Lesson video

Hi, thanks for choosing to learn with me today.

In today's lesson, we're gonna be looking at how maths can help us with our everyday lives.

Let's get started.

In today's lesson, we're going to be learning how we can critically analyse graphs.

In other words, we're gonna be thinking about what conclusions we can draw from different types of graphs, and if those conclusions are actually valid.

Are all graphs drawn the same way? What's the point of them? Let's investigate.

Before we get started, we're just gonna recap some keywords that we're gonna use in our lesson today.

Now, these words might be familiar to you and that's great if they are, but if not, you might want to pause the video now, and just have a quick read through these.

And here's the next one.

This one's about scatter graphs.

So again, if this isn't familiar, feel free to pause the video while you read through.

Today's lesson is split into two parts, and we're going to begin with, "Misleading Graphs." Now, graphs are used regularly in all sorts of industries.

What industries can you think of which use graphs? Pause the video while you either have a discussion with your partner or think about this yourself.

So what did you come up with? Well, any company's gonna use graphs to track trends or profits, and many companies use graphs for advertising as well.

Did you think of these ones? Maybe you came up with this one.

Public services, journalists and politicians all use graphs to convey information to the public.

Where do you see graphs in your life? Again, pause the video while you think about this.

Welcome back, what did you come up with? Well, you might have said on the news, you might have said on the packaging of products, you might have even said adverts.

You might have come up with many more.

Sofia points out that her smartwatch creates graphs out of her fitness data.

And her phone uses graphs to track her data usage.

You can go into settings on your phone and have a look at your data usage, if that's something that interests you.

But why are graphs useful? Well, they can be easier to read than lots of lines of data or information.

They can sort the data into an order, which makes it easier for us to comprehend.

They can be a lot easier to interpret and draw conclusions from.

They can compare sets of data or show trends over time, and they can help predict future events.

Graphs are often used because they're visually appealing, and this means people can draw conclusions without needing to analyse the data thoroughly.

However, this can lead to the misuse or misinterpretation of graphs, and that's what we're going to look at in this part of our lesson.

So here we have a graph and it shows, average daily ice cream sales.

And we've got Jacob.

Jacob says, "This graph shows that the average daily ice cream sales for my local ice cream shop more than doubled from 2022 to 2023." Hmm.

Did they? What do you think? Pause the video and have a discussion now.

If you said no, you're spot on.

Did you see the axis there for frequency? Double 55 would be 110, and the 2023 figure looks to be around 65, maybe only 62.

It's hard to tell because of the scale.

It's definitely not as high as 110 though.

But why do you think Jacob said this? Well done if you said the second bar is more than double the height of the first bar.

Hmm, does that mean the graph was drawn wrong? Because at the moment it does look like the sales have jumped considerably in 2023.

Why might that be? Did you spot where the frequency axis starts? The scale starts at 50 and not zero.

This skews the scale and can make small differences look more significant than they are.

Let's check you got that.

A local newspaper writes this headline, "Local schools sees devastating drop in number of pupils studying History at GCSE." Huh, sounds pretty bad, doesn't it? But let's look at the graph.

Can you write down a reason why the graph may not support this headline? Pause and do that now.

Now, there are lots of things you could have come up with.

This is just a suggestion.

You could have said that the y axis has a broken scale.

So although the graph seems to drop, this is only actually eight fewer pupils than 2022 and actually only four less than 2020.

So you might have made a comparison between the year groups.

You might also have considered that some year groups have fewer students in any way.

Not every year group has the same amount.

And equally the word devastating in the headline, it's used to create emotion and it's not actually related to this data at all.

Now, using a broken scale, inconsistent scale, or even no scale at all on the y axis, that can be misleading, especially when we're comparing values on a graph.

There are times when using a broken scale may be necessary if we want to analyse trends and variation in the data, especially when the data has very large values.

However, it is important not to make conclusions based on the proportion if the graph does not start at zero and have a consistent scale.

You need to keep an eye out for information or adverts, which use a broken scale, because they might be trying to influence your interpretation of the data.

Now, sometimes we see graphs which are not just a bit misleading, but are actually drawn incorrectly to try and influence your opinion.

Here we have a pie chart showing the percentage of votes for two main parties in an election.

But what's the problem with this graph? Pause and have a discussion now.

Welcome back, did you spot it? That's right, pie charts have to add to 100%.

45% and 38% do not make 100%.

Ah, this is what they've done.

They've ignored all the votes for the other parties.

When we add those back in, A, our pie chart is now correct, and B, we can now see that the difference between the Pink party and the Blue party looks a lot less.

Jacob has created this pie chart about how long he spends on each activity on a Monday.

"Wow, Jacob, you watch far too much TV on a Monday." Well, I have to agree, Sofia, certainly the pie chart makes it look like that.

In fact, it looks like he spends the most amount of time watching TV, more time than he does spend sleeping.

Why do you think that might be though? Has this pie chart been drawn in a way that makes us think that? You're right, the pie chart is at an angle and TV is at the front.

This makes the sector look a lot bigger than it actually is.

It's really difficult to compare the different proportions of a 3D pie chart.

They can be used therefore to mislead a reader.

If we used a 2D pie chart, the data would look like this instead.

Ah, that looks a lot different, doesn't it? So Sofia points out, she still thinks Jacob watches too much TV, but actually now I can see that the sleep sector is actually a lot bigger than the TV sector.

Yeah, it looks a lot more reasonable now.

Sofia's collected data about people's favourite colours and we can see them here.

What's the problem though with how this pie chart has been drawn? Pause and have a think.

All the bars are different widths.

Did you spot that? They've been drawn so they have the same width as the colour name.

Now, why could that be misleading? Well, your eyes are drawn to the wider bars, which make them look a lot bigger than they actually are.

So let's do a quick check.

Which of these could be ways to draw misleading graphs? Pause the video and make your selection now.

Well done if you said B and D.

Now, sometimes with C, labels are deliberately left off to make it harder for people to interpret, not put on.

It's time for your first task.

For question one, I want to know why the graph in this advert might be misleading.

Pause the video and have a go at this question now.

Welcome back.

Question two, this chart shows the profit of a company in thousands over a five-year period.

What I'd like to know is what elements of this chart might make it misleading and why do you think the company chose to draw it this way? Pause the video and work on this now.

Question three, Jacob records the average temperature in one of his classrooms each month and draws this graph.

What I'd like to know is what conclusion might he be trying to convey with this graph and why is his graph misleading? Pause the video and work on this now.

Question four, Sofia collects this data on how long she spends on different activities on a Saturday.

For part A, I'd like to know what type of chart Sofia could draw to display this data.

And then in part B, Sofia wants to make it look like she spends lots of time exercising.

How could she manipulate her chart so that she can imply this? Pause and work on this now.

It's now time to go through our answers.

So why might the graph in this advert be misleading? Now, these are just some suggestions and you could have written something different, but it'd be equally valid.

You might have said, "There's no scale on the y axis, so the difference in height could be anything." "The premium toothpaste bar is much wider, which makes it look like it represents a bigger number." "There's no label on the y axis, so we don't actually know what we measured here." And the, "Use of colour highlights the premium toothpaste." Well done if you said any of these or if you wrote something else that's equally valid.

Question two, I asked you what elements of this chart might make it misleading.

Well, it's a 3D bar chart, which can make bars in the foreground look much bigger than they are.

It's very hard to read the exact numbers off this axis.

So why draw it this way? Well, to me, it looks like the yearly profit increase is really impressive.

By year five, they've definitely made way more money than year one.

But is that the case? In question three, we wanted to know what conclusion Jacob might be trying to convey from this graph.

Well, to me, it looks like he's trying to imply that July was a significantly hotter month than all the others.

So why might it be misleading? Well, did you spot the broken scale? So he can't use proportions here.

The scale is stretched.

If we actually look at the actual values, then the actual difference in temperatures is only 2.

25 degrees centigrade.

He misses out August as well, which might make that peak in July look a little bit more significant.

Did you spot that? Did he jump from July to September? Question four, what type of chart could Sofia draw? Well, she could do a bar chart, pie chart, pictogram.

We have quite a lot of options here.

Now in part B, what is she going to do to manipulate her chart? Well, if she does a 3D pie chart, she can make sure that exercise is at the front and put the pie chart at an angle.

If it was a bar chart, she might decide to leave the numbers off the axis.

And if it was a 3D bar chart again, she could put exercise in the foreground to make the bar appear bigger.

Well done.

It's time now to move on to the second part of our lesson, "Misinterpreting Graphs." Now, we've seen how graphs can be drawn to be misleading, but it can also be very easy to misinterpret graphs or make incorrect conclusions based on a lack of understanding about the data.

Well, what do we mean by that? Well, for example here, we have a time series graph showing the number of passengers at Oakfield Station.

A local newspaper article states, "The number of people using the station is at an all-time low." So does the graph support this headline? What do you think? Pause and have a discussion.

Well, the graph only has the data for the last year.

Did you notice it starts in March, 2023? So it's incorrect to state that this is the all-time low for passenger numbers.

We don't actually know what it was like before.

It may be the case that winter months always have fewer passengers.

What about this graph? This graph shows the number of ice creams sold and the number of skateboarding accidents over a series of days.

"That's crazy," says Alex.

"The number of skateboarding accidents increase when ice cream sales increase as well." Whoa, is Alex trying to imply that if we sell more ice creams there's going to be more skateboarding accidents? Oh, Sofia thinks so.

She says, "People must be so distracted eating their ice creams that it's causing them to fall off their skateboards." Do you think that sounds right? Pause the video and have a discussion now.

Exactly, it is unlikely that ice creams are causing the skateboard accidents.

It's also unlikely that skateboarding accidents are causing people to go and buy ice cream.

Can you think of a reason though, why we might be seeing a similar pattern here? Pause and have a discussion.

What did you come up with? Personally, I've gone for weather.

I think if it's a sunny day, you're more likely to buy ice cream.

You're also more likely to be out skateboarding.

Now, the day of the week could be a factor as well.

More people are likely to buy ice creams at the weekend, because they're more likely to be out.

And the more people out on their skateboards, the higher the chance of accidents.

So this is an example where two things are correlated, but one does not cause the other.

In this case, there's likely to be a third factor, which affects both in the same way.

For example, we talked about the weather just a minute ago.

So it's possible for things to be correlated without there being any connection between them.

And it's often possible to manipulate data to show a correlation between two completely unrelated variables.

This can be a particularly tricky idea for scientists to navigate when they're testing chemicals or procedures or the effectiveness of medical treatments.

Let's do a quick check.

If two things are correlated, then one must cause the other.

Do you think that's true or false? Pause now, while you make your choice, Well done if you said it's false.

Now, think about that.

Why is this statement false? Pause the video and write a justification now.

Welcome back, did you remember that there could be a third factor affecting both or it may be a coincidence or a result of manipulating the data in a certain way? These are all valid reasons.

Let's move on to our last task.

In question one, Alex drew this chart from data he collected about how pupils travel to school.

In part A, the school newspaper reports, "Only 50 pupils walk to school in the 2022 to 2023 academic year." Is this supported by Alex's chart? And then in part B, the headline of the article states, "Fewer young people choose to walk to school now than in the past." Is this supported by Alex's chart? Pause the video while you work on this now.

Question Two, this time series graph shows the number of shark attacks and the number of heat stroke sufferers throughout the year.

Jacob concludes that, "The more shark attacks there are, the less likely people are to go in the sea, so they're more likely to get heat stroke.

Shark attacks cause heat stroke." Does the graph support the claim that shark attacks cause heat stroke? And then in part B, suggest a possible reason why there's a similar pattern between these variables? Pause and do this now.

And question three, here is a sketch of a scatter graph showing the number of times people visit restaurants and the value of their car.

Sofia concludes, "All I need to do is eat out more, then I'll be able to get a fancy car." Does the graph show correlation between the number of times people visit restaurants and the value of their car? And how do you know? And then the second part, explain why Sofia's statement is incorrect.

Pause and do this now.

Our last question, this chart shows the number of people who live in rural areas compared to urban areas.

Alex says that, "This shows twice as many people live in urban areas than rural." And Sofia says, "I think it shows that four times as many people live in urban areas than rural areas." Now, I'd like you to give a reason why Alex may be correct, then give a reason why Sofia may be correct, and then give a reason why both might be incorrect.

Pause and do this now.

Time to go through our answers.

So for part A, is this supported by Alex's chart? Well, no, the y axis is percentage of pupils who walk to school.

50% of pupils walk to school, but we don't know how many this actually is.

And then in part B, the chart does seem to show a decrease in the percentage of pupils walking to school, however, this chart only goes back as far as 2020.

So we can't say, "In the past," because we don't know about the previous years.

Well done if you spotted that.

And question two, does the graph support the claim that shark attacks cause heat stroke? Well, the variables follow a similar pattern, but this does not mean that one causes the other.

This could be coincidence or there could be a third factor.

Well, what could it be? Well, the weather and the holidays, are likely to influence both variables in a similar way.

The warmer it is, the higher the likelihood of heat stroke and the more people will be swimming in the sea, so the higher chance of shark attacks.

Now, Sofia here was thinking that if she eats out more, she'll be able to buy a fancy car.

Well, I can see why she thinks that, because the graph does show correlation.

From this limited data, we see what looks to be positive correlation.

The points are lying reasonably close to a linear relationship.

However, just because there's correlation, it does not mean that one variable causes the other.

In this case, people with more money are likely to be able to eat out more and therefore buy more expensive cars.

And our final question.

Now, Alex might be correct, because the height for urban areas is twice that of the height of the picture for rural areas.

Sofia might be correct because the area of the picture for urban areas is four times that with the picture for rural areas.

Now of course, they might both be wrong, because there's no scale on that y axis.

And if the scale doesn't start at zero, then comparing the proportions is incorrect.

It's impossible to interpret what this graph is showing.

Let's sum up what we've learned today.

Graphs are often used to convey information because they're visually appealing.

They are sometimes drawn to deliberately mislead the reader or influence their opinion.

Some graphs are drawn accurately, but interpreted incorrectly.

And just because there appears to be a correlation between two variables does not mean that one causes the other.

Well done, you've done a great job today.

I look forward to seeing you for another lesson.

Bye for now.