Lesson video

Hello, I'm Mrs. Lashley, and I'm really looking forward to working with you throughout this lesson.

So today's lesson is all about the median, thinking about what it is measuring, how it is doing it, and finding the median from data presented in a list.

Words that you've met before, but we will be using during the lesson, are on the screen now.

So just take a moment, pause the video, and re-familiarize yourself with those and their definitions.

We do have a new word today and that is median.

This lesson is all about the median and so we're gonna explore it during the lesson.

Our lesson has got two parts to it, two learning cycles.

The first part is thinking about finding the middle in small data sets, and the second part is finding the middle in large data sets.

So we're gonna make start on that first learning cycle, which is about finding the middle, and the middle will become apparent in a moment, from small data sets.

So on the screen there are three relatively low-valued numbers, and which number is the middle value? Three.

Three is the middle value.

Now we've got three more, again, which number is the middle value? 30.

Which number is the middle value here? 45.

Which number is the middle in this set? 63.

So on this one they were not ordered, so you might argue that the middle would be 92.

It is in the middle of, but here we're thinking about middle from a mathematical point of view, that the middle number, the number that's in the middle, would be 63 if you think about them on a number line.

18 is the least, 92 is the largest, so 63 is the middle value.

If you think about yourself, perhaps you're not an only child and there's more than just you, then you can be the middle child.

You're the middle child if you were the one that was born second, if there were three of you.

Here, you've now got more numbers.

Which one of those is the middle? Well, it's actually 58, but now it's 28.

So as the numbers are growing, as they're sort of getting larger and they're more spread and they're not ordered, it becomes more difficult to sort of gauge where the middle is.

And this is the median, and the point of the median, so the median is a type of average.

And so because of this, it's a measure of central tendency.

So, the measure of central tendency is trying to find a single value that attempts to describe the set, the data set, by finding central value.

The median as an average is the central, the middle, piece of data when the data are in numerical order.

So, it is not about it being exactly middle, central, midpoint.

It's like that analogy of an order of siblings, the middle child, if there were three children, the middle child is the second child.

Regardless if there is a big age gap between the first and the second or the second and the third, that second child is the middle child.

Here are seven Oak Academy pupils, and Sophia is in the middle.

Does she have the median height? So no, the median needs to be representing the sort of typical height of the group.

And that's why the order is really important.

So Sam is now the middle, and their height would be the median height for this group, it's more typical for the dataset.

And you can see that Sophia is actually the shortest in the group.

So if Sophia was meant to be the middle or the typical height, that isn't representing the data at all.

So, what has Sam done wrong when they have tried to get the median for this list of data? Pause the video whilst you think about that.

And then when you're ready to come and check what you think the answer is, press Play.

So, Sam has not put the data in order, and their value of the median, 182, is the the largest value in the data set.

So once again, it's not gonna be a measure of central tendency if it's the largest dataset.

We're trying to find a single value that describes the data, the central part of the data.

So here are some Oak Academy pupils and their favourite colours.

And underneath we've got their favourite colours.

So, what is the median favourite colour? Lucas is in the middle and his favourite colour is blue.

However, the pupils could have lined up in any order, and that would mean that the median would change depending on who ends up being in the middle.

So because colours do not have a defined order, there is not a median.

So the median is the middle, the sort of central, piece of data when ordered.

If you cannot order the data, then you cannot have a median.

So, check on that.

For which of the following data sets can you find the median? Pause the video whilst you consider your answers and then when you're ready to check it, just press Play.

So, B and C, you could get a median value, because you can put them in order.

The order could be ascending, it could go from smallest to biggest, or it could be descending and go from biggest to smallest.

There would still be an order and you'd still get to that middle piece.

A is a set of colours, and yes, they're in the order of the rainbow, but that's not a defined order that it has to be, and so we can't claim that to be our median.

Here we've got some data collected about total sunshine hours in a year for Dunstaffnage.

So what is the median number of sunshine hours in a year? So first of all, we need to put it in order so that we can locate that middle piece.

So here we've got it in order running from the least amount of sunshine up to the greatest amount of sunshine.

And then Jun suggests that if we work in from either end and crossing off the data, we're gonna locate that middle piece.

So if we get rid of our smallest, get rid of our largest, get rid of our next smallest and our largest, then we locate the central piece of data, and that is 1,256 hours of sunshine a year.

So, we firstly ordered them.

Needs to be in order first, and then we've located our middle by crossing off and moving in towards the centre.

Another data set is shown on this dot plot.

So, dot plots, each point, each dot, represents a a data value.

And if there is repeated data values, then they sort of stack up, as you can see on the 37s If we use Jun's method of sort of working from the outside in to locate the median, crossing off the dots as we go, so we'd get rid of the 26 'cause that's our least, then now 40 'cause that's the biggest, we end up crossing them all off.

There is not a piece of data in the central position.

However, the two either side of them are both 37, and so the median will be 37.

So, if you end up crossing all of the data off, if you use that method of moving from the outside in, and you end up crossing them all off and you haven't left a single value, if the two that you crossed off either side, the last two that you crossed off had the same value, then the median is going to be that value.

We've got another data set on a dot plot and there are 12 pieces of data, which means once again, if we cross off, we're gonna not end up at a single value.

And so we end up, I haven't crossed them all off here, I've left the two that will be next to go.

So, you end up with 29 and 31.

So it feels like there's three options for the median.

It could be 29, it could be 30, or it could be 31.

And so which is most average, 'cause remember that the median is a type of average.

So, 30 is the most average.

It's the mean of the two central data points.

So if you do 29, add 31, and divide by two to find the mean, you get 30.

It is the midpoint of the two central data points.

It is exactly halfway between those remaining two.

Here we've got a dot plot again, and we're gonna try and calculate, try and locate the median for these data.

And by crossing off, moving from the sort of largest and the smallest, and moving in towards the centre.

So, we are left with 42 and 45.

The mean of those two middle points is 43.

5, and that means that the median is 43.

5.

We've got another set of data here.

Remember, we're working with small data sets in this part of the lesson.

So, this is monthly mean daily maximum temperatures.

So there's only 12 data points, and what is the median temperature? So we need to order it at first, we need to get that into order.

We can cross off from end to end and because of the 12, we are gonna end up having two points that are at either side of our median value.

So to calculate where that median value is, we need to get the mean of the two central points, and in this case it would be 16.

15 degrees Celsius.

So, we've worked in from the outside, like Jun suggested, crossing them off, but because we would end up crossing all of them out, our median is not a a piece of data in the dataset, but instead is the mean of the two central data points.

So, a check, the median will always be a value in the dataset, and justify your answer.

Pause the video whilst you are considering your answer to that, and then when you're ready to check it, press Play.

So, that's false, and that's because when there are even number of data points like we just saw with the 12 months of the year, the mean of the two central pieces is calculated.

So you're now gonna do some practise on finding the middle in smaller data sets.

Question 1 and 2 are on the screen now.

So have a go at those, press pause, and then when you are ready to have a go at the next question, come back and press Play.

So this question, Question 3, "If possible, find the median for these data sets." Pause the video whilst you're doing that, and then when you come back we'll go through the answers to all three questions.

Question 1, had some cupcakes on a tray, what was the median amount of cupcakes? Four cupcakes.

So, if you reordered them from the least amount of cupcakes on a tray to the most amount of cupcakes on a tray, there were four in the middle.

Question 2 was a dot plot and you needed to find the median value, and it was six.

So you could do that by crossing off from the smallest values going into the central, and six would've been the median value.

Question 3, "If possible, find the median for these data sets." So, Part A, you needed to order them, and then crossing off, go moving towards the centre.

There were two central pieces of data, or the most central, but they were actually the same value.

So the median is 29.

If you find the mean of 29 and 29, it comes back as 29.

So, when those two most central pieces are the same, that will be the median value.

Part B, you couldn't find the median.

There is no way of ordering those different fruits and vegetables.

And the last one, negative data can also be ordered.

So we've got -10, that's our smallest, up to -1.

And then working from the outsides in we get to the median value of -3.

The second part of the lesson continues with the finding the middle, but this time we're finding the middle for larger data sets.

So up till this point, our data sets have not included many pieces of data, but realistically data is often tens and thousands of pieces of data, large data sets.

So how do we find the middle for large data sets? So, some data regarding number of attendees at a team practise.

So, there was 15 on one day, and 13 on another, et cetera.

To find the median number of attendees, first of all, we need to order the data first.

We saw that in the first learning cycle.

So here it is reordered from smallest to biggest, and then we can work from the ends in towards the middle.

This method becomes more difficult when data is spread across multiple lines.

It's doable, the median is 15, but as the as the data becomes your hands, if you were doing it with your fingers, rather than crossing 'em out, they're going in opposite directions.

That can cause a little bit of confusion.

So the median is 15 for the number of attendees at team practise.

Here is data regarding the mean maximum daily temperature in Cardiff over a period of time To find the median, by working towards the middle, it starts to become impractical as our dataset increases.

So firstly, we've got it on multiple lines, and it needs to be on multiple lines because of how much data there is.

And if we were then going to work inwards from our outside, it becomes difficult.

So, is there another way to do this? As data sets increase, become larger, and it becomes impractical to sort of move in towards the middle by crossing off either end, is there another way to do this? So let's simplify and go back to small data points.

If there are three and they are in order, which position is the median? Well, the second.

If there were five, which position would the median be? The third.

If there were nine data points, in which position would the median be? The fifth.

So, is there a connection between the number of data points and the position of the median? So, just got it in tabular form now, what we've just seen, so when there were three data points, the position of the median was two.

When there was five, it was the third position.

When there was nine, it was the fifth position.

So, Jun has noticed that if you half the number of data points and then add 0.

5, or a half, you get the position.

So let's see if that works.

So, if we half three, you get 1.

5.

If you add 0.

5, you get two.

And that is the position.

If we try that with the nine, if you half nine you get 4.

5.

If you add 0.

5, you get five.

So, it has worked for two of the three cases.

Laura has seen that if you add one to the number of data points, and then half, you get the position.

So, three add one is four, and half of four is two.

Nine add one is 10, and half of 10 is five.

So, both of their methods seem to be working.

And that's because they're both correct.

So, both of their methods are valid.

So let's increase the number of data points.

We've got 19 data points now.

Which position is the median in? Well, using Jun's method, we're gonna half it.

We get 9.

5, we add 0.

5, which gives us 10.

So, Jun has worked out, has calculated that the position of the median, so not the median value, but the position of the median, is the 10th position.

Laura, her method was to add one and half.

So, add one gives you 20, half it gives you 10.

The median is in the 10th position.

So both of their methods have worked for 19 data points as well.

So what if there were n data points? So, n could be any number, any integer number.

Which position would the median be in? So Jun's method where he divided by two, halved it, and then added a half, would be in the n over two plus half position.

Laura added one first, and then halved that total.

So, hers would be n plus one over two.

So Jun has fractions with a common denominator of two.

So, as a single fraction, so working without knowledge of fractions, when you've got a common denominator, you can add the numerators, you can combine them to be a single fraction.

It's actually the same as Laura's.

So these, it's an identity, n over two plus a half is equivalent to, is identical to n plus one over two.

So, they're two methods.

Both work because ultimately they're the same thing.

So, a quick check.

Using Jun or using Laura's method, a dataset has 431 pieces of data.

So, a large dataset.

What position would the median be? So pause the video whilst you are working that out.

And then when you're ready to check your answer and to move on with the lesson, press Play.

So the method that we tend to use is the one that Laura suggested, and they were both equivalent, if you remember.

So, add one, divide by two.

So, adding one gets you 432, and then if you half it you get 216.

So, the median would be in the 216th position.

So, a dot plot back representing data.

And there are 11 data points.

So the median would be in the sixth position.

So, this isn't a large dataset, not yet.

But to use this method that we've just found that, we can locate the median, we can locate the position of the median by n plus one, when n is the number of points, and then divide it by two.

So, we know that the median is in the sixth position.

So if we now just count, we're not crossing off, we're just counting.

So that's the first piece of data.

That's the first position.

Second, third, fourth, fifth, sixth.

So, we know that the median is 34.

The sixth piece of data is on that line of 34.

So now we've got a slightly larger data set.

We've got 20 on the dot plot.

So the median is gonna be in the 10 and a half position, or the 10.

5 position.

That's because we've got an even amount.

So, going back to the first learning cycle, if we were to work there from the outside in, we would end up crossing everything off.

The two most central pieces of data is what we're gonna use to find the median.

So, if it's the 10 and a half position, we actually are looking for the 10th and the 11th position, because the median is the midpoint of those two.

So again, we can just count through our points.

And so 74 was our 10th position, and 75 was the 11th position.

And that means that we want to find the mean of those two, and the median would be 74.

5.

So if when you add one and you divide by two, you end up with this decimal, you end up with the half, that's because you've got an even set of data.

And so you need to locate the two either side of it.

So, a dataset has 5,320 pieces of data.

So now we really into large datasets.

Between which two pieces of data does the median value lie? Pause the video whilst you're working that one out, and then press Play to check your answer.

So when you add one and divide by two, you get 2660.

5.

So you're looking for the 2660th piece of data and the 2661st piece of data.

Those two pieces of data is what you're gonna use to calculate the median.

So, you are gonna get through some practise on the large data sets.

The first question is to work out the position of the median for the five cases.

Pause the video whilst you're doing that, and then when you're ready for the next question, press Play.

Question 2, there are two dot plots and you need to work out the median for each dot plot.

Pause the video whilst you calculate the median for each dot plot, and then when you're ready to move on, press Play.

So Question 3 and Question 4 are both on this screen.

So question 3, there are 97 pieces of data, what is the median value? And Question 4, there are 2000 items in the dataset in total.

This part of the list shows you that starts at the 998th entry.

What is the median for this dataset? Pause the video whilst you're answering those two questions, and then when you come back we will go through the answers.

So, Question 1 was about using that formula of adding one and dividing by two to to locate the position.

So it's not the median, you're not working out the median, you're working out the position of the median.

So you've got the answers there on the screen.

d and e were even sets.

And so they came out as a decimal part, and so, to actually calculate the median, you would be finding the mean of the values either side.

Question 2, the median from each dot plot, so 77 was for a, and 83.

5 was for b.

Question 3 and 4, so, these were large data sets, 97.

I'm hoping that you didn't do it by crossing off from either end.

It would work, but it becomes impractical.

The larger the list, it becomes impractical.

So instead you knew there were 97 pieces.

So if you add one and divide by two, you know that the position of the median was 49, and then you could just count 49 pieces.

And that locates 13.

9.

Question 4, again, 2000 items, hard to show that on the screen, so the list started on the 998th.

If you added one to 2000, you get 2001.

If you half it, you get 1000.

5.

So you were looking for the thousandth piece of data, and the thousandth and one, which is the ones in the box, -4.

2 and -3.

9.

And then doing the mean of those two, because the median would be central, would be the midpoint of those two.

And that is -4.

05.

So, in summary about this lesson, understanding the median, the median is a measure of central tendency.

It is the middle, or the average middle, depending whether it's an odd amount or an even amount, in an ordered data set.

The position of the median can be found using n plus one over two, where n is the number of data values.

Well done today.

I hope you've enjoyed, and I look forward to working with you again in the future.