Lesson video

Good day everyone.

Thank you for joining me for today's maths lesson.

I'm Mr. Gratton, and today we will be considering different statistical representations and the insight that they may give to a dataset.

For most of this lesson, we'll be focusing on pie charts and bar charts.

Pause here to familiarise yourself with the definitions of these charts.

To start, let's compare some pie charts and bar charts.

Both bar charts and pie charts are used to visually represent data that has been collected.

When looking at these charts briefly at a quick glance, that is, they can both show key insights to this dataset, which insights are more easily understood from this quick snapshot look at each of the charts, the bar chart and the pie chart? Well, a quick look at a bar chart can help compare between each subgroup of data.

For example, it rained on more days than it was dry.

A quick look at a pie chart can help compare one subgroup against the whole dataset.

For example, it rained on a majority of the days last month.

We can describe these comparisons in more detail by reading the heights of the bars, rain occurred roughly three times as often as it was dry.

From looking at the pie chart, dry looks just over 90 degrees, a quarter of the whole dataset.

Looking at the modal activity, which is the activity that occurs most frequently, you can see that the most frequent activity is sports, both by quickly looking at the pie chart, sports has the biggest sector and the bar chart, sports has the tallest bar.

On the other hand, finding out how frequent sports was.

Well, we can read this quite easily on the bar chart by looking at the height of the bar.

From the height of this bar, we can see that sports occurs 18 times, but a pie chart cannot tell you any frequencies without any further information being provided.

But to pie chart's benefits, we can more easily tell that board games appears a quarter of the time, a quarter of the total frequency of the data collected.

This can be seen on the pie chart because a quarter is represented by that 90 degree right angled sector.

We can also figure this information out on a bar chart, but that would require quite a few calculations and so is a much slower process than the easy spot with the pie chart.

And finally, if we're looking to find how many more students did homework versus film club, we can read the frequencies of homework and film on the bar chart and then do a subtraction.

But again, because this question talks about frequencies, the pie chart really isn't useful.

If a frequency for one sector was given or the total frequency that the pie chart represents was given, it would be possible using proportional reasoning.

But this would again take much longer than simply using the bar chart and one simple subtraction between the heights of two bars.

Here's a quick check, which of these questions can be quickly answered using this bar chart? Pause here to consider each question.

A and D can be answered quickly as these questions require you to look at a frequency, similar again for this pie chart, which of these questions can be easily answered? Pause here to consider each of these four questions.

The answers are B and C as these questions check for a proportion rather than frequency.

Okay, but what about bar charts and pie charts that have many more subgroups? In the case of months, 12 subgroups, one for each month, for bar charts, it is still quite easy to gain that insight.

November is clearly the modal month 'cause it has the highest bar.

It is easy to read that February has a frequency or a height of 24 on a bar chart, and we can see that the lowest height is June and so that is the least frequent birth month.

However, for a pie chart, it may lose some of its benefits.

You can tell November is the modal month, but it isn't very clear and it's definitely not as clear as the bar chart, but even though proportion is the specialty of a pie chart, it is quite hard to tell if any month had more than 10% of the students representing it.

We can find this information out using a protractor, but we certainly cannot do it by looking at a quick snapshot at this pie chart.

It is also tricky to interpret anything for low frequency sectors.

Notably when there are a handful of sectors that are all low frequency, meaning they all have small angled sectors on the pie chart that are tricky to distinguish by sight.

Furthermore, it is highly impractical and impossible without a protractor to calculate any frequencies as you would need further information and use proportional reasoning calculations to find these frequencies out.

Finding out the number of people born in October cannot be done at a quick glance and it would require far more calculations than using a bar chart.

To the defence of pie charts, broader insights can still be possible.

We know that more than 25% of students were born in these winter months of November, December, and January.

I personally think it is much quicker and easier to spot this type of information from a pie chart compared to the same question on a bar chart, is it pretty obvious to you that November, December and January using a bar chart represents 25% of the data? It's not as obvious.

So for dataset with many subgroups, bar charts retain its specialty in reading frequencies, but to gain insight into proportion, you would have to add up all of the heights of all of the bars to find the total frequency first.

Pie charts retain some of its specialty in reading proportion, but you would either need a protractor or at least one frequency to gain any further detailed insight using a pie chart.

Okay, two quick checks.

First one from only a quick glance at the pie chart and bar chart, which mode of transport was most frequently used and which graph or both is it possible to answer part A? Pause to check both of these charts.

The answer is bus, and it was possible to tell this from both charts because bus had the largest sector and was the tallest bar on the bar chart.

Similar question again, did more, less, or equal to 25% of students walk to school and with which graphs could you answer that question? Pause to give this a go.

The answer is more than, and it is much clearer to answer that using the pie chart than the bar chart.

This is because we can see that the walk sector had a sector size greater than 90 degrees whilst tricky to calculate, it is still possible to find this information from a bar chart, just not with a quick glance at the two charts.

Time for some practise, put a tick by the statements that were easily gathered using this pie chart.

Pause to look through all five of these sentences.

For question number two.

For each of these five questions, put an answer in the pie chart column or the bar chart column or both.

Pause now to do this for parts A, B, C and then pause again for D and E.

So pause now to answer questions A, B, and C.

And again for questions D and E.

For question number three.

For each of these questions, write down a Q if the question can be answered quickly by a quick glance, write down an S if you can answer that question after one single calculation such as a subtraction or M if you need many calculations in order to answer that question.

Afterwards, if you can, then answer each question, pause to give yourself some time to do this.

Onto the answers B, D and E were possible to gather from this pie chart and for question number two, the mode of raincoat can be easily gathered from both the pie chart and the bar chart.

For parts B and C however, you can only easily gain this information from the pie chart, however it is still possible to do from the bar chart if you spent time doing the calculations.

For parts D and E, the frequency answers are both found using the bar chart and for question number three, here are the answers.

Pause here to check your answers against the ones on screen.

For this cycle, we'll be looking at a completed chart and comparing it to an incomplete chart of the same data in order to attempt to complete it.

Such as here, this pie chart is complete but the bar chart is incomplete.

We can use a protractor to measure the angles on the pie chart and use this information to complete the bar chart.

Notice how coffee is half the angle of water.

This means that coffee on the bar chart will be half the height of water and so as we can see, the height of water was six and so the height of the coffee bar must be three.

The same applies for juice being half the height of tea.

If tea had a height of 10, then juice must have a height of five.

Here's a check which two drinks were equally frequent.

Pause here to find which chart says this information clearly.

the answers are water and juice.

They both have the same frequency as seen by them both having a 90 degree sector on the pie chart.

Using this information, what should the height on the bar chart be for water? Pause to make the link across these two charts.

The answer is nine squares, the same as the height for juice.

Here are the angles for milkshake and fizzy.

How tall should the bar for milkshake be? Pause to make the comparisons using these angles.

The answer is 12 squares.

This is because milkshake was double the angle of fizzy and so the bar must also be double in height.

Onto some practise questions.

Complete the bar chart using the information in this pie chart, pause to take in all of this information.

Question number two is similar to question one, but this time you'll need to use a protractor to measure the angles in the pie chart for yourself.

Pause to find the angles in this pie chart and then complete the bar chart.

Question three is the other way around the bar chart is complete this time, but the pie chart is incomplete.

Pause, define the total frequency of the bar chart and use this information to complete the pie chart.

Here are the answers, five times the number of people voted for dinner than lunch, therefore the bar should be five times as tall, giving you a total height of 10 compared to two for lunch.

For question number two, first of all, here are the angles.

Any angle measured tow within around two degrees of the ones that you see on screen is completely fine.

Pasta should be the same height as sandwich at a height of six whilst snack should be one third the height of roast dinner at three rather than the nine of roast dinner on screen.

For question three, here are the angles for the three sectors of this pie chart.

The only representations we've looked at today so far are pie charts and bar charts, but what about the other less familiar charts and graphs? Here's some examples and there are many of them.

Do any of these look familiar to you either from TV or from a different lesson such as science? Most ways of visualising data are variations on the basic graphs that you have either already met or will meet very soon, which means you have the skills to read them even if they look unfamiliar to you at the moment.

The two that we will look at in more detail are the population pyramid on the left and the core choropleth map on the right.

Starting with the population pyramid, they are great at comparing between two different populations.

The Y axis on a population pyramid shows the population subgroups, for example, age and yearly income.

This one shows age split into five year intervals, zero to four, five to nine, and so on.

The X axis will always represent frequency no matter what the Y axis represents.

A frequency of zero always lies in the centre of the two parts of the population pyramid with positive frequencies growing out, both left and right from the zero to represent the frequencies of the two populations we are comparing.

Whilst population pyramids might look a little unfamiliar to you, you have all the skills that you need to interpret and understand a population pyramid.

Why? Because they're basically just two bar charts stacked to the left and right of each other as you can see.

Let's answer some questions about this population pyramid.

In which location, Oakfield or Rowanwood were there more zero to four year olds? Well, by looking at this bottom bar that represent the zero to four year olds, we can see that Oakfield had about 14,000 zero to four-year-olds.

Whilst in Rowanwood they had just over 12,000.

Therefore, Oakfield had more zero to four-year-olds than Rowanwood.

In which location were there more people over 80 years old? To do this, we need to look at the highest or the top bar of this population pyramids.

In Oakfield there were about 1000 people whilst in Rowanwood, just over 3000 people.

Therefore, Rowanwood had more 80-plus-year-olds.

And the next graph is a chorochoropleth map, or at least this is a simplified version of one.

It is used as an overlay for maps.

It describes how populated an area is.

The darker the area, the more populated, numbers and, or symbols can also be used to support the visualisation of the choropleth map.

In the choropleth maps, we will look at today the higher number, the larger the population of that area.

It is essential for a choropleth map to have a key like this one.

This is so we can understand the scale of the populations of the lighter shaded areas and the darker shaded areas.

For example, the darkest areas of one choropleth map might represent a couple thousand people, whereas the darkest areas of other choropleth maps might represent tens or hundreds of thousands of people.

The key is important to understand that scale.

The darker the part of the map, the more densely populated, that means more people in the same area.

We can describe location on a choropleth map like this.

For example, at the top middle of the map or the bottom left of the map or the middle right of the map, how many people live in a region of a map? Maybe this region is described by how densely populated it is or by its location, bottom left, top right north, southeast, or west for example.

In this particular example, it's asking for the smallest population In that highly densely populated area, we can use the key to inform us.

Smallest means using the smaller number in the range that the key gives us for that shade.

Five squares have the darkest shade and 10,000 people is the smallest number of people for each square, so five times 10,000 equals a 50,000 people minimum population in that densely populated area.

Okay, onto the last few checks, which location had more 50 to 54-year-olds, Oakfield or Rowanwood? By giving your answer to the nearest 1000 people, how many more people of this age group were there in that location? Pause here to study this population pyramid.

The answer is Rowanwood by around 2000 people.

If we looked at this population pyramid in more detail, we would find the precise answer to be approximately 1,800 people.

For this choropleth map, describe where the most populated area is.

Pause to have a look at this map and think of how you would describe that area.

The answer can be as simple as the middle right of the map.

There is a new densely populated settlement shown by four squares, by using the key, what is the minimum number of people in this new settlement? Pause here to look at the key and the map.

There are two fours and two threes.

The minimum number of people in a three square is 500 and the minimum number of people in a four square is 1000.

Therefore 1000 plus 1000 plus 500 plus 500 is a total of 3000 people minimum in this settlement.

Here are the final few practise questions.

For questions one and two.

Have a look at this population pyramid and compare between the yearly salaries in 1000 pounds of Oakfield and Rowanwood.

Pause here to try this.

For question three and four, compare between the modal salaries of both locations.

Pause here to give yourself some time to do this.

Lastly, for questions five and six, answer these questions using the choropleth map and its key, pause here to do these last two questions.

Okay, onto the answers, for question number one, just under 5,000 people between 8,000 and 11,900 pounds in Rowanwood, the answer to question two is Oakfield by just over 1000 people.

The frequency in Oakfield for this interval is just over 2000 people whilst in Rowanwood it is just under 1000 people.

For question number three, Oakfield's modal group was 32,000 to 35,900 and Rowanwood's modal group was 20,000 to 23,900.

Do note that the correct answers are not, for example, 32 to 35.

9.

The Y axis scale for this population pyramid is in thousands of pounds and so your answer must show that most simply by multiplying the numbers on the Y-axis scale by 1000 each.

And for question number four, by comparing the frequencies of both Oakfield and Rowanwood's modal subgroups, we can see that Rowanwood had more by 6,000 people.

For question number five, you would describe the densely populated area by saying the bottom right of the map.

For question number six, the minimum number of people is 8,000, and the maximum number of people is 18,995 people.

A very well-done if you got any of those questions correct with these unfamiliar graphs.

In today's lesson, we have compared both pie charts and bar charts and used one to complete the other.

We've also looked at less familiar graphs and used the skills we have learned with other graphs to help us interpret these new ones.

Very, very good work in today's lesson.

Thank you so much for joining me, Mr. Gratton, and I hope to see you soon for some more maths.

Have a great day.