Lesson video

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

Our lesson today, we're gonna be working with the median, and we're gonna be calculating the median from frequency tables and different graphs.

Not just from a normal list of data that you may have looked at before, but instead looking at when data is being represented, how we can still calculate the median.

On the screen there's some words that you should already be familiar with.

I would suggest you pause the video now just to re-familiarize yourself, as we're gonna make use of those during the lesson.

Our lesson today has got three learning cycles, the first one is finding the median from a line graph.

The second part is gonna be finding a median from an ungrouped to frequency table, and the last part is to find the median from a bar chart.

So we're gonna make a go at the first part, which is about line graphs.

Data has been collected about the number of days of frost in February over nine different years.

It runs from 1941 to 2021 and it's about the number of days of frost.

The same data can be shown on this line graph, or a time series graph is another name that you will come across for it.

Where is the median? Is the median the middle of the vertical data? So our data runs from zero to 16, the middle is therefore eight.

Is that the median? Is it the middle of the horizontal data? So our years went from 1941 to 2021, which puts 1981 in the middle.

Or, is it the middle of the ordered raw data? Well, it's actually C, it's the median is the middle of the ordered raw data.

If you've got a line graph, the median is not easily seen through the line graph.

The median comes from the original ordered data.

So if we've got that data on the line graph and we want the median, we need to take the days of frost off, the number of days of frost, off of the line graph in an order.

We're gonna go from the least to the most.

Zero days of frost in 1961.

And 16 in 1991.

Now we've got our nine pieces of data, that was nine years that it was recorded in order from the least amount of days of frost up to the most amount of days of frost.

And the median number of days of frost is nine, and that's because it's in the fifth position.

And we can work out the position by doing nine pieces of data.

Add one, gives us 10, and dividing it by two tells us the median would be the fifth data point.

And in this case that is nine, so nine days of frost is our median.

Here's a check, where is the median on a line graph? Is it the middle of the horizontal axis, the middle of the vertical axis, or the middle of the ordered raw data? Pause the video whilst you make your decision, and then when you're ready to move on come back and press play.

So it's C, it's the middle of the ordered raw data.

Here is a line graph that shows the time taken for someone to travel to work over the course of one week.

There are five pieces of data, so we know that the median value is in the third position, it's the third piece of data when ordered.

So it's not Wednesday necessarily because Wednesday is the middle of the week, It is the third piece of data in order.

Friday had the least amount of travel time, so that would be our first piece of data.

Monday was the next lowest amount of time taken and our third was on Thursday.

The median time taken to travel to work would be 23 minutes.

A check for you.

This is Camborne's maximum temperature rounded to the nearest degree Celsius on the first of every month in 2022.

What position is the median? Pause the video whilst you come up with your answer and then press play to check it and ready for us to move on.

There are 12 data points because it was once a month, so January to December is 12.

The position therefore would be the 6.

5 position because if you add one and half it, you get 6.

5.

To calculate the median, we would actually need to find the sixth piece and the seventh piece of data, and then do the mean of those two.

You're gonna do a bit of practise working with line graphs and calculating the median.

For question one, you need to work out the median minimum temperature for both Aberporth and Lowestoft.

Pause the video whilst you do that and then when you are ready to check your answers, just press play.

Aberporth had a median temperature of nine degrees, and Lowestoft had a median temperature of eight degrees.

We're now onto the second part of the lesson and the second part of the lesson is finding the median from an ungrouped frequency table.

We've got some data from a ticket booking company, and they collect data about how many tickets are in each order for a new theatre production.

And so you can see the data there.

We could take that data and represent it as a dot plot.

We can now see how many of each group size has been ordered for this theatre production.

A different way to organise the data is a frequency table, and often data in a data cycle would've been collected and then represented as a frequency table before any other representations are made.

But this dot plot will allow us to see, it's what a frequency table actually means.

There were six dots above the two, which means that there were six orders of two tickets.

The frequency is six.

Likewise, there were five different orders where people had bought three tickets.

Because there were five different orders the frequency is five, it's the amount or the number of.

And then how many orders had four tickets in them? There were seven of those.

And we can do the same for every single group size.

Nobody brought tickets as a group of nine, but we still need a zero in that position.

And then there were two groups of 10, one group of 11 and one group of 12.

So the frequency is the amount of or the number of, you're doing the same thing.

As a check for me I want you to write the frequency for the data shown in this dot plot.

Pause the video whilst you are adding in the numbers to that frequency table, and then when you're ready to check them press play, and then we'll continue with the lesson.

14, there were four dots, which means there was four of them.

The frequency is four, then eight, two, four, two.

Here we've got that same dot plot about the tickets for the theatre production, and our frequency table that we previously created using that data.

We've got an extra row which is last position, and we're gonna go through what that means in a moment.

There were 37 orders for this theatre production, and we can come to that value by adding up the frequency row, that would add up to 37.

With the dot plot we can just count how many dots there are and you'll get 37.

The median is the 19th position, we know that by adding one and dividing by two.

37 add one is 38, and half of 38 is 19.

The median number of tickets purchased in an order would be in the 19th position.

Now we're gonna try and think about where is the 19th position.

Group sizes of two, people that just bought two tickets, has the first piece of data up to the sixth piece of data.

If you imagine this as the list again, in an ordered list there would be six twos next to each other, two, two, two, two, two, two, because there was six groups of two.

And so that means that the first position is a two, the second position is a two, the third position is a two, the fourth position is a two, the fifth position is a two.

And lastly, the sixth position is a two.

That last position of a number two is the sixth position.

Group size three would have the seventh piece of data up to the 11th piece of data, and that's because group size two finished at number six.

The sixth position in our list of 37 was a two, the seventh position is number three, and so the last position there is 11.

We've got six pieces of data that are number two, and we've got five pieces of data that's number three.

And so we know it goes up to the 11th position because six and five makes the 11.

Now if we include that group size as a four, 'cause remember we're trying to find the 19th position, we're trying to locate where the 19th position would be.

Well, the group size four starts at 12, because group size three finished at 11.

We've started at 12 and we're gonna go up to 18, because there are seven pieces of data, a group size of four.

If we're starting on 12, that would go up to the 18th position.

So now we know where the 18th position is, and that's in the fours, that means that the 19th position of data is a five.

There are 18 pieces of data up to the four, and so the five is the start where it starts on 19.

The median group size of ticket orders is five.

Here's a check.

The frequency table shows data about how many houses there are in a number of streets.

How many streets have 14 or 15 houses on them? Pause the video whilst you think about that, and then when you're ready to check your answer just press play.

There are 12, four of the streets had 14 houses, and eight streets had 15 houses, which means that there were 12 streets that had 14 or 15 houses on them.

Another check, same frequency table.

How many streets have at least 17 houses? Pause the video whilst you think about that, be careful with the word at least, or the phrase at least.

When you're ready to check just press play.

There are six streets that have at least 17, there are four streets that had 17, and then there were two streets that had 18 houses on them.

18 is more than 17, so it's got at least 17.

The data here is about group sizes at tables in a restaurant, and that's what we can see in this frequency table.

How many tables were empty? Well, that means that the group size would be zero, 'cause if it's empty there's nobody sat at it.

And so we can see from the frequency that that is one table.

How many tables had a group size of one or less? Well, that would be two.

There was one table of one, and there was the empty table as well.

So together, if we collect them together, that makes two.

How many tables had a group size of two or less? Four tables, because we've got two tables of two, then we've got one table of one, and one table of zero, together collectively that is four tables.

How many tables had a group size of three or less? Six tables.

It's two or less was four, and then we've got two extra tables that had three, which makes us six.

How many tables had a group size of four or less? This makes nine tables, because there were six tables that had three or less and now we want four or less.

All we've gotta do is add on those tables of four, and there were three tables of four, so six add three gives us the nine.

How many tables had a group size of five or less? 11 because we're gonna add on the extra two tables.

And so here we've got the frequency table that we started with, and our answers to all of those questions, we can redraw our frequency table.

And this size, our group size, is collecting the ones below it as well.

We've got an empty table, zero, then we've got one or less, two or less, three or less, four or less, five or less.

And they include the data from the rows before, because it's or less.

So using the original frequency table, I could write it out as a list of data.

It would be naught, one, two, two, three, three, four, four, four, five, five.

The naught comes from there.

And then if we are asking for two or less, then it's zero, one and two, and there are four of those.

And that's what that frequency means in that column.

If we want four or less, then it's the nine, it's the first nine pieces of data because they go up to the four.

Just a quick check with that, there's a frequency table on the left and then there's a gathering frequency on the right.

And so what is missing in the frequency on the right? Pause the video whilst you figure that one out and then when you're ready to check it, press play, and we'll go through the answer.

So hopefully you went for eight, and you can calculate it to be eight in two different ways really, but it was two or less.

So if we go to the the frequency table on the left, that means it would be the five for the two, the two for the one, and the one for the zero.

And, five add two, add one gives you the eight.

The other way that you can do this is work backwards, that three or less was 14, and there were six tables of three.

You could reduce the 14 by six and you'll get back to the eight.

That frequency table on the right is a running total as the group sizes increase, but you are also allowed to have the ones below as well.

Another check, and you're gonna go the other way this way time.

So you've got the full frequency table on the right, where we've collected the ones below, and need to figure out what the missing frequency on the left is.

Pause the video whilst you're doing that, and then when you wanna check press play.

I'm hoping you got six, and you can see that by it went from two or less being ten, three or less was 16.

So the gap of six was because of those extra group sizes of three.

Now we've got both of these frequency tables complete, what is the position for the median for these data? Well, there's 25 pieces of data in total, and we can see that from the bottom of the right hand frequency table when it had all the data involved.

Five or less meant it could be four, it could be three, it could be two, it could be one, it could be zero, it could be five.

That was everything, so 25 tells us the total.

The other way we could have found the total is by adding the first frequency table, and the first frequencies and you'd get 25.

So the position of the median, we add one and we divide by two is the 13th position.

So now we want to find where is the 13th piece of data, what is the median group size? So the 13th piece of data, and this is where the second frequency table comes into its own.

We know that there are 10 pieces of data that are two or less.

Well, we are looking for the 13th we know it's above the two.

The median is above the two, because the two or less goes up to the 10th piece of data, because its frequency is 10.

The 13th piece of data is a three, the median group size is three.

And so we know that because there were only 10 up to and including two, and then there were 16 pieces of data up to and including three.

And therefore the 13 is within that extra six pieces of data.

So we know that the medium would be a three.

Here are some data that we've seen previously still about group sizes of tables in a restaurant.

We've got it written as a list because there's not that many pieces of data.

We've got the frequency table on the left where we have collected as we went, the median group size, there are 11 pieces of data.

So the median would be in the sixth position 'cause 11, add one is 12 divided by two is six.

And we can see that there are six pieces of data that are three or less.

There were four pieces of data that were two or less, which means that there is, if there's only four pieces of data and you can see that from the list more clearly.

That if there are four pieces of data which are two or less, when we are looking for the sixth piece of data, then we need to go above, and there are six pieces of data that are three or less.

The median is three.

This running total frequency allows us to see where the locations of the data are, and then we can use the group size to work out our median value.

The frequency table here shows about goals scored each match in a football season by all the teams in one club.

And so we've gone from the frequency table on the left where we've got them as just groups.

There were 19 games where no goals were scored, there were 15 games where three goals were scored.

And the the right hand frequency table is that running total frequency, where we can now see that there was 53 matches where there was one or less goals scored.

That would mean there's either been one goal scored or there's been no goal scored.

And that 53 has come by doing the 19 of zero, and then adding on the additional 34 matches where one goal was scored.

We know there's 100 matches in total, we know that by adding the frequency on the left, or looking at that bottom right cell on the right.

And so the position of the median would be the 50.

5 piece of data.

And that will be in this group here, the 53, if this was written as a list there would be 19 zeros, and then we would have 34 ones.

Which means that between zero and then up to one there are 53 pieces of data.

We are looking for the 50 and the 51st in order to find the mean of those two to get our median, our average middle.

And so we know that the 50th and the 51st belong in the group in the goal scored of one.

The median number of goals scored is one, because the 50th and the 51st piece of data, the two middle pieces.

Are in that group of one.

There was only 19 zeros, which means that the first number one is in the 20th position, and then the last number one is in the 53rd position, so our 50th and our 51st are in the ones area of the data.

A check for you to have a go now.

Using the frequency tables, what is the median star rating? Pause the video whilst you think about that, and then when you're ready to check press play, and then we'll go through it.

The median is four stars.

And we can use our right hand frequency table, that grouped, the collected one.

We know there's 134 pieces of data in total, if we want to work out the position of the median, add one, divide by two.

Then we work out we're looking for the 67.

5 piece of data, so actually we need the 67th and the 68th.

Using right hand frequency table, we can see that there are only 51 pieces of data up to the three.

Ones, twos and threes accounts for 51 pieces of data.

And then we get the fours, and the fours take us from the 52nd piece of data up to the 89th piece of data.

And so our median is definitely in that group, and that's why our median is four stars.

You're gonna do some practise for median from an ungrouped frequency table.

Question one, you need to complete the missing frequencies, it's about the same data.

The first one is a is a ungrouped frequency table, and the second one is where we're collecting data as we go up the star ratings.

Pause the video whilst you complete those frequency tables, and then when you're ready for question two press play, and we'll move on.

Question two, you need to work out the median for these data sets.

So you've got two frequency tables, and then for part C, you need to justify why your answers are sensible in part A and part B.

If you realise that your answer is not sensible, then you probably want to have a go again.

Pause the video whilst you do that, and then when you come back we'll go through the answers to question one and to question two.

Question one, you needed to complete the missing frequencies.

These were connected tables, so it was about the same dataset, so you needed to use them both to be able to fill 'em out.

The first missing frequency would be 16, we know that because on the right hand one the star rating of one had a frequency of 16, so the star rating of one would still have a rating of 16.

The frequency for a star rating of three is 34, and there's a couple of ways that you could have done this.

Probably the easiest way is you know that the total frequency is 160, and we know that from the other frequency table.

If I add up the other four frequencies and subtract it from 160, I get the remainder which is 34 and that must be the three.

To get the star rating for three or less, we're gonna use the four or less 129, and remove 54 from the frequency, that 54 is all of the fours.

And so that leaves you with only the three or less, which is 75.

You could have worked with a 75 out before the 34, and then if you've run the 75, then you could have worked out the 34 using a similar method of taking off or adding on.

Question two, you needed to work out the median for these data sets.

It might be that you've worked out the position and not told the median.

That's the extra step on a median calculation is locating the position of it, and then actually writing down the median.

The median will always be in the context, so for part A, it's about goals scored, and for part B it's about tube lines.

Making sure that you're actually giving your answer to the question, rather than just leaving midway through a working out.

A, the median goal scored is two, and for B the median number of tube lines is one.

For each of your answers justify why your answers are sensible.

Part A, the data is fairly symmetrical, so if you look, if you imagine it as a dot plot, it's gonna look fairly symmetrical in its distribution.

The median as a measure of central tendency, and the data being pretty symmetrical feels right to be a two.

That's the number of goals scored.

And for part B, the median looks for the central, the middle, and so over 50% of the data is in that category of one.

When you are thinking, oh okay, how many have I got here? If I add that up and then divide by, add one and divide by two to find the position, you are finding that middle piece, that 50%, and that's gonna be in number one.

If that data was plotted on a dot plot, it would be heavily loaded towards number one, and so the median lying within there is not a surprise.

But now up to the last learning cycle, this part is about median from a bar chart, here we've got a bar chart that shows the number of each table type in a restaurant.

How many diners per table? The data values are either a zero, a one, or two, a three, a four, five or a six.

Jun has suggested that the median will be one of them, is Jun, correct? He might be, it's not a definitive yes or no here, he could be, but it could also be the mean of two of them.

Because the median is either a piece of data when it's an when it's odd, or if it's even, and the two pieces either side are the same.

Or it's the average middle, and the average being the mean, then sometimes our median value isn't part of the dataset.

Aisha said, "Three is the middle bar, so it's the median." Here we have a bar chart and a three is in the middle, so that is the median, is Aisha correct? Again, she could be correct, but not for the reason of it being the middle bar.

Three might be the median, but not because it is the middle bar.

We've got a dot plot overlaid on top of the bar chart, and the total frequency represents the amount of data values.

And so we can do that by adding up the heights of the bars, or with the dot plot we can count the dots.

And so there are 13 pieces of data.

If there's 13 pieces of data, then we know that the median is in the seventh position, and we can do that by adding one and dividing by two.

If we count through the dots.

Then the median is three.

Aisha said the median is three, but her reason, her justification was because it was the middle bar, that's not the reason The median is three.

The median is three because it is the seventh piece of data, it's the middle piece of data in 13.

A check, the median will always be in the middle bar, true or false? And then justify your answer.

Pause whilst you're thinking about that, and then when you're ready to check it press play.

This is false, and it's because the medium will be in the middle, the central position.

And so it just really depends on the distribution of the data.

If the data is fairly symmetrical, then yes, it's probably the middle bar.

But if the data is not distributed evenly or symmetrically, then there's no reason that it's gonna be in the middle bar.

This bar chart shows us reviews of a restaurant, so they've been asked to review the restaurant from one star to five star.

Sam says there are 40 pieces of data, the median is the 20.

5 piece of data.

Here we've got an even amount of data, if we were to go all the way back to that idea of a list and crossing off to the to the middle, we'd end up crossing them all out.

We need to look for the two most central pieces, and use them to get our median, we'll do the mean of those.

With the dot plot overlaid, we can see that there are 19 pieces of data in bar one, bar two and bar three combined.

'Cause there are five in the first bar, which is about star rating one.

There are nine in the second bar, and there are five in the third bar, and so that is 19 in total.

There are 19 ratings of three stars or less, the median must be four stars.

Because we are looking for the 20.

5 piece of data, and if there are 19 up to here, then the 20th and the 21st must be in the next bar, which is the four star rating.

A check, Jacob has calculated the median to be 12, why is this not correct? Pause the video whilst you think about that, and then when you've got your reason, come back to check it.

No restaurant has a food hygiene rating of 12, the median will be within the data.

The median is a measure of central tendency, it is trying to find that single value that describes the central piece of data.

It needs to be within the dataset to be central to it.

12 is not part of the dataset, this data is a food hygiene rating from zero up to five, so a median could not be 12.

Now you are onto your final part of practise, and this is all about the bar charts.

For question one, answer the three questions using the bar chart, pause the video whilst you get on with that.

And then when you're ready for the next question, press play.

We've got a comparison bar chart here, and it's looking at two different towns and the food hygiene ratings for restaurants within each of the towns.

Calculate the medium food hygiene rating for each, pause the video whilst you're doing that.

And then when you're ready to go through the answers, press play and we'll start with question one.

Question one was about food hygiene ratings at a group of local restaurants.

The first question asked you, how many restaurants is this data about? You needed to add up the total frequency, take the frequency from each bar, so that's the height, and then add them up.

And you should have got 85 restaurants.

What's the position of the median? If we've got 85 pieces of data, we're gonna add one, divide by two.

The 43rd data point is the position of the median, 43 cannot be your median value because our data only runs from naught to five.

But the 43 is the position, so don't get confused between the position and the actual value.

What is the median food hygiene? We are looking at which bar would the 43rd piece of data be in.

And so the food hygiene rating would be three, and that's because there are 41 pieces of data in the first three bars.

For zero, a rating of zero as a rating of one and a rating of two, if you add up their frequencies, you get to 41.

We are looking for the 43rd, and that would therefore be in the next bar.

Question two, you needed to calculate the median for both town A and town B, on this screen we've got town A, there were 101 ratings in total.

Again, just adding up the heights of each of the blue bars, the left hand bar.

The median would be the 51st value, because if I add one and divide by two.

There are 45 restaurants that have a food hygiene rating of two or less.

And there are 64 restaurants that have a food hygiene rating of three or less, which means that the median would be a rating of three.

We are looking for the 51st, and so 45 is too little, we've not got to the 51st piece of data yet.

And at the end of the hygiene rating of three we get to 64, so it is within them threes.

So the food hygiene rating would be three for town A.

Town B, there was 113 ratings, so adding one divided way two, we find the position as 57.

And there were 41 restaurants which had a hygiene rating of two or less.

There were 65 restaurants with a rating of three or less.

So our 57th is within that 42 to 65.

In that bar of three, so the median for town B is also a rating of three.

To summarise what we've covered in this three part lesson.

The median can be calculated from a line graph by ordering the raw data points.

If you're given the the line graph, it's just a group of data, take the data off, put it in order, find the position, and that will give you the median.

An ungrouped frequency table, by considering how many piece of data has been accounted for previously.

And the median from a bar chart is how much data as you go through the bars.

The bar chart and the frequency table, very similar, with this idea of how much data have you already passed when you're trying to find the position that you're looking for.

Well done today, and I look forward to working with you again in the future.