Lesson video

Hello.

My name is Dr.

Rollinson and I'm excited 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 "Calculating Summary Statistics "from stem and leaf diagrams".

And by the end of today's lesson, we'll be able to calculate the mean, median, mode and range from stem and leaf diagrams. Here are some previous keywords that we're going to use again during today's lesson, so you may want to pause the video if you want to remind yourselves what these words mean before press and play to continue.

This lesson contains two parts.

In the first part, we'll be calculating the median, mode and range from a stem and leaf diagram.

And then the second part, we're going to be calculating the mean from stem and leaf diagrams. Let's start off with median, mode and range from stem and leaf diagrams. Here we have a stem and leaf diagram.

Stem and leaf diagrams can be useful for finding summary statistics such as the median, mode and range.

And Alex says "The stem and leaf diagram is helpful "because it presents the data in order" and that is particularly helpful for median, mode and range, which we'll see during this lesson.

For example, Alex is finding the range from the stem and leaf diagram we can see here.

Alex says "The range is the difference "between the two extreme points in the data".

And because the data in the stem leaf diagram is an order, the lowest is the first data point and the highest is the last data point.

But Alex says, "I'll need to refer to the key "to interpret the values".

So the range will be 175, subtract 151, to get 24.

Alex is now finding the mode from a stem and leaf diagram.

He says "The mode is the most frequent value.

"I can see three fives in the stem and leaf diagram".

Oh, but don't be too hasty Alex, because each of them represent a different value.

The first five represents 155.

The second one represents 165.

The third one represents 175, so looking for the most frequent leaf in the whole stem and leaf diagram will not get you the mode.

Alex says, "I need to look for leaves "that appear frequently in the same row.

"These all represent the same value".

And we can see that here, with these two ones which are both in the same row.

These both represent 161, so the mode is 161.

An extra data point is introduced to the data.

So we have an additional leaf which says nine, and that is in the second row of this stem and leaf diagram.

How will that affect our mode? Alex finds the mode from the new data sets.

He says, "There are now two modes, "161 and 169" So let's check what we've learned.

Here we have a stem and leaf diagram and we have Sofia.

Sofia has made a mistake when finding the mode from this stem and leaf diagram.

She says, "The mode is three".

Explain what mistakes Sofia has made.

Pause the video and write down a sentence to explain her mistake and press play when you are ready for an answer.

So what mistake has she made? She has identified the most frequent leaf, rather than the most frequent value.

And this is wrong in this case because each three represents a different value.

So what is the mode then? Find the mode from the data in the stem and leaf diagram.

Pause the video while you write it down and press play when you're ready for an answer.

3.

6.

We can see that in the first row there are three sixes in the leaves and a key tells us to interpret that as 3.

6 kilogrammes with the context.

And here we have Lucas.

Lucas has made a mistake when finding the range from this stem and leaf diagram.

He says "The range is eight".

Explain what mistake Lucas has made.

Pause the video while you write down a sentence to explain what mistake he's made and press play when you're ready for an answer.

Well, it looks like Lucas has found the difference between the highest and the lowest leaves, but these are not the highest and lowest values of the data.

So, what is the range? Find the range for the data in this stem and leaf diagram.

Pause the video while you do that and press play when you're ready for an answer.

The answer is three, which we get from doing 6.

3, subtract 3.

3 and our units are kilogrammes in this case.

A stem and leaf diagram can also be helpful for finding the mediam.

For example, here we have a stem and leaf diagram and Alex is finding the median.

Now, there are a couple of different ways Alex could approach this.

He says "I could start at the two extreme ends of the data "and work my way in, "until I find which data point is in the middle".

So, he starts by marking the first and last data point, and then the second and second to last data point, and then the third, and third to last data point and so on, until it gets to the data point in the very middle, which in this case is this data point here.

This represents 161.

Alex says, "Another method could be to find which position the middle data point is in first.

So, you could do that by using this expression here, N plus one divided by two, where N represents the number of data points in the data.

Alex says "There are 11 data points, so N equals 11".

He substitutes 11 in for the N, so he is got 11 plus one divided by two, which gives six.

Now this doesn't mean that the median is six.

Alex says "This means I'm looking for the 6th data point".

And so he does that.

He marks the first data point, the second, and so on, until he gets to the 6th, which represents 161.

One data point has now been removed from the stem leaf diagram.

It was the number 159, which was represented by a nine in the first row.

Alex finds a median for the new data sets.

He does the same thing again, but this time substitutes the number 10 in for N.

10 plus one divided by two is 5.

5.

He says, "This time I'm looking for "the five point 5th data point, which is the midpoint "of the 5th and 6th values.

So we can see in the stem and leaf diagram, that the first five data points have been marked.

The 5th one is 161, the 6th one is 165, and Alex's median is the midpoint between those two values.

He says, "I need to find the midpoint of 161 and 165".

Now, you might just spot what the midpoint of those two numbers are, but if not, you can calculate it by adding them together and dividing by two.

And in this case it gives 163.

Now rather than find the midpoint of 161 and 165, Alex says, "Another method could have been "to find the midpoint of one and five, which is three, "and then use the key to obtain the median".

And a three in that row would mean 163.

So let's check what we've learned.

Here we have a stem and leaf diagram.

Please find a median for the data in this stem and leaf diagram.

Pause the video while you do that and press play when you are ready for an answer.

The answer is 4.

3 kilogrammes, which is this 7th data point.

One data point has been removed.

It was the 6.

3.

Find the median for the data in the stem and leaf diagram now.

Pause video while you do that and press play when you're ready for an answer.

The answer is 4.

25 kilogrammes.

It's the midpoint of the 6th and 7th data points.

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

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

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

You need to find a range, mode and median for this data.

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

And here is question two.

A group of children participate in a shot put competition.

The stem leaf diagram shows the distances that each person through the shot put rounded to one decimal place.

And once again, you need to find the range, mode and median.

Pause the video while you do that and press play when you are ready for question three.

And here is question three.

This time we have two groups of runners who complete a 400 metre race.

The stem and leaf diagram shows the race times given to the nearest second.

You can see that you've got group A on one side and group B on the other side of this stem and leaf diagram.

You need to first find the range, mode and median for each group, and then use these summary statistics to answer the questions in parts D and E.

Pause the video while you do that and press play when you are ready to go through some answers.

Okay, let's now go through some answers.

In question one the range is 34 minutes.

The mode is 27 minutes and the median is 30 minutes.

Question two, the range is 4.

9 metres.

For the mode, there are two modes, 9.

7 metres and 10.

8 metres.

And for the median it is 10.

75 metres.

And then question three, in part A, you needed to find the range for each group.

For group A, it was 28 seconds.

For group B, it was 35 seconds.

In part B, you need to find the mode for each group.

For group A, it was 59 seconds, and for group B, it was 71 seconds.

For part C, you need to find the median for each group.

For group A, it was 63 seconds, and for group B, it was 72 seconds.

You then had two more questions to answer.

In part D, which group was faster on average? And justify your answer Well, group A was faster on average, and you can justify it using either the median, or the mode.

They are both types of average.

The median for group A, which is 63, was less than the median for group B, which was 72.

Or you could say the mode for group A, which is 59, is less than the mode for group B, which was 71.

And then part E, which group had the most consistent race times? Justify your answer.

This again was group A.

The justification this time, is using the range.

The range for group A, 28, is less than for group B, which is 35.

Okay, so far, so good.

Now let's move on to the second part of today's lesson, which is finding the mean from stem and leaf diagrams. Here we have Alex, who is calculating the mean from this stem and leaf diagram.

Now, what Alex is not going to do, he's not going to add together one, five, nice, zero, and so on and divide by 11, because those values are not the values from the data.

Alex says, "I'll need to use the key "to help me interpret the actual values of each data point".

Now what he could do is he could do 151, add 155, add 159, add 160, and so on, and then divide by 11.

But Alex has a different idea.

He says, "I could do the process in small steps "by first finding the sum of each group".

In other words, find a sum of the data points in each row of the stem and leaf diagram.

So for the first row, he would do 151, plus 155, plus 159, to get 465 and write that down.

And then if you add together the values in the second row, we get 984.

And if we add together the values in the 3rd row, we get 348.

Alex then says, "I'll then find the total "for the entire data set by summing my previous answers".

So, the overall total is 1,797.

He says, "I'll then need to divide the total "by the number of values".

So, all the data in the stem leaf diagram sums to 1,797.

There are 11 data points.

So we'll divide 1,797 by 11 to get 163.

4 if we've rounded it to one decimal place, and it's always worth just double checking, that that number fits with the rest of the data.

We can see that the lowest value in this data set is 151 and the highest is 173 and 163.

4 is between these two numbers.

Also, we can see that the majority of the data points are in the 160's and so is our mean here.

So, the mean seems to make sense.

So, how helpful was the stem and leaf diagram for finding the mean? Well, Alex says, "Stem and leaf diagrams are helpful "for finding the median, mode and range, "because they present the data in order, "but they're not as helpful for calculating the mean".

There is not necessarily an advantage to putting the numbers in order when it comes to calculating the mean, because with addition, doesn't matter what order those numbers are in.

Guess the main advantage here is that it allows you to break it into small steps and add up small sets of numbers at a time.

Let's check what we've learned.

Here we have Jacob, who has made a mistake when finding the mean from this stem and leaf diagram.

Explain what mistake he has made.

Pause the video while you write down a sentence that describes his mistake and press play when you're ready for an answer.

Well, he's got those totals, 21, 17, 12 and three, but they are totals of the leaves, rather than the totals of the actual values for the data.

Here we have Aisha.

Aisha has made a mistake when finding the mean from the stem and leaf diagram.

Here's Aisha's working.

What mistake has Aisha made? Pause the video while you write something down and press play when you're ready for an answer.

Well, it looks like this time all of Aisha's totals are correct, but the mistake she's made is she's divided by four.

She's divided by the number of groups or strata, rather than the number of values.

There are more than four values in this dataset.

So, here's the same stem and leaf diagram, and we can see that the mean has been partially calculated, but what you need to do is finish the calculations to find the mean and give your answer rounded to three significant figures.

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

To get the mean, you need to divide 58.

3, which is the total by 13, which is a number of values, and you get 4.

48 kilogrammes when rounded to three significant figures.

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

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

A group of children participate in a shot put competition.

The stem leaf diagram shows the distances that each person through the shot put to one decimal place, and what you need to do, is calculate the mean and give your answer rounded to one decimal place.

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

And here is question two.

We have a stem and leaf diagram, which represents data from two groups of runners who complete a 400 metre race.

You need to first calculate the mean for each group, given your answer to one decimal place, and then use that to answer the question in part B.

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.

For question one, your mean should be 10.

5 metres.

For question two in part A, had to calculate the mean for each group in this stem and leaf diagram.

For group A, the mean is 63.

8 seconds.

For group B, it's 72.

2 seconds.

Then part B, which group was faster on average? Justify your answer.

Group A was faster on average and our justification is that the mean for group A, which is 63.

8, is less than the mean for group B, which is 72.

2.

Fantastic work today.

Now let's summarise what we've learned.

A stem and leaf diagram can be helpful for finding the median, mode and range because it presents the data in groups and in order.

The mean can also be calculated from the stem and leaf diagram, although it is a bit time consuming to calculate.

Either way, it can still be done.

Thank you very much.

Have a great day.