Lesson video

Hello there.

My name's Dr.

Rowlandson, and I'm delighted to be guiding you through today's lesson.

Let's get started.

Welcome to today's lesson from the unit of "Comparisons of numerical summaries of data." This lesson is called "Constructing Stem and Leaf Diagrams," and by the end of today's lesson, we'll be able to construct stem and leaf diagrams. This lesson contains a new keyword, which is stem and leaf diagrams. A stem and leaf diagram is a systematic way to visually organise and represent numerical data.

There's an example on the screen here with the stem and leaves labelled, but we'll go into this in more detail during today's lesson.

The lesson contains two learning cycles.

In the first learning cycle, we're going to be plotting stem and leaf diagrams for a single sample of data, and then in the second learning cycle, we'll plot back-to-back stem and leaf diagrams. Let's start off with stem and leaf diagrams for a single sample of data.

Here we have Andeep and Sam.

Andeep and Sam design a game that gives each player a score and they want to pilot the game so they can investigate how highly players tend to score.

Andeep invites all of his classmates to play their game once.

This sounds like an experiment.

They collect their data from the experiment by asking each person to write their score on a slip of paper and hand them in at the end of the game.

And here's what they get.

They attempt to organise and display their data on a poster, which we can see here.

Andeep says, "It's difficult to analyse the data when it looks like this." So how could they organise their data so that it's easier to analyse? Pause the video while you think of some suggestions for what they could do with this data, and then press play when you're ready to continue.

Well, they have a lot of options, but let's see what they do.

Sam says, "we could rearrange the scores so that they are in order," and so they do.

They sort the slips of paper so that the scores are in ascending order.

It's looking a bit clearer now.

Andeep says "This is clearer, but it's still just a long list of numbers." So how could they organise their data so that it's even easier to analyse? Pause the video while you think about any other suggestions you might have, and then press play when you're ready to continue.

Sam says, "We could arrange the scores into four groups.

Scores that are in the 30s, scores that are in the 40s, scores are in the 50s, and scores are in the 60s." So they do that, or they try to do that.

They try to sort the data into four groups according to their tens, but it looks like they've had a little bit of trouble.

Andeep says, "Nice idea, but they don't all fit on the poster like this." Hmm, I wonder what they could do? Let's see what Sam thinks about this.

Sam says, "I've got an idea for how to display the scores more succinctly.

Each number has two digits, so we could write the first digit at the side of the poster," which we can see here on the poster.

Andeep says, "These numbers are the first digits of each score." All the scores are in the 30s, 40s, 50s, or 60s.

So the 3, 4, 5, and 6 we can see here represent the first digit of each score.

Sam then says, "We could then cut each slip of paper in half and just use the second digit for each number," like so.

"So this piece of paper represents a score of 32," Andeep says because the 3 on the poster tells us that it's 30 something and the 2 on a slip of paper tells us that it's 32.

This next piece of paper represents a score of 34.

The 3 which is written on the poster represents the 30, and the 4 which is written on the piece of paper represents the number of ones, so that's 34.

And they continue as so until all the data is laid out like this.

Sam was right.

It does all fit onto the poster much better now, but Andeep spot's a problem.

Andeep says, "We know what score each piece of paper represents, but how will other people know?" Sam says, "We could write a key to explain how to interpret each data point." Something a bit like this.

This key says 3 with a line and 2, which is what we can see with the first data point in the top left, 3 and then a line and 2, that means 32 points, and a reader could use that to interpret the rest of the data points.

Now that they've finished this, they consider what observations they could make from their data.

I wonder what observations you can make from this data? Do people tend to score high numbers or low numbers? Where do people tend to score? Pause the video while you think about that, and press play when you're ready to hear from Andeep and Sam.

Andeep says, "Most of the scores were in the 40s," and we can see that because the 4 on the poster, which represents 40, has the most number of slips of paper next to it.

And Sam says, "Overall, most scores tended to be towards the lower end of the range of data." The data goes from 32 all the way up to 66, and we can see that the majority of these slips of paper tend to be in the 30s and 40s, at the lower end of that range.

What Andeep and Sam have made here is a stem and leaf diagram.

A stem and leaf diagram is a systematic way to organise and represent numerical data by splitting each value into a stem and a leaf.

And we can see an example of a stem and leaf diagram on the screen here.

Let's look at each part and turn.

We have some numbers written vertically above each other with a line on either side.

That's our stem, and that represents all but the last digit of each data point.

In this case, the 3 represents the 30.

And then if you look to the right on the stem and leaf diagram, you'll notice that each number in the stem has a list of numbers written after it.

Well, they are the leaves.

The leaves are made from the last digit of each value, and the leaves are written in ascending order.

We can see, for example, on the top row, the numbers 2, 4, 5, 6, 7, 7, 9, and 9.

Those are in ascending order from smallest to largest.

And at the bottom of the stem and leaf diagram, you'll notice a key.

A key is required to inform a reader how to interpret each data point, and we're going to see some examples of some different keys and how it makes a difference to how you interpret data.

In this case, the key says 3 line 2.

Well, that represents the first data point we can see in the stem and leaf diagram.

The 3 is in the stem, the line is what divides a stem from the leaves, and the 2 is in the leaf.

So 3 with a line and a 2 means 32 points.

That way, a reader can use that key to interpret any data point in this stem and leaf diagram.

The values can be interpreted from a stem and leaf diagram based on which part of the stem they are in line with and the key.

Let's look at some examples.

The highlighted data point here represents 32 points.

That's because it's in line with the 3 from the stem, the leaf is a 2, and the key says 3 line 2, or a stem of 3 and a leaf of 2, means 32 points.

This data point represents 34 points.

It's in line with the same part of the stem, so it's 30 something, and it's 34 because it's a four leaf, and that's how the key tells us to interpret it.

This one represents 35, but if we move them to a different part of the stem, this data point represents 46 because it's in line with a 4 on the stem, so it's 40 something, in this case 46.

And this data point represents 52 because it's in line with the 5 from the stem.

However, not all stem and leaf diagrams have the tens in the stem and the units in the leaves.

It can be different depending on the key.

Changing the key changes what each data point in a stem and leaf diagram represents.

Currently, the key says 3 line 2, or a 3 in the stem and a 2 in a leaf, means 32 points.

So this data point represents 51 points, but what happens if we change the key? If the key says 3 line 2 means 3.

2, then the highlighted data point means 5.

1.

If the key says this, 3 line 2, a three in the stem, a 2 in a leaf, means 0.

32 points, then the data point which we've highlighted now means 0.

51.

So it's absolutely vital that when you construct a stem and leaf diagram, you'd write a key, and it's important that when you read a stem leaf diagram, you refer to the key.

Here we have Izzy.

Izzy is representing some data in a stem and leaf diagram.

The first three data points are below.

We've got 7.

9, 8, and 9.

20, and here's the start of Izzy's stem and leaf diagram.

We've got a stem with the numbers 7, 8, and 9, those are the first digits of each of these numbers, and we've got a key to explain that 7 in the stem and a 9 in the leaf means 7.

9.

Izzy then starts to put the data into the stem and leaf diagram.

7.

9 goes here, just like it says in the key, but then Izzy's got a problem.

She says, "I don't know what to write for 8 because it's not 8 point something, it's just eight.

Should I leave a blank space?" She decides to leave a blank space for the time being, but then she has another problem.

She says, "I also don't know what to write for 9.

20.

Should I write 2 zero in the leaf?" Hmm? So this is what Izzy has written so far.

What do you think about this? What problems are there with what Izzy has written so far and what could Izzy do differently? Pause the video while you think about this, and press play when you're ready to continue.

Well, the problem seems because by the way that the numbers are written in her list of data.

When drawing a stem and leaf diagram, data should be truncated or extended so that all data points terminate in the same place value column.

Currently with Izzy's numbers, she has 7.

9, 8, and 9.

20, which all terminate in different places, but that can be fixed by writing numbers like this, 7.

9, 8.

0, and 9.

2.

All these numbers now terminate in the same place value column in the tens, and this now becomes much easier to organise into a stem and leaf diagram.

We can put the 9 next to the 7, a zero next to the 8, and a 2 next to the 9.

So let's check what we've learned.

Here, we have a stem and leaf diagram with a key and one data point has been highlighted.

What value does the highlighted data point represent? Pause the video while you write it down, and press play when you're ready for an answer.

The answer is 62.

In the context, it's 62 points.

The key has now been changed and another point has been highlighted.

So what value does the highlighted data point represent here? Pause the video while write it down, and press play when you're ready for an answer.

The answer is 0.

047, and with the context, it's kilogrammes.

So let's now look at constructing a stem and leaf diagram from start to finish.

Jun is drawing a stem and leaf diagram to represent the numbers below, and he's gonna talk us through how he does it.

He says, "First, I'll make a rough draught to put each data point in the correct row, and then I'll make a final version with the leaves in order." This sounds like a great idea because there's a lot going on when you construct the stem and leaf diagram.

You need to create your stem.

You need to put all your data points in the right part of the stem, and also they need to be an ascending order.

So it looks like Jun is breaking this process into two stages.

A draught version where he just tries to put the numbers on the right row, and then a final version where they are in ascending order afterwards.

Let's continue together.

He says, "I need to draw a stem where I'll write all but the last digit of each number." He does this.

In his stem, he has 3 two-digit numbers, one 5, one 6, and one 7.

He then says, "I'll go through each number in the list and write its last digit in the correct row on the diagram." And he does this working from left to right in the list.

For 165, he puts a 5 in the second row, for 175, he puts a 5 in the third row, and for 159, he puts a 9 in the first row, and he continues with the rest of the numbers.

So he's got something that looks a bit like a stem and leaf diagram now, but what you'll notice is that the numbers in the leaves are not in ascending order.

So he says, "I'll now redraw my stem and leaf diagram with the leaves in ascending order." And so he does that, and here's his final version.

Now, Jun didn't have to do a draught version before the final version.

He could have just gone straight to the final version.

However, the entire dataset has 10 numbers, but each row of a stem and leaf diagram has a maximum of five numbers.

In the case of the top one, it's only got three leaves.

So by doing a draught version first, it means you're only sorting small groups of numbers into ascending order at a time, but Jun isn't finished yet.

He says, "Finally, I need a key.

I'll use the first data point as an example for the key." You don't have to use the first data point, but it's a good one to use, and there's his key.

How you present your stem and leaf diagram will affect how easy it is to analyse and make interpretations from it.

Equally spacing the leaves makes it easier to compare the frequencies for each group or strata within the stem and leaf diagram.

Here are two stem and leaf diagrams. On the left, we have an example of a stem and leaf diagram with leaves are equally spaced.

On the right, those leaves are not equally spaced.

Now, both of these stem and leaf diagrams have the most number of data points which are in the 40s, but which stem and leaf diagram shows this more clearly than the other? It's the one on the left.

With the stem and leaf diagram on the left, it's really quick and clear to see which groups have greater frequencies than others.

It almost acts like a bar chart where we can see very clearly that the most number of data points are in the 40s, and then the 30s, and then the 50s, but with the stem and leaf diagram on the right because the leaves are not equally spaced, comparisons of frequency are less clear.

You can still do it by counting the leaves, but that requires more effort, and it is definitely not as immediately obvious to see.

So let's check what we've learned.

What is wrong with this stem and leaf diagram? Pause the video and write a sentence, and then press play when you're ready for an answer.

The leaves are not written in ascending order.

They should be.

What is wrong with this stem and leaf diagram? Pause the video while you write it down, and press play when you're ready for an answer.

The problem here is that there is no key.

How do you know what these data points represent? What is wrong with this stem and leaf diagram? Pause the video while you write it down, and press play when you're ready for an answer.

The problem here is that the leaves are not equally spaced.

Okay, it's over to you now for Task A.

This task contains two questions, and here is question one.

A group of runners complete a five kilometre race, and the data shows each runner's race time rounded to the nearest minute.

Represent this data in a stem and leaf diagram.

Pause the video while you do that, and press play when you're ready for question two.

And here is question two.

A group of children participate in a shot put competition.

The data shows the distance in metres that each person threw the shot put rounded to the one decimal place.

You need to, please, represent this data in a stem and leaf diagram.

Pause the video while you do that, and press play when you're ready for some answers.

Okay, let's see how we got on with that then.

For question one, this is what a stem and leaf diagram should look like for this data.

Some things to check.

Do you have your leaves in ascending order? Are they equally spaced? And do you have a key? You might have used a different example in your key, but you do need to have a key in there.

Pause the video while you check this with your own, and press play when you're ready for question two.

And question two.

When you represent this data in a stem and leaf, it should look something a bit like this.

Once again, check that your leaves are in ascending order, they are equally spaced, and you have a key.

Great work so far.

Now let's move on to the second part of today's lesson, which is about drawing back-to-back stem and leaf diagrams. Here we have two stem and leaf diagrams. These stem and leaf diagrams show the scores that Andeep and Sam's classes achieved on a game.

We have Sam's class in the left and Andeep's class on the right.

Sam says, "We have data here from two samples." Andeep says, "Could we show the data in a single stem and leaf diagram?" We're going to do that together in a second, but pause the video while you think about how you might approach this, and then press play when you're ready to continue.

There are plenty of options we could do with this data, but one idea could be to put it into a back-to-back stem and leaf diagram.

Data from two samples can be displayed together in a back-to-back stem and leaf diagram, like we can see here.

We have the same stem going through the middle of the stem and leaf diagram with 3, 4, 5, and 6, but we have the data for Andeep's class on the right and the data for Sam's class on the left.

Now you may notice that the leaves for Andeep's class are in ascending order as you go from left to right, but for Sam's class, if you read the numbers from left to right, they're in descending order.

The leaves on the left hand side of a stem and leaf diagram are in descending order.

But in general, think of it this way.

The values of the leaves increase as you read outwards from the stem.

So for Andeep, you're moving away from the stem as you go from left to right, and the numbers will increase as you go in that direction.

But for Sam's data, as you read the numbers from right to left, they go further away from the stem, so the numbers increase in that direction from right to left.

So the further away from the stem you are, the higher the numbers will be in this value.

Sam and Andeep then compare the scores from the two classes.

We'll hear from them in a second, but perhaps pause the video and think about any observations you can make about the differences between Andeep's class and Sam's class with the data, and then press play when you're ready to continue.

I wonder what observations you made.

Let's hear from Sam and Andeep.

Sam says, "My class seems to be split between high scores and low scores." And we can see that from the stem and leaf diagram because with Sam's class, it looks like the majority of the scores are either in the 30s or in the 60s.

Andeep says, "My class tend to score relatively low, but mostly in the 40s." And we can see that with Andeep's side of stem and leaf diagram because most of the leaves are in line with the 3 or the 4, and in fact, most of them are in line with the 4.

So a back-to-back stem and leaf diagram can be helpful for representing two samples of data, but also for making direct comparisons between those two samples of data, like we can see here.

So let's check what we've learned.

The stem and leaf diagram below shows 200 metre race times for a group before and after completing a training programme.

The left hand side of the stem and leaf shows their times before they start the training, and then the date on the right hand side of the stem and leaf diagram shows their times after they complete the training.

What's wrong with this diagram? Pause the video while you write it down, and press play when you're ready for an answer.

The issue here is that the leaves on the left hand side should be in reverse order.

They should be getting greater as they move further away from the stem.

We now have a data point highlighted.

What value does this highlighted data point represents? Pause the video while write it down, and press play when you're ready for an answer.

It represents 38 seconds.

Now let's interpret the graph.

Were the group generally faster before training or after training? Pause the video while write it down, and press play when you're ready for an answer.

The group were generally faster after training, and we can see that because on the left hand side of the stem and leaf diagram, which represents the times before training, all the leaves are in line with a 3, 4, and 5 on the stem, which represent times in their 30s, 40s, and 50s.

But on the right hand side of the stem and leaf diagram, which represents those times after training, all those leaves are in line with a 2, 3 and 4 in the stem, which represent times in the 20s, 30s, and 40s.

So they were faster after training.

Okay, it's over to you now for Task B.

This task contains one question, and here it is.

Two groups of runners complete a 400 metre race.

The data below shows the race times given to the nearest second.

You need to first plot this data in a back-to-back stem and leaf diagram and then use it to answer parts b and c.

Pause the video while you do this, and press play when you're ready for answers.

Okay, let's see how we got on with that.

So when you draw your stem and leaf diagram for this, it should look something a bit like this.

Some things look out for.

Your leaves should be equally spaced.

On the left hand side of the stem and leaf diagram, your leaves should be in descending order as you read them from left to right, but on the right hand side, they should be ascending order as you read from left to right, or a better way to think of it is for both sides they should be an ascending order as you move away from the stem, and you also need to have a key.

We then need to use our stem and leaf diagram to answer some questions.

In part b, in which group was the fastest runner? That was in group B.

It was the runner who ran 49 seconds.

And which of the groups tend to be fastest overall? It would seem that group A tended to be the fastest.

Okay, group B had the fastest runner, but the majority of data in group A seems to show faster race times than the majority of data in group B.

Wonderful work today.

Now let's summarise what we've learned.

A stem and leaf diagram is a systematic way to organise and represent numerical data by splitting each value into a stem and a leaf.

The stem is made of all but the last digit of each value, and the leaves are made of the last digit.

Writing the leaves in order and equally spaced makes it easier to interpret the stem and leaf diagram, and it is important that place value is preserved in a diagram by providing a key.

Thank you very much.

Have a great day.