Lesson video

Hi there.

My name's Ms. Lambell.

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 Interpreting a Cumulative Frequency Graph and that's within the unit Graphical Representations of Data.

And we're gonna be looking at cumulative frequency and histograms. By the end of today's lesson, you'll be able to interpret a cumulative frequency graph.

You should already know how to draw one.

A key word we'll be using in today's lesson is median.

And you should be really familiar with what we mean by median.

It's the central piece of data when the data are arranged in a numerical order.

It is a measure of central tendency representing the average of the values.

And like I said, that should be really familiar to you.

Some other words that probably won't be familiar to you, and you'll see them here written in bold, are percentile, lower quartile, and upper quartile.

Don't worry if you're looking at that now thinking, "Oh, I don't know any of that." It's probably like I said, new to you.

And that's what we're going to be focusing on in today's lesson.

So promise you everything will be fine.

But let's just run through what they are.

A percentile is a value on a scale of 100 that indicates the percentage of distribution that is equal to or below it.

The lower quartile is the value under which 25% of the data points are found when they are arranged in increasing order.

It's also known as the first quartile or sometimes Q1.

The upper quartile is the value under which 75% of data points are found when they are arranged in increase order.

Also, this can be known as the third quartile, referenced as Q3.

Today's lesson is in two separate learning cycles.

In the first one, we will look at concentrating on finding the median and the quartiles from our cumulative frequency graph.

And in the second one, we will look at problems that ask us about something that might be greater than or less than.

Let's get going with that first one then, finding the median and the quartiles.

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

The results are shown on this cumulative frequency graph.

Remember, when you're plotting a cumulative frequency graph, you plot the upper point with the cumulative frequency and you join those points together with a nice smooth curve.

But you should be really familiar with that.

We are going to estimate the median time using this graph.

And remember, it's an estimate because we don't know the exact values, we just know that a certain number of pupils were in a particular part time period.

That's why it's an estimate.

The total number of pieces of data is 30.

There were 30 pupils.

And we can also see this as the total cumulative frequency, 30, where that final point is plotted.

Therefore, the median value will be when the cumulative frequency is 15.

We find 15 on our cumulative frequency axis.

And we draw a nice ruled line, making sure that we are parallel to the time axis, and we draw it across until we get to our cumulative frequency curve.

And then we can read off that value.

The median time pupils could hold their breath for was 30 seconds.

So we read off the value that was halfway through the data.

So the 15th value would give us a median time of 30 seconds.

This is a cumulative frequency graph of the heights of 150 members at a running club.

What is the total number of pieces of data represented on this graph? It's 150.

We can see this from the question and also the final point on the graph, like I mentioned in the previous question.

Which cumulative frequency value will we read off to find the median? That's right, it'll be the 75th.

There are 150 members.

Half of that would be 75.

So we want to read off the 75th value.

So I find 75 on my cumulative frequency axis and I draw my line across, remember, making sure it's parallel to the height axis, and then I read my value off.

An estimate of the median height of the members of the running club was 168 centimetres.

I'd like you, please, to use this graph to work out an estimate for the median score of this class' test scores.

Pause the video, work out your answer, and when you're done, you can come back.

How did you get on? Great work.

There are 30 pieces of data in total.

Therefore, median is going to be the 15th value.

So draw your line in, making sure it's parallel to the test score, and then read off.

And I can see here that my answer is about 11.

5.

As long as you have any answer between 11.

2 and 11.

8, that would be acceptable because it's not an exact value here.

This is a cumulative frequency graph representing the test scores of a class.

The median test score is 9.

5.

What mistake has been made? Pause the video, make your decision about the mistake that's been made, and then come back.

It'd be even better if you impress me by correcting the mistake.

Pause the video, good luck, and I'll see you in a moment.

So what was the mistake? They have assumed the highest value on the cumulative frequency axis was the total frequency.

So we can see the highest value on the cumulative frequency axis was 30.

But, actually, if we look at the final point, it only had a cumulative frequency of 25.

Now like I said, you may have already done this 'cause I said you'd impress me if you could correct the answer.

So, firstly, what value should have been read off to find the median? So we know it's not 15.

What should it have been? It should've been half of 25.

So, therefore, it should've been the 12.

5th value.

And we read that off.

We can see that the median was approximately 8.

3.

When comparing data, it's also useful to identify other values.

These are most often quartiles and percentiles.

How many parts do you think the data/graph will be split into if we are working with quartiles? We split it into four equal parts.

If you think about it, quartiles, quarters, we can see that quar part tells us we're splitting into four equal parts.

How many parts do you think the data or graph will be split into if we are working with percentiles? We would split it into 100 equal parts.

And, again, that percent, percent, we know that means per 100.

Here we have a cumulative frequency graph.

Total frequency is 40.

We need to find 1/4 and 3/4 of the total frequency.

What is 1/4 of 40? That's right, it's 10.

We need to find the 10th piece of data.

So we find 10 on our cumulative frequency axis, we draw a line across, making sure that it's parallel to the test score, and then we can read off that test score.

And we can see that reading this value, we get a test score of 11.

This is the lower quartile.

What is 3/4 of 40? It's 40 multiplied by three, divided by four.

You may have done that in a different order or you may have taken your previous answer and multiplied that by three.

Anyhow, you should've got the 30th piece of data.

Let's read off what the test score is for the 30th piece of data.

We draw a line in, and we read off the test score, and we can see that reading off this value gives us a test score of 18.

5.

This is the upper quartile.

Here, we had our lower quartile.

Then we had a median.

And then the upper quartile.

Notice, we have split the graph into quarters.

That's what we said we were going to do.

That's what quartiles means, split it into four equal parts.

Sometimes you will see these referred to using the letter Q.

The first quarter, the lower quarter, Q1.

The second quarter, the median, is Q2.

And the third quartile, the upper quartile, may be referred to as Q3.

Q1 is the lower quartile and it is 1/4 or 25% of the way through the data.

Q2 is the median and is half or 50% of the way through the data.

And Q3, the upper quartile, is 3/4 or 75% of the way through the data.

This is a cumulative frequency graph representing test scores of a class.

I'd like you, please, to find the median, lower, and upper quartiles.

Pause the video and then come back when you've got your three answers.

Good luck.

Remember when you're reading off your values to make sure that your line going across is parallel to the test score.

Good luck, I'll be here waiting when you get back.

Great work.

Let's check those answers for you.

So the lower quartile, you should've read off a cumulative frequency of five.

The lower quartile was 10, or Q1 was 10.

The median was halfway through the data.

Well, there were 20 bits of data in total.

Half of 20 is 10.

And if we read off that, we can see that that is a test score of 13.

So Q2 or median is 13.

And then 3/4 of the data, 3/4 of 20 is 15.

And we read off that test score, we can see that the upper quartile or Q3 is 17.

Is that what you got? Brilliant, well done.

Let's have a look and see what Laura's saying here.

Laura says, "I thought this graph was supposed to be split into quarters! But if you look at the arrows between each section they are not the same size." What has Laura misunderstood? It's the frequency that we're splitting into quarters, not the test score.

The frequency is what we're splitting into quarters.

If we look here, we can see now they are four equal parts.

Like I said, now we can see that all of the parts are the same size.

A company decides it wants to give 25% of its best sales representatives a bonus.

Oh, lucky them.

This graph shows the sales of companies employees for the last month.

What number of sales should the company choose to reward? Here we want to work out what the best 25% of representatives did.

So we're going to need to find 75%.

We're gonna need to find the upper quartile because we know that below the upper quartile is 75%.

So we'll be left with the 25% that are over.

What is the total frequency? That's 30.

What is 3/4 of 30? And that is 22.

5.

So I'm going to draw my line from the cumulative frequency from 22.

5 and then I'm going to read off what the number of sales are.

And I can see that any sales above 31 should get the bonus.

It's now time for task A.

For question 1a, you're gonna find the median, the lower quartile, and the upper quartile.

Pause the video, and then when you're done, come back.

Well done.

And part b, same thing again.

Can you please find me the median, lower and upper quartiles.

And question two, a company decides it wants to give 25% of its best sales representatives a bonus.

This graph shows the sales of the company's employees for last month.

What number of sales should the company choose to reward? Again, pause the video, and then come back when you're done.

And let's check those answers.

1a, the lower quartile was roughly 113.

So somewhere around there.

The median was 120.

And the upper quartile was 130.

Part b, the lower quartile was 118, the median was somewhere around 123, and the upper quartile was 128.

And then finally for task A, question two, any sales above 32 should get a bonus.

Now we can move on then to looking at questions that are greater than or less than problems. And, actually, that company problem was one of those.

So we've done that inadvertently without me even making you aware of it.

We've got our graph back (indistinct).

And it's a graph showing the number of seconds that 30 pupils were able to hold their breath for.

And the results are shown on this cumulative frequency graph.

The teacher wants to know how long 80% of the class could hold their breath for.

The total frequency is 30.

Remember, we can see that from the question.

We're told class of 30 pupils, or as the highest cumulative frequency.

What is 80% of 30? 80% of 30 is 24.

Of course, that's what you said, didn't you? So we need to read off 24.

It's really important here that we consider the scale.

From zero to 10 is five squares.

Each square is worth two.

So 24 is going to be here.

I'm going to read off what that time is.

And I can see that 80% of people could hold their breath for up to 42 seconds.

The teacher now wants to know how many pupils could hold their breath for less than 36 seconds.

So this time, we've got a time rather than the number of pupils.

We find 36, and then we read off.

And we can see that 20 people could hold their breath for less than 36 seconds.

Remember, when we're reading values from the cumulative frequency graph, we're reading off values less than.

I'd like you, please, to have a go at this one.

Very similar to what we've just done.

The teacher wants to know how many pupils could hold their breath for less than 18 seconds.

Find 18 on the seconds, and then across.

And we can see that six people could hold their breath for less than 18 seconds.

This is a cumulative frequency graph representing test scores of class.

The teacher is going to ask pupils who scored less than seven marks to retake the test.

And that's a bit mean, isn't it? How many pupils will be asked to retake the test? We find seven on the test score because we want to know less than seven marks.

And then we can read off this as a cumulative frequency.

And we can see that three pupils will be asked to retake the test.

So we've got cumulative frequency graph.

This time the teacher is going to be a bit meaner and they're gonna ask the pupils who scored less than 12 marks to retake the test.

Your job is to tell me how many pupils will be asked to retake the test.

Pause the video, and come back when you are done.

And how many people are gonna be disappointed this time? Okay, we read off 12 marks.

And we can see eight pupils are going to be asked to retake the test.

Here, we have a cumulative frequency graph representing the improvement in the number of marks a group of pupils makes on a test.

The head teacher wants to give a certificate to any pupil that made an improvement of more than 23 marks.

How many certificates do they need? We find 23 marks, and we're gonna read off our cumulative frequency.

Alex says they will need 37 certificates.

So Alex has read off that the arrow is pointing at 37.

So 37 certificates.

Do you agree with Alex? Notice now I've highlighted the fact that it says more than 23 marks.

Alex says, "No, that is how many people made an improvement of 23 or less." Remember, when we're reading values from a cumulative frequency graph, it's giving us the number of less than.

How many certificates does the head teacher then need? The head teacher needs three certificates.

We know the total number of pupils was 40.

We know that 37 scored less than 23.

So, therefore, what's the difference between the two? The head teacher is going to need three certificates.

This is a cumulative frequency graph showing the score in a competition.

To progress to the next round of competition, you have to score more than 22 points.

How many people progress to the next round of competition? Pause the video 'cause this is your check for understanding and then come back when you've got your answer.

Great work.

Right, 22 points, and we read off the value, and that gives us that 33 people scored less than 22 points.

Total number of people is 35.

The difference between 35 and 33 is two.

Two people progressed to the next round of the competition.

Now task B, you're gonna use the cumulative frequency graph showing the mass of some apples to answer the following.

A, how many apples had a mass of 125 grammes or less? B, how many apples had a mass of more than 145 grammes? Pause the video, make sure that you draw your lines nice and accurately, and then when you come back, we'll reveal the next question.

Well done.

And question two, use the cumulative frequency graph showing the heights of a group of primary school pupils on a trip to a theme park.

A, the height restriction on a ride at a theme park is 1.

25 metres.

How many pupils cannot go on the ride? And B, how many pupils are between 1.

15 metres and 1.

3 metres in height? Pause video, and then when you're done, you can come back.

Great work.

Question three.

This is a cumulative frequency graph representing the improvement in the number of marks a group of pupils made in a test.

I'd like you to find out the following, please.

A, how many people made an improvement of more than 13 marks? B, what percentage of people made an improvement of 19 marks or less? And C, complete the statement.

55% of people made an improvement of more than how many marks? Now, these questions are a little bit more challenging, particularly B and C, because I'm asking you to draw on other areas of your mathematical knowledge, but I know that you've got this.

So pause the video and then come back when you're done.

Great work.

Let's check those answers.

Question 1a, the answer was 26 apples.

B, the answer was four apples.

And we can see there on my graph, I've shown you how I've got those answers.

So if you need to pause the video and check those, you can.

Question two, a, approximately 12 pupils, and, b, 21 pupils.

And, again, there are my working lines on my graph should you need to check those.

And three, the correct answers were, a, 26 people, b, 77.

5 people, and c, the answer was 16 marks.

And, again, if you need to pause video and take a look at those lines, if you've made any errors, which I'm sure you haven't, then you can do that, obviously.

Now let's summarise our learning from today's lesson.

Cumulative frequency graphs can be used to find the median, quartiles, and percentiles.

That's what we've really focused on during today's lesson.

Q1 is the lower quartile and it is 1/4 or 25% of the way through the data.

Q2 is the median and it is half or 50% of the way through the data.

And Q3 is the upper quartile and this is 3/4 or 75% of the way through the data.

Really important to remember, when you read a value off from a cumulative frequency graph, it's telling you how many did less than that.

So if a question asks you for more than, you need to subtract your cumulative frequency from the total frequency.

Superb work today, well done.

Look forward to seeing you again, and I hope to do that really soon.

Take care of yourself, goodbye.