Lesson video

Hi there.

My name's 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 "Constructing Box Plots", and that's within the unit Graphical Representation of Data, focusing on cumulative frequency and histograms. By the end of this lesson, you will be able to construct a box plot.

Quick recap are some keywords that we are going to be using in today's lesson, and those words are median, lower quartile, and upper quartile.

The median is the central, middle piece of data when the data are in numerical order, it is a measure of central tendency and it represents the average of the values.

The lower quartile is the value under which 25% of data points are found when they are arranged in increasing order, also known as the first quartile and referred to sometimes as Q1.

The upper quartile is the value under which 75% of data points are found when they are arranged in increasing order, also known as the third quartile, or Q3.

An unfamiliar word, or possibly unfamiliar words to you today are this term box plot.

Now this is the main focus of today's lesson, so if you've not heard of it before, don't worry.

Here's just a quick overview of what it is, but we are going to be covering all of this in the lesson, so please don't worry.

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

For today's lesson, I've split it into two separate learning cycles.

In the first one, it will draw box plots from a cumulative frequency graph, and in the second one we will draw box plots from a list of data.

So let's get going with that first one, concentrating on box plots from cumulative frequency curves.

This is a cumulative frequency graph representing test scores of a class.

We are going to draw a box plot.

We need to know these five pieces of information.

We need to know the lowest score, the lower quartile, the median, the upper quartile, and the highest score.

Now, here we don't actually know the lowest and the highest because it is grouped data, but we assume it is the beginning and end points.

So the lowest score is going to be zero, and the highest score is going to be 25.

Remember, we're talking about the scores here, we're talking about the data, and the data was test scores, so it was 25.

The highest test score anyone could get was 25.

Now, you know how to find the median and quartiles from a cumulative frequency graph.

We find half of the data, which is half of 20, which is 10, and we read off that test score.

That gives us a test score of 13.

The median is 13, the lower quartile is a quarter of the way through the data.

There are 20 pieces of data in total, and so therefore a quarter of this is five.

So we read off.

This gives us our lower quarter and we can see that the lower quartile is 10, and now we calculate the upper quartile from the graph.

Three quarters of 20 is 15.

I'm gonna read off my value, my test score, I should say, and the cumulative frequency is 15 and that gives me a test score of 17.

Now we've got this information, we can put this into a box plot, and this is what a box plot looks like.

We start off with the lowest value and we mark this on our box plot.

So you can see here I've got my scale and I've got my test scores on the bottom.

Right, I'm then going to move on to drawing my line for my lower quartile, which is a 10.

Now, notice here that this line is a bit taller than the previous line.

It doesn't matter how tall you draw your lines, but your lowest value should be shorter than your lower quartile.

Now we're going to draw the line in for the median, and that's at 13.

Notice my line is lined up and is the same height as my lower quartile.

Now we move on to the upper quartile, which is 17.

So the same length line, but at 17 and then the highest value is 25.

Highest value is 25.

Notice the lowest and the highest are at the same height and the same position.

We then complete the box plot by drawing a box between the lower and the upper quarter with the median within there.

And then we draw the lines.

Sometimes people refer to this as a box and whisker plot, with the box obviously being the box in the middle and the whiskers being those two bits that stick out either side.

A box plot is used to represent these five statistics about a set of data.

So we had the lowest value, the lower quartile, the median, the upper quartile, and the highest value.

Those are the five different points that are represented on a box plot.

Here we've got a cumulative frequency graph representing test scores of a class.

We're gonna draw a box plot.

The lowest score, zero, and the highest score was 30.

Now, you may choose to move on and find the lower quartile.

Personally, I always like to find the median first.

Total number of pieces of data, it's 40.

Half of 40 is 20.

So I draw my line in making sure, remember it's parallel to the test score and I read my value off, and I can see that the median is 18.

Now I need to find a quarter, a quarter of 40 is 10, or I could find half of 20.

That's the same thing.

So this time my lower quarter is 14, and then my upper quarter is three quarters of the way through the data.

So that is at 30.

And if I read off my upper quartile, I can see that that gives me a test score of 22.

Again, we can put this into our box plot.

So we start with our lowest score of zero, lower quartile is at 14, the median is at 18, the upper quartile at 22, and the highest score at 30.

And then we need to complete our box.

And then our lines joining our box to our highest and lowest values.

I think we're ready to have a go at a check for understanding now.

I'd like you please to label each of the parts of the box plot.

So a, b, c, d, and e, what do they each represent? Well done.

A is the lowest value, b, the lower quartile, c, the median, d, the upper quartile, and e, the highest value.

Did you get them all right? Well done.

Now I'd like you please to decide which part of the box plot is incorrect.

And I'd also like you to fill in the missing values in the table.

I'd like you to pause the video and then come back when you are done.

Great work.

A was correct.

If we look at the table, the smallest value or the smallest height was 110 centimetres.

B was incorrect.

The lower quartile looking at the table was 132, and we can see here that I've plotted it at 133.

So that was my error.

It should be at 132, not 133.

C, while we were missing this, this line in the middle of the box, not necessarily central to the box, but in between the upper and the lower quartile is the median and we can read that value off as 148.

The upper quartile was 162, which was correct.

And then the greatest height we can see from the box plot was 190.

So the error was plotting my lower quartile incorrectly, and then we could find the other missing values.

Well done.

Ready now then for Task A.

You're going to draw a box plot for the information.

Pause the video, make sure you draw all your lines ruled with pencil and a ruler.

And when you're done, you can come back.

Well done.

Question number two, I'd like you to draw a box plot of the data represented.

This time I've not told you what the values are, you need to use the cumulative frequency graph to work out each of the five values represented in a box plot.

Well done.

Let's check those answers.

So box plot with the information, you should have your lower point or your lower mass, lowest mass at a hundred, then your lower quartile at 114.

The median is at 122, the upper quartile at 135, and the highest at 160.

And making sure you've joined your box together and then drawn the lines between the box and the highest and lowest values.

And question two, we should have read off a value of 7.

5 for the lower quartile, which gave us 1.

5.

For the median, we should have read off a value of 15, which gave us a value of 2.

2.

And for the upper quarter, we should have read off a value of 22.

5, which gave us a value of 2.

95.

And then I transferred all of that information onto my box plot.

The lowest value is zero and the highest value was four.

Let's now then move on to the second learning cycle for today's lesson.

And we're going to be looking at a box plot from a list of data.

Sometimes we have the raw data rather than in a frequency table or represented on a cumulative frequency graph.

Here are the test scores for a group of 15 pupils.

Here are my test scores.

To draw a box plot, we need to identify the following.

What do we need to identify? Yeah, we need the lowest value, the lower quartile, the median, the upper quartile, and the highest value.

Let's take a look at how we go about doing that.

Well, the lowest score, that's easy, isn't it? It's the first number in the list, or the lowest number in the list, which is two.

And the highest score also nice and easy.

That's the highest score that anybody got, which was 25.

I'm now gonna find the median.

So I'm crossing off one from each end until I end up with one value in the middle.

Remember to cross off an even number from the left and the right, and I always like to double check.

I've crossed off seven to the left of 17 and seven to the right of 17.

So I know I've not made an error.

This has given me the median.

The median is 17.

Now I need to find the lower quartile.

I'm looking at the data to the left of the median and effectively finding the median of that.

So again, we're gonna cross off until I end up with one and sometimes two in the middle.

We've got here a value of 10.

That's our lower quartile.

Our lower quartile is 10.

We're gonna repeat that now for the second half of the data, the data to the right of the median, again, crossing off one number from each side until we're left with one in the middle.

And that is 24.

The upper quartile for this data is 24.

We can then put that information onto our box plot.

And I've drawn the box plot there.

The lowest value was two, the lower quartile was 10, median was 17, the upper quartile 24, and the highest value was 25.

Here is a list of the number of times a group of 10 pupils were able to banks a basketball in 10 seconds.

Draw a box plot to represent this data.

Izzy says, "I know how to do this.

First find the median.

That's halfway between 12 and 14 as they are in the middle of the list." Do you agree with Izzy? Izzy is partially right by saying the median is halfway between the two pieces of data in the middle of the list, but what crucial step has Izzy not remembered? She's forgotten that the list needs to be in order before finding the median.

Remember, it's the middle of an ordered list.

Here I have now ordered the times that the people could bounce the basketball for.

Now we can find the middle, and like Izzy said, we had two numbers left in the middle this time.

Let's just check.

I've crossed off four on the left and four on the right.

The median lies halfway between 10 and 12, which we know is 11.

But remember, if it's not obvious, we can find the mean of those two values.

So 10 add 12 divided by two is 11.

The median is 11.

I'm gonna draw a line in to show where my median is, 'cause this is gonna help me 'cause I now need to find my lower quartile.

My lower quartile means finding the median of the values to the left of the median.

So we're crossing one off of each side until we end it with one or two in the middle.

We can see that the lower quartile is seven.

Repeat this for the second half of the data.

So I'm crossing off 12 and 23, 12 and 21, and I'm left with 14.

And that is my upper quartile.

What was the lowest number of bounces? Two, and the highest number of bounces, 23.

It doesn't matter which order you decide to fill the table in.

We can then plot this onto a box plot.

Your turn now.

I'd like you please to fill in the table of values using the data here, and this data is the number of votes 10 people receive to become a year group representative at the Oak Academy.

So pause the video and then when you're done, you can get back to me.

Please tell me you did remember to put them in order first.

Of course, you did.

Thank goodness for that.

Let's have a look then.

So we wrote them in order first.

The lowest number of votes was eight and the highest number of votes was 49.

The median was halfway between 23 and 32.

Now, that's not immediately obvious to me.

So I'm gonna find the mean of those numbers, which is 27.

5, 23 add 32 divided by two.

And then I'm going to find the lower quartile.

So halfway between the lower half of data, which is 16.

And then repeat that for the second half of data, giving me 44.

Right, we're ready.

We are ready for Task B.

So this is our final task for today's lesson.

You are going to draw a box plot of the test scores below.

You're gonna draw a box plot.

So remember you need to find those five values, find them really, really carefully.

Make sure that you are checking that you've crossed off the same number from each side.

And then when you've found those five values, draw me a nice neat box block with your pencil and ruler, please.

Pause the video and then come back when you're done.

And question two, same thing again.

I'd like you please to find the five different values and then draw a box plot.

Pause a video and I will be here waiting when you get back.

Well done.

Question number three, again, find the values and draw me the box plot, please.

Super.

Here are our answers then.

So first thing we wrote the night in order, the lowest value we could then see was 12 and the highest was 38.

The median was 24, 'cause you're really useful to find the median before finding your quartiles.

Then the middle of the lower half of data was 18, and then the upper quartile was 34.

And then we put that onto a box plot.

Remember, it doesn't matter how tall your whiskers are, if you're gonna call 'em box and whisker plots, it doesn't matter how tall your bar is.

What's important is where each of those vertical lines are made.

Question two, the lowest temperature is 10, lower quartile 12, median 18, upper quartile 26, and highest temperature was 33.

And that's what it looks like on a box plot.

And question number three, the lowest number of bounces was one.

The lower quartile was at seven, the median was at 17.

5, upper quartile at 21, and the highest number of bounces was 24.

And there is your box plot.

Now, obviously, if you need to, you can pause the video and go back and look carefully at where each of the median, lower, and upper quartiles were.

If you've made any errors, I suggest you go back and watch the example that was similar to the one that you got wrong.

We went through each of these different types in our lesson.

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

A box plot is used to represent these five statistics about a set of data.

So we have a scale across the bottom, remembering the height of the lowest value and highest value lines and the height of the box.

It doesn't matter, you can choose whatever you want, but our highest and lowest values should be shorter than the height of the box.

If the lowest value is the first value, then the lower quartile is the left hand side of the box.

The median is somewhere within the box, not necessarily halfway.

And we can see that here.

The upper quartile is the right hand side of the box.

And then the highest value is on the very right hand side.

And we join the box to those two extreme points, like I said, sometimes known as whiskers are the lines that we use to draw them.

Box plots can be drawn from cumulative frequency graphs or from a list of data.

Remembering if you're doing it from a list of data to make sure that your data is in order first, you firstly identify the median and then you identify the median of the data to the left of the median and then the median of the data to the right of the median.

And this will give you your other three values to put on your box plot.

And that's your lower quartile, median, and upper quartile.

Superb work today.

Well done.

Now you know how to draw a box plot and you'll be able to use this in your future learning.

Like I said, thank you so much for joining me.

What a fantastic choice you made and hopefully I'll see you again really soon.

Take care of yourself.

Goodbye.