Lesson video

Hi there, my name is 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 Interquartile Range, and this is within the unit, Graphical Representations of Data and, in particular, Cumulative Frequency and Histograms. By the end of this lesson, you will be able to calculate the interquartile range, sometimes referred to as IQR, like it is here, and interpret what this means.

Keywords that we will need to use in today's lesson are box plot, median, and range.

All of these will be very familiar to you, but I always think it's worth a recap just to make sure that we've got those in our minds ready for this lesson.

A box plot is a diagram that clearly shows the minimum and maximum value of a set of data, along with the three quartiles.

And remember, those are the upper and lower quartile and the median, sometimes referred to as Q1, Q2, and Q3.

The median is the central middle piece of data when the data are in numerical order.

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

The range is a measure of spread.

It is found by finding the difference between the highest and the lowest values.

Like I said, you'll be really familiar with each of those terms. You may not, however, be familiar with the term interquartile range.

It is calculated by finding the difference between Q3, the upper quartile, and Q1, the lower quartile.

And this is because it is the range from the first to the third quartile.

For today's lesson, we've got two separate learning cycles.

In the first one, we will just concentrate on calculating the interquartile range.

And then in the second one, we will look at using the interquartile range and, in particular, in a context.

Let's get going with the first one, so calculating the interquartile range.

We know that the mean and the range can be used to compare data sets.

There are some disadvantages, however, of using these two measures if we include all of the data.

And that is that they can be affected by extremes, or outliers.

Using the median, we can find an average which will not be affected by outliers.

The interquartile range is also not affected by outliers, as it only considers the middle 50% of the data.

The interquartile range is the difference between the upper and the lower quartiles.

To calculate the interquartile range, and like I said, often referred to as IQR, we subtract the lower quartile from the upper quartile.

Or you may see it written as Q3 subtract Q1.

If we take a look at what that looks like on our box plot, the interquartile range is the width of the box.

We can see that that is the difference between the upper and the lower quartiles.

The interquartile range can be calculated from a box plot, cumulative frequency graph, and a list of data.

And we will consider each one of those in turn in a moment.

Let's start by looking at the interquartile range from a box plot, because that's what we just looked at in that diagram.

The interquartile range from a box plot is the width of the box, or the length of the box.

That is an interquartile range.

Q1 is the lower quartile, that's represented by the left hand side of our box.

And here, that is 114 grammes.

Q3 is the upper quartile, and this is represented by the right hand side of our box.

And here we can see that Q3, for this box plot, is 135 grammes.

The interquartile range is the difference between the two.

135 subtract 114 gives us an interquartile range of 21 grammes.

I'd like you please to calculate the interquartile range from this box plot.

Here you have some masses, and they are given to you in kilogrammes.

Please make sure you include a unit with your answer.

Pause the video, you may use a calculator if you want to, and then come back when you're done.

Super work, well done.

Let's check that for you.

Q1 is here, and that is a value of 1.

5 kilogrammes.

Q3 is here, the right hand side of the box, and that is a value of 2.

85 kilogrammes.

So the interquartile range is the difference between the two.

2.

85 subtract 1.

5 equals 1.

35 kilogrammes, the interquartile range is 1.

35 kilogrammes.

Now let's take a look at the interquartile range from a cumulative frequency graph, the upper and the lower quartiles.

Now, you are familiar with how to find these.

But if you need a quick recap, the total data is 40, so the lower quartile is a quarter of the way through this, so at 10.

And the upper quartile is three quarters of the way through, so at 30.

Here we can see that the interquartile range is the difference between the two, and it's represented by this on our graph.

This is a cumulative frequency graph, and it is showing us test scores of a class.

We are going to find the interquartile range.

Let's start with Q1.

Q1 is 25% of the way through the data, or a quarter of the way through the data.

There are 20 pieces of data, a quarter of this is five,.

So we're going to read off the test score when the cumulative frequency is five.

We can see that from our arrows.

And I can see that Q1, the lower quartile, is 10.

Now we'll calculate the upper quartile, Q3.

That will be 15, so we read off the test score when the cumulative frequency is 15.

And this gives us an upper quartile, or Q3, of 17.

The interquartile range is the difference between them.

17 subtract 10 is seven.

Remember this is seven marks, we must refer to the data.

We don't just leave our answer as seven.

Your turn now then.

Please could you find the interquartile range for these test scores? Pause the video, and then come back when you're done.

Well done, let's check those answers.

So this time, our total frequency was 40.

So you should have been reading off for the Q1.

You should have been reading off what the test score was when the cumulative frequency is 10, which was 11.

For Q3, we should have been reading off when it was 30.

And that gave us a test score of 18.

5, meaning our interquartile range is 18.

5 subtract 11, giving us 7.

5 marks.

And again, making sure that we refer back to the context of the question and we give a unit to what the 7.

5 is.

And lastly, we can calculate the interquartile range from a list of data.

Here are the test scores for a group of 13 pupils.

Now, then the most important thing, I've given them to you in order, that's really important.

We're gonna find, firstly, the median by crossing one off of each side.

And I always double check, I've crossed the same number on the left and the right.

And I have, I've crossed six to the left of 17, and six to the right of 17.

This is the median.

I don't necessarily need to make a note of what the median score is.

I just need to identify it so I can then find the lower quartile, and that's halfway through the lower half of the data.

So crossing one number off each end, I end up with seven and 10.

So my lower quartile is halfway between seven and 10.

And then I do the same for the upper quartile.

And I can see that my upper quartile is halfway between 23 and 24.

The lower quartile then is seven add 10 divided by two, which gives me 8.

5.

And my upper quartile is 23 add 24 divided by two, which is 23.

5.

And for that second one, you probably didn't need to do that calculation 'cause it's obvious 23.

5 is halfway between 23 and 24.

It's less obvious that 8.

5 is halfway between seven and 10, however.

So our interquartile range is 23.

5 subtract 8.

5, and that is 15 marks.

So the interquartile range of this group of pupils is 15 marks.

Let's take a look at another one.

So again, I've been very kind here, I've given you the list in order.

We need to identify where the median is.

So crossing one off from each side, and I end up with two in the middle.

So my median is halfway between the two.

Now, I can find the middle of the left hand side of my data.

So crossing off, I can see my lower quartile is here.

And then crossing off, I can see that my upper quartile is here.

My lower quartile then is halfway between 10 and 13.

And if you're not sure, you can add those two together and then divide by two, giving us 11.

5.

And then the upper quartile, we find the sum of the two values it's between and then divide that by two, giving us 23.

5.

The interquartile range then is 23.

5 subtract 11.

5, and that gives us 12 marks.

The interquartile range for this group pupils is 12 marks.

Your turn now, I'd like you please to work out for me the interquartile range of this data.

And its data of 10 pupils, and they were asked how many times they could bounce a ball in 10 seconds.

Pause the video, and then when you've got your answer, come back.

Great work, let's check the answers then.

So we should have, you should have identified that the median was halfway between 10 and 12.

And then your lower quartile was seven, and your upper quartile was 14.

Giving us an interquartile range of 14 subtract seven, which is seven bounces.

Now we can have a go at filling in some missing information using all of the things that we've learned up until now.

Let's start with the range, we know that the range is the highest score subtract the lowest score.

And we're told in this table of values that the range is 36, and the lowest score is 12.

So the range 36 is the highest score, which we don't currently know, subtract the lowest score, which is 12.

This means the highest score is 36 add 12, so the highest score is 48.

Now we can take a look at the interquartile range, and we know that this is the upper quartile subtract the lower quartile.

From the table, we know that the interquartile range is 15, the upper quartile is 32, but we don't know what the lower quartile is.

To calculate the lower quartile, we need to do 32 subtract 15, giving me 17.

The lower quartile is 17.

So we're able to find missing values if we know the range and the interquartile range.

Please could you have a go at this one for me? Pause the video.

If you need to go back and rewatch that example, you can, 'cause I have only given you one example this time, but I'm sure you are ready to have a go at this one.

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

Super work, let's check your answers.

We know the range is the highest subtract the lowest, and we're told in the table that the range is 30.

And we're told that the highest score is 38, and we don't know what the lowest score is.

But if we rearrange this, we end up with the lowest score is 38 subtract 30, which is eight.

The lowest score was eight.

And we know that the interquartile range, or IQR, is upper quartile subtract lower quartile.

What do we know from the table? Well, we know that the interquartile range is seven.

We know that the upper quartile.

No, actually, we don't know what the upper quartile is, but we know that we are going to subtract the lower quartile, which is 18.

And then if we rearrange this, we end up, the upper quartile is seven add 18, which is 25.

And you could always do a little check here.

Just check, is 25 subtract 18 seven? It is indeed.

Task A then, you're gonna calculate the interquartile range for each box plot, pause the video, and then come back when you are done.

Well done, and question number two, calculate the interquartile range of the graph showing the number of seconds to complete a puzzle.

Great, and question number three, I'd like you please to find the interquartile range of these sets of data.

Well done on those.

And question four, fill in the missing information.

Great work, well done on that.

Quite a lot of questions there weren't there? Let's check those answers then.

A, we should have Q1, 1.

6 kilogrammes, Q3, 3.

2 kilogrammes, giving an interquartile range of 1.

6 kilogrammes.

B, Q1, 123 centimetres, Q3, 148 centimetres, giving an interquartile range of 25 centimetres.

Question two, Q1 was 10, Q3 was 17, giving us an interquartile range of seven seconds.

Question three, Q1, 20, Q3, 38, interquartile range of 18.

And 3B, Q1 was 12, Q3 was 29.

5, giving an interquartile range of 17.

5.

And finally, question 4A, the missing lower quartile range was 107 centimetres, and the tallest was 152 centimetres.

And then on B, the lowest mass was 125 grammes, and the upper quartile was 158 grammes.

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

Let's move on then to using the interquartile range in a context.

The interquartile range is a useful way of comparing sets of data.

It's important to consider the context of the data when comparing sets of data.

Below are the median and interquartile ranges of the time taken for a group of adults and children to complete a 5K park run.

So the adults' median is 32 minutes, and the children's is 38 minutes.

The adult's interquartile range is 21 minutes, and the children's interquartile range is 16 minutes.

Jun says, "Children did better at the run as their median is higher." Do you agree with Jun? Jun is right that the children's median is higher, but he has not considered the context of the question.

Here, the times represent the time taken to run a race.

So the lower median is better.

Really important to consider the context of the data that you are working with.

Jun says, "Of course, the adults were quicker, but the children's times were more consistent." Why does Jun say that the children's times are more consistent? And the reason he says that is because the interquartile range, which shows the range of the middle 50% of data, is lower and therefore more consistent.

True or false, the range is the best measure of spread or dispersion for this data, is that true or false? And also, as always, please give me your reason.

What did you decide? And it's false.

And the reason for that is because the lowest score is considerably lower than the lower quartile, which implies it would probably be an outlier.

This is why we generally use the interquartile range as a more reliable value to represent the spread or dispersion of the data.

We can see that most of the data is above 62.

So to use that score of one might not be the most sensible thing to do.

In general, when comparing sets of data where we may have extreme values, we use the median as the measure of central tendency and the interquartile range as the measure of spread or dispersion.

It is really important to refer to the context of the data when making those comparisons.

Below are the medians and interquartile ranges for the heights of two different classes in year 11.

Compare the distribution of heights of the two different classes.

In this type of question, you should give two comparisons.

One using the median, and one using the interquartile range.

Laura says, "Class 11B have a higher median and a higher interquartile range." Jun says, "The pupils in class 11 B were taller on average, the heights of the pupils in class A were more consistent." Whose response would you choose if you were answering this question? And you should have said Jun's answer, 'cause Jun has included the context in his response.

Laura has just said which is higher, whereas Jun has actually looked at, this is talking about heights of year 11s.

And so, therefore referred to the heights.

And also what the interquartile range is showing us, and that's showing us consistency of the data.

Check for understanding now then.

Laura and Jun have both been growing tomatoes, below is a table of their masses.

Compare the distribution of the masses of Laura's and Jun's tomatoes.

We've got Laura with a median of 123 grammes, and an interquartile range of 15 grammes.

And Jun is 128 grammes is the median, and the interquartile range is 12 grammes.

Pause the video, I want two lovely sentences please, bit of a clue there.

And when you're done, come back.

Super, and you should have written something like, Jun's tomatoes are heavier on average.

And you may have said because the median is higher, the masses of Jun's tomatoes are more consistent than Laura's.

And again, you may have said there, because the interquartile range is lower.

And now, we're ready for the final task of today's lesson.

There are only two questions in this task.

I'd like you please to compare the distributions of the data I've given you.

So for A, I've given you data about the heights of year 7s and year 11s.

And for B, I've given you the times taken to complete a puzzle for adults and children.

So you're going to please write me some lovely sentences about the comparison of those distributions.

Pause the video and then when you're done, come back and we'll check those answers for you.

We're nearly there now, so just keep with me just for a little bit longer.

Well done, you should have something like this.

The pupils in year 11 were taller on average.

The heights of the pupils in year 7 were more consistent.

Or on the flip side of that, you could have, the pupils in year 7 were shorter on average.

And the heights of the pupils in year 11 were less consistent.

And then B, on average, the children completed the puzzle quickest.

The adults times were more consistent.

Or, on average the adults took longer to complete the puzzle and the children's times were less consistent.

Let's summarise our learning from today's lesson then.

The interquartile range is the difference between the upper and lower quartiles.

To calculate the interquartile range, sometimes referred to as IQR, we subtract the lower quartile from the upper quartile.

And sometimes you'll see this referenced as Q3 subtract Q1.

The interquartile range is a useful way of comparing sets of data.

When looking at the interquartile range on a box plot, it is the width of the box.

It is important to consider the context of the data when comparing sets of data.

In general, when comparing sets of data where we may have extreme values, we use the median as the measure of central tendency and the interquartile range as a measure of dispersion or spread.

It is really important to refer to the context of the data when making comparisons.

And you can see how important that is, 'cause I seem to have written it twice.

Remember, the median tells you who is better, who is quicker, who is taller, who is shorter.

And the interquartile range tells you which set of data is more consistent.

The smaller the interquartile range, the more consistent that data is.

Well done on today's lesson.

Thank you for joining me, and I hope to see you again really soon.

Please do take care of yourself and, like I said, hopefully I'll see you again soon, goodbye.