the second group." Wondering if you can see what Sam may have spotted? Sam says, "When you get to the third group, you already know the total of the first two groups, therefore you can just add the third frequency to the previous cumulative frequency." If we take a look at the next one then, we've got the previous cumulative frequency is seven and then we're going to add the next group seven, add eight.

I'm wondering if that's what you said.

Did you spot that little shortcut? Well done.

That gives us 15.

To find the next one then, we take our previous cumulative frequency of 15 and we add in the next group.

That gives us a total of 23.

And then instead of doing four, add three, add eight, add eight, add seven.

I know the total of four, three, eight, eight and seven is 23.

So I just need to take the previous cumulative frequency of 23 and add on the next group of seven, which is 30.

From now on, we will use this more efficient method to complete our cumulative frequency tables.

Here we have a table showing the heights of 150 members at a running club and we can see that their heights range from 155 to 185 but not include in 155 remember.

We're going to draw a second table.

So here I've written my groups because I want each group to go up to and include the next group.

The first cumulative frequency is always the same as the frequency, it's 35.

To find the next one, we add the previous cumulative frequency to the next group.

So we're gonna do 35, add 26, which is 61.

Take the previous cumulative frequency, which is 61, and add on the next group, which is 23.

61 add 23, is 84.

And we repeat this all the way through the table.

84 at the next group is 24, giving us a total of 108.

108, add 16 gives us 124.

And for the final group, we've got a cumulative frequency of 124.

We add the next group of 26 giving us 150.

Now what should you be doing? Well done, yeah, check in.

We've got 150 as our total frequency and we were told that these were the heights of 150 members at the running club.

Time for you now to have a go at this check for understanding.

So Sam has filled in the cumulative frequency table.

They know they have made a mistake as the final cumulative frequency is not the same as a total frequency of 100.

So well done Sam, for remembering to check your answer.

I'd like you please to find Sam's mistake and correct it.

You're gonna pause the video and then come back when you're ready.

Good luck with this.

And did you spot Sam's mistake? You did and correct it? Of course you did, well done.

Let's take a look then.

First one, remember the first cumulative frequencies is always the same as the first frequency.

The next one, we're going to take the previous cumulative frequency of eight and add on the next frequency of 12.

Eight add 12 is 20, so the second row was correct.

Now we're gonna take the previous cumulative frequency of 20 and add on the next group, which is 10.

That gives us 30.

Then we're going to do the same again.

The previous cumulative frequency of 30 add in the next group of 37 is 67, so that's right.

Previous cumulative frequency of 67, add 26 is 93.

That's where Sam's mistake is.

Is that what you've got? Well done.

So it shouldn't have been 83, it should have been 93.

And then we're going to take the 93 and we're going to add in the final group of seven giving us 100.

And now we can see that the total is correct.

So it's definitely worth always doing that double check.

Only if though you've been given the total number in the question.

Sometimes that might not be there.

And in which case then I would really recommend that you go back through and check each of your totals carefully.

And like I said, we can see that it matches the 100.

Here we have two tables and they are showing the ages of 100 employees at a company.

We need to fill in the missing values.

On the left we have the frequency table and on the right we have the cumulative frequency table.

We'll start by filling in the missing ages because that's a little bit easier than the missing frequencies or cumulative frequencies.

We can see that the first group goes between 20 and 30, but not including 20.

The next group goes from 30 to, and if we look at what the next group starts at 40.

So the missing number here is 40.

And then what do you think the missing number is at the bottom one on the left hand side? That's right, 60.

If we look at the cumulative frequency column, the lower boundary is always the lowest in this case, age, which is 20, but remember not including 20.

So that's gonna be 20.

If we look at the third row in the frequency table, we can see that the upper boundary of that group is 50.

So this is gonna be anything up to 50.

The missing value here then is 20.

And then the missing value here is 70.

Let's start by looking at this missing value.

This is anyone 30 or younger.

How many people with 30 or younger? Or we can see this from the cumulative frequency table.

The first groups are always the same in a frequency table and a cumulative frequency table.

So the missing value is 18.

Now let's take a look at this one.

This wants to know anyone that is 40 or younger.

If we look at the group, we can see that A is greater than 20, but also less than or equal to 40.

How many people were between 20 and 40? But remember not including 20.

And that is the sum of these two frequencies.

18 and 24, which gives us 42.

Now we'll take a look at this one.

This is anyone between 40 and 50, but excluding 40.

How are we gonna work that one out? We're actually going to find the difference between these two groups here.

We're gonna find the difference.

We know that there are 42 people up to and including 40, and we know that there are 74 up to and including 50.

Therefore, the difference between them must be those that are aged between 40 and 50, but not including 40.

The difference between those two values is 32.

Now let's take a look at this missing value.

This is anyone 60 or under.

To find anyone who is 60 or under, we are going to take the previous cumulative frequency and add in the people who are up to 60.

74 and 19, we find the sum of those, which gives us 93.

And then finally the last missing value.

This is anyone between 60 and 70, excluding 60.

We're finding the difference between 93 and 100.

We're finding a difference and that gives us seven.

Now you can have a go at this one, please.

I want you to do exactly the same thing.

I'd like you to find out please, what numbers my purple boxes are covering.

Also a video, and I'll be waiting when you get back.

Great work on that.

Now let's check your answers.

So in the first table, missing ages were 50 and 60.

And the second table, the missing ages were 40 and 20.

The missing frequency in the first row was 14.

In the second row, the cumulative frequency was 45.

In the fourth row, the missing cumulative frequency was 97.

And the missing frequency for the final group was three.

How did you get on? Superb, well done.

Now then you can have a go at Task A.

You're gonna pause the video and all I'd like you to do please is to fill in the cumulative frequency column.

Good luck, pause the video and then I'll be waiting when you get back to reveal the next question.

And part B, fill in the cumulative frequency table for these 50 masses.

Question number two, you need to fill in the missing values.

So where there's a line, there's a missing value and you need to fill that in please.

Well done, and then finally, for Task A, question two B, again, fill in the missing values.

Great work, now let's check those answers for you.

The cumulative frequencies, I'll just read them down from the column.

So we've got, first cumulative frequency is 14, then 51, then 80, and then 94, and then 100.

Part B, you should have six, 18, 37, 48, and 50.

2A, I'm not going to read out the answers to this 'cause it might get a little bit confusing.

So I'm gonna ask you to pause the video, check your answers, and then when you are done come back and I'll give you the answers to part B.

And here are the answers to part B.

Again, pause the video, check your answers, and then come back and we'll move on to the second learning cycle for today's lesson.

How did you get on? Superb work, well done.

I knew you'd get it.

Now then, we can move on to the second learning cycle.

We know how to find cumulative frequency.

We're now going to actually construct some cumulative frequency graphs.

One of the reasons for finding the cumulative frequency is to enable us to interpret the data.

One of the ways we can do this is by drawing a cumulative frequency graph.

When constructing a cumulative frequency graph, we use the upper boundary of the interval.

The plotted points must be joined with a curve, not straight line segments.

That's really important.

Here is the data we looked at earlier.

So we've got the times and we've got the cumulative frequency.

Firstly, we plot the data points.

Here's our graph, and I've plotted the points already, but let's make sure that we're happy with where those points have come from.

The first one we can see I've plotted at ten, four.

We look at the scale on the cumulative frequency.

Each square represents two.

The reason I've plotted ten, four is 'cause the upper boundary of the cumulative frequency is 10 and the cumulative frequency is four.

If I look at the next one, my upper boundary is 20, and the cumulative frequency is seven.

The next group is a cumulative frequency of 15 plotted with the upper boundary of 30 and then 40 with 23.

And finally a time of 50 with 30.

I then join my points together remembering to do that with a smooth curve.

So join my points together with a smooth curve.

This table shows the heights of 150 members at a running club.

It's not necessary to draw a new table.

We can simply add an extra column.

So we're gonna add our cumulative frequency column and we can see those cumulative frequencies so you don't have to draw out a separate table.

Now I can remove the frequency column, and again, I plot those points.

So the first one I plot is 160 with 35, and then 165 with 61, 170 with 84, and so on.

And then I join those points together with a nice smooth curve.

Your turn to have a go at this one.

Spot the mistakes on the cumulative frequency graph.

So the table shows the mass of 60 apples.

I'd like you please to spot the mistakes.

Notice plural, so there is more than one.

Pause the video and then come back and let me know what mistakes you find.

What did you find? Well, the first one was dead easy to spot.

The points are joined together with line segments and not a smooth curve.

You must join the points together with a smooth curve.

And what was the second mistake? That's right.

A really core mistake is to plot the data points against the mid points of the group.

And remember, it should be the upper boundaries because the cumulative frequency takes us right the way up to that upper boundary.

So those were the two mistakes in that cumulative frequency graph.

Now you are ready for your final independent task of today's lesson.

For each of the following, you are going to construct a cumulative frequency graph, pause the video, and when you've done question A, you can come back and I will reveal question B And part B.

Well done, and the final graph for today's lesson is graph C.

Great work well done.

Let's check those answers.

So your graph should look something like this.

It may slightly different if you've used a slightly different scale, but remember, you must have plotted 30 with 15, 40 with 50, 50 with 80, et cetera, and joined your points together with a nice smooth curve.

And B, pause video, check your answer, and then come back when you are done.

And C, again, making sure that you've plotted 30 with six, 40 with 44, et cetera.

Join those points together with a lovely smooth curve, which of course I know you've done.

How did you get on with those? Great work well done.

Now we'll summarise our learning from today's lesson.

Calculating the cumulative frequency means accumulating the frequencies as we move through the table.

It is useful to think of this as a running total and remember the link between the two words, cumulative and accumulate to help you remember what we're doing when we're finding the cumulative frequency.

When constructing cumulative frequency graph, we use the upper boundary of the interval.

Remember, if the group goes from zero to 20, we are including anybody up to and including 20, which is why we need to plot at the upper point.

And the final one, very important, the points must be plotted and then join together with a smooth curve.

You must never join your points with straight line segments.

Fantastic work today.

Well done, really enjoyed working alongside you with these cumulative frequency graphs.

I hope to see you again really soon.

Take care of yourself.

Goodbye.