Lesson video

Hello there and thank you for choosing today's lesson.

My name is Dr.

Rowlandson and I'll be guiding you through it.

Welcome to today's lesson from the unit of comparisons of numerical summaries of data.

This lesson is called "Checking understanding of summary statistics from a frequency table." And by the end of today's lesson, we'll be able to find the mean, median, mode and range from a frequency table.

Here is a previous keyword that we'll use again during today's lesson.

So you may want to pause the video if you want to remind yourself what this word means and then press play when you're ready to continue.

This lesson contains three learning cycles.

In the first learning cycle, we're going to find the mode and range from a frequency table, and then we'll find the median from a frequency table.

And then finally we'll find the mean from a frequency table.

Let's start off with mode and range from a frequency table.

Here we have some data that has been collected about the number of sports a group of people play and it has been presented as a frequency table.

Let's take a look at what this frequency table tells us.

This row represents the 12 people who play no sports.

This row represents the six people who each play four sports.

And this row is the total frequency, in other words, the size of the sample.

So let's check we've got that.

Here's another frequency table.

Use it to fill in the blanks with these sentences here.

Pause the video while you do this and press play when you're ready to go through some answers.

There were 20 pupils who achieved grade eight.

Six pupils achieved grade one.

There were 234 pupils who took the test.

96 pupils achieved a grade four or less.

And 190 pupils achieved at least a grade four.

Here we have a set of data that is presented as a list.

In this list of data, it is clear to see that 34 appears more than any other value.

So the mode is 34.

Here's the same data now presented as a frequency table, and some might say that when data is presented as a frequency table, it can be less clear than when it's presented in a list.

However, a frequency table is a more practical way to represent data when there's a large amount of it.

For example, here's a frequency table that shows the number of reviews for each rating for a film.

So what can we say about this? Well, Lucas says there are two thirteens, so the mode is 13.

But Jacob says there are two thirteens, so the modes are one and two.

But Alex says the rating with the highest frequency is five, while Andeep says the highest number is 30, so that is the mode.

Are any of them correct here? Pause the video while you think about this and press play when you're ready to continue.

Well, in fact, Alex is correct here.

The rating with the highest frequency is five, so five is the mode.

If you imagine writing each rating out the number of times it says in the frequency, you would write one 13 times, you'd write two 13 times, you would write three 19 times, you would write four 26 times, and you'd write five 30 times.

That means you would write the number five more than any other number.

So five is the mode.

It's the one with the highest frequency.

So let's check what we've learned.

Here we have three frequency tables.

Which frequency table shows data with a mode of one? Pause the video while you make a choice and press play when you're ready for an answer.

The answer is A.

In that table, one has the highest frequency, so the mode is one.

But that's not the case in the other two tables.

Here we have a frequency table that shows the colours of pet collars.

Let's see what we can learn from this frequency table.

We can see here that the highest frequency is 21, but it appears twice.

So this data is in fact bimodal.

There are two values that have the highest frequency.

The modal colours, therefore, are red and green.

Now if we look at the frequencies, we can actually see that 21 appears twice and so does 16.

So why is it not bimodal, because the 16 and 21 both appear twice? Pause the video while you think about this and press play when you're ready to continue.

We need to remember that the 21 and 16 are not data values in themselves.

They are frequencies.

21 is the highest frequency, so red and green both appear more times than blue and yellow.

So let's check what we've learned there.

Here we have a frequency table, which shows data about 188 games and the score received out of five each time.

And Jacob says the data is bimodal.

A score of one and four is the most common score.

So is Jacob correct here? Pause the video while you think about this and justify your answer, and press play when you're ready for an answer.

No, Jacob is not correct here.

The highest frequency is 53, so the score of three was the most common score.

Here we have another frequency table, which shows the number of sweets per individual pack.

Let's see what we can interpret from this.

What was the smallest number of sweets found in one pack? Well, that would be seven sweets.

What was the biggest number of sweets found in one pack? That would be 11 sweets.

What was the range of the number of sweets found in one pack? Hmm.

Maybe pause at this point and think about how we might work the range out and then press play when you're ready to continue.

What you're not going to do here is subtract seven from 25.

Because even though 25 is the biggest number you can see in the table, 25 is not a data value in itself here.

That 25 is a frequency, and it means that the number nine is included 25 times in this dataset.

So the lowest value in this dataset is seven and the highest value is 11.

So you'll be doing 11 subtract seven, which is four.

If another individual pack was added to the sample and it contained nine sweets, would the range change? Pause the video while you think about that and press play when you're ready to continue.

No, it wouldn't change.

The least and the most number of sweets would still be seven and 11, so the range would still be four.

So let's check what we've learned.

Here's a frequency table that shows the number of pupils per class.

What is the range of the number of pupils per class here? Pause the video.

I'll let you do this.

And press play when you're ready for an answer.

The answer is seven, which we get from doing 32 subtract 25.

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

This task contains two questions and here is question one.

You have three frequency tables, and you need to find the mode and the range for each frequency table.

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

And here is question two.

You have six frequency tables, and what you need to do here is match the frequency tables that have the same mode or have the same range.

Pause the video while you do this and press play when you're ready to work through some answers.

Here are the answers to question one.

In part A, the mode for the first table is cat, the second table is one and two, and the third table is four.

In part B, the range for the first table cannot be worked out because they are not numerical values.

But for the second table, the range is three and the third table is also three.

Then question two.

If you match the tables together by their modes, you would match A with D, B with C, and E with F.

And if you match the tables together by their range, you would match A with B, C with F, and D with E.

Great work so far.

Now let's move on to the second learning cycle, which is finding the median from a frequency table.

A frequency table is usually ordered.

This makes it easier to calculate the range but also the median as well.

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

Let's take a look at an example.

Here we have some data about the number of passengers in a cable car compartment that is shown in a frequency table.

Let's interpret a few things from this table.

For example, how many cable cars were empty? Well, if we look at the row that has zero passengers, we can see that the frequency is one.

So there was one cable car that was empty.

How many cable cars had a group size of one or fewer? Well, if we look at the rows that have zero and one passengers, and look at the frequencies of those, we can see that there are two cable cars altogether that have a group size of one or fewer.

So let's check we've got that.

Here we have a frequency table that 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 while you work this out and press play when you're ready for an answer.

12.

There are five with 14 and there are seven with 15.

How many streets have at least 17 houses? Pause the video while you work this out and press play when you're ready for an answer.

The answer is seven.

There are five with 17 and there are two with 18.

Here we have our frequency table again about the number of passengers in a cable car compartment.

How many cable cars had a group size of two or fewer passengers? Well, if we look at the first three rows and add up all the frequencies for those three rows, we'd see that there are four cable cars with two or fewer passengers.

So that means the third piece of data in this table represents a cable car with how many passengers? Hmm.

Well, the first piece of data has zero passengers.

The second piece of data had one passengers.

And then the next two pieces of data had two passengers.

So that would include the third piece.

So the third piece of data in the table represents a cable car with two passengers.

The eighth piece of data in the table represents a cable car with how many passengers? Well, we could do the same thing again.

We could start at the top of the table and count the frequencies until we figure out where the eighth piece of data is.

The first piece of data has zero passengers.

The second has one passenger.

The next two, the third and fourth, has two passengers.

The next two will be the fifth and sixth.

They have three passengers.

So the next three would be the seventh, eighth, and ninth.

So the eighth piece of data must have four passengers.

Now the method that got that answer was to keep a running total of the frequency, which we're doing verbally out loud then, but this can be trickier to do when there is a lot of data and big numbers in the frequency.

An alternative method could be to rewrite a table so it shows a running total as we go through the data.

In this case, we could rewrite this table like this.

We can now see that the passenger column keeps a running total of all the different combinations we've had so far.

Zero, and then one or less.

That includes zero.

And then two or less is the combination of zero, one, and two.

And three or less is a combination of zero, one, two, and three, and so on.

So let's fill in the frequency column now of this table.

The frequency for cable cars with zero passengers was one.

The frequency for one or less is any data that are zero or one.

So that means the frequency in the second table will be the sum of the first two rows in the first table.

That's two.

The frequency for two or less will be the sum of one, one, and two, which is four.

The frequency for three or less will be the sum of one, one, two, and two, which is six.

Now rather than always adding up all the frequencies from the top row down to where I'm up to, what I could do is just add on the next frequency each time.

So I can see so far for three or less, the frequency is six.

To get four or less, I can add on the next frequency, which is three, and I'll get nine.

And for five or less, I can add on the next frequency, which is two, and I'll get 11.

As the data is ordered, this helps to know the data in each position.

Let's take a look at this rewritten frequency table in a bit more detail.

And what we have on the right-hand side is the same data presented as a list.

If we take a look at two or less, for example, we can see that in the list, that includes zero, one, two, and two.

So that is why the frequency is four because it includes the first, second, third, and fourth data points.

So what about the next row? Three or less is any data that are two or less or three.

This frequency is two more than that of two or less.

So the fifth and the sixth data points must both be three.

And then when we include four or less, that's an additional three in the frequency that includes the seventh, eighth, and ninth data point.

And when we include five or less, that is an additional two in the frequency.

So that includes the 10th and 11th data positions.

So let's check what we've learned there.

We have a frequency table on the left and it is being rewritten as a frequency table on the right.

Complete the missing frequency.

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

The answer is nine.

What is the group size of the seventh piece of data then? Pause the video while you work this out and press play when you're ready for an answer.

The seventh piece of data is a one.

The last zero is the third piece of data and the last one is the seventh piece of data.

What is the value of the 22nd piece of data then? Pause the video while you do this and press play when you're ready for an answer.

The answer is three.

The last two is the 16th piece of data and the last three is the 24th piece of data.

So the 22nd one must have been a three.

Here we have a frequency table about the goals scored each match in a football season by all teams in a club, and it's been rewritten as a new frequency table on the right where it includes one or less, two or less, three or less, and so on.

Let's now think about how we would find the median from this data.

The median is the middle piece of data, and what we can see here is that there are 100 pieces of data altogether.

So that means the position of the median is 100 plus one divided by two, which is the 50.

5th.

Now what that really means is it's the midpoint between the 50th and the 51st piece of data.

Either way, where would this piece of data be? Well, if we look at the right-hand frequency table, we can see that the frequency for the first row is 19.

So we definitely haven't found the 50.

5th data here yet.

But when we get to the second row, we can see the frequency is 53.

So that means the 50th and the 51st piece of data must have been in that second row.

Therefore, the median number of goals scored is one.

So let's take a look at another example and then you go do one very similar yourself.

Here we have a frequency table and we want to find the position of the median.

Well, we would need to first work out how many pieces of data there are as we move through this table.

We could do a running total.

In the first row, there are 12.

By the time when we get to the second row, there have been 30 pieces of data altogether.

By the time we get to the third row, there have been 51 pieces of data altogether.

And the next row, there have been 89 pieces of data altogether.

And then by the last row, there's been 134 pieces of data altogether.

So how would we use this to find the position of the median? Well, if there are 134 pieces of data, the median is in the 134 plus one divided by two, which is 67.

5th position.

Your turn now.

What is the position of the median for the data in this frequency table? Pause the video while you work this out and press play when you're ready to continue.

There are 141 pieces of data.

So the median is in the 141 plus one divided by two, which is the 71st position.

So back to my example.

If we know that the median is in the 67.

5th position, what is the median star rating? Well, we need to find which row the 67.

5th piece of data would be in.

If we work from the top downwards, we can see that in the first row, we had 12 pieces of data that hasn't included the 67.

5th piece.

By the time we get to the second row, we've had 30 pieces of data.

We still haven't found the 67.

5th piece yet.

By the time we get to the third row, we've had 51 pieces of data altogether.

We still haven't had the 67.

5th piece of data yet.

But when we get to the fourth row, we've now had 89 pieces of data altogether.

That means we must have met the 67.

5th piece of data in that fourth row.

So that means the median is four stars.

Your turn now.

What is the median star rating for your table? Pause video while you work this out and press play when you're ready for an answer.

Well, you are looking for the 71st piece of data.

It's between the 54th piece and the 88th piece.

So that means the median is four stars.

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

This task contains four questions and here is question one.

Given the number of pieces of data, calculate the position of the median.

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

And here is question two.

You have a frequency table that is being rewritten as another frequency table for the same data.

What you need to do is complete the missing frequencies 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 questions three and four.

And here are questions three and four.

You need to find the median number of cars in question three and find the median mode and range in question four.

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

Okay, here are the answers to question one.

Pause the video while you check them and press play when you're ready for more answers.

Here are the answers to question two.

Pause while you check and then press play to continue.

And here are the answers to questions three and four.

Pause while you check and press play when you're ready for the next part of today's lesson.

You're doing great so far.

Now we're onto the third learning cycle, which is about finding the mean from a frequency table.

A frequency table organises data into groups, or stratas.

And this is much more practical than having a long unordered list of data.

For example, this data is about the number of items per order for an online retailer.

Let's take a look at what we can see from this table.

We know that this row tells us that there were five orders of two items. So how many items is that altogether? Well, there were five orders where each order had two items. So that means there would be five lots of two, which would be 10 items altogether.

How many items were sold in orders of four? Well, let's take a look at the row with four items. We can see that's six orders of four items. Six times four would be 24 items. How can we use the frequency table to calculate the mean number of items per order from this online retailer then? Well, Jun's going to help us with this one.

Jun says we need to find the total number of items sold across all of the orders.

Let's look at it row at a time.

This first row shows that there are three orders of one item, so that is three items altogether.

The second row shows five orders of two items. That's 10 items altogether.

The third row shows six orders of three items. That is 18 items altogether.

The fourth row shows six orders of four items. That's 24 items altogether.

And the fifth row shows five orders of five items, which is 25 items altogether.

So that means, in total, there must have been 80 items sold.

Hmm.

How do we use this then to help us find the mean? Jun says, then we need to divide it by the number of orders received.

Let's think about how many orders were received then.

There were 25 orders received altogether.

If we wrote out the number one three times and the number two five times and three six times and so on, we would write 25 numbers and those numbers would add up to 80.

So to calculate the mean, we would do 80 divided by 25 to get 3.

2 items per order.

Let's take a look at another example and then you go do one very similar yourself.

Here we have a frequency table.

What is the total number of goals scored? Well, let's do it row by row.

On 17 occasions, no goals were scored, so that's zero.

On 25 occasions, one goal was scored.

That's 25 goals altogether.

On 25 occasions, two goals were scored.

That's 50 goals altogether.

And on 13 occasions, three goals were scored.

That's 39 goals altogether.

So in total, there must have been 114 goals scored altogether, which is the sum of 0, 25, 50, and 39.

Your turn now.

What is the total number of goals scored in your frequency table? Pause the video while you work this out and press play when you're ready for an answer.

125.

And there's the working.

So back to my example.

We know the total number of goals scored altogether, but what is the mean number of goals scored per match? Well, I need to work out how many matches there were.

That is by adding up the frequencies.

There were 80 matches altogether, and in those 80 matches, 114 goals were scored altogether.

So the mean would be 114 divided by 80, which is 1.

425 goals per match.

So your turn now.

What was the mean number of goals scored per match? Pause the video while you work this out and press play when you're ready for an answer.

The answer is 1.

5625 goals per match, and the working is on the screen for you to see.

What you may do with this is round it to, say, three significant figures to 1.

56.

Let's check one more thing before we start the next task, and that is about making sure our means seem sensible.

Here you have a frequency table and you have three possibilities for the mean.

Now you don't want to work the mean out exactly.

I want you to think about is, which of these mean values seem reasonable for the data that you've been presented? Pause the video while you think about this and write down an answer, and then press play when you're ready to see what the answer is.

The answer is B, 3.

5.

The reason why is because the data can be seen in the star ratings column.

The frequency column tells you how many times each piece of data occurs.

And what we can see from the star ratings column is that the lowest is one and the highest is five, and you'd expect the mean to be somewhere in between those.

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

This task contains four questions and here is question one.

A class take a quiz, marked out of six, and the frequency table shows the results for the class.

What you need to do is use that table to answer parts A, B, and C.

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

And here is question two.

The frequency table shows the number of bedrooms in different houses within an area of a town, and what you need to do is calculate the mean number of bedrooms per house.

And don't forget to check your answer is sensible at the end.

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

And here is question three.

The frequency table shows the number of leaves on the stems for 100 roses.

Use the frequency table to answer parts A, B, C, and D.

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

And here is question four.

The frequency table shows data about the number of letters in the first name of a group of people.

The mean number of letters per first name is 5.

4, and what you need to do is find a missing frequency in that table.

Pause the video while you do this and press play when you're ready to go through some answers.

So in question one, how many pupils are in the class? There were 30 pupils.

In question B, how many marks did the class get as a group? There were 90 marks.

And for question C, what is the mean mark per pupil? It was three marks per pupil.

Question two, here are the answers.

Pause the video while you check it with your own and press play when you're ready for question three.

Here are the answers to question three now.

Pause the video while you check these and press play when you're ready for the answers to question four.

So what about question four? Well, if we call that frequency x, what we can do is find an expression for the number of people.

That is adding up all those frequencies, and that will be 42 plus x.

And then we need to find an expression for the total number of letters.

That's doing three times four is 12, four times x is 4x, five times 12 is 60, and so on, and adding all those together.

And then the mean is the total number of letters.

That's 238 plus 4x divided by the total number of people, which is 42 plus x.

And we know that when we do that division, it gives an answer of 5.

4.

So what we need to do is then solve that equation.

If we solve that equation by rearranging, eventually, we get an answer of x equals eight.

So that missing frequency must be eight.

Fantastic work today.

Now let's summarise what we've learned during this lesson.

It is more practical to organise large datasets into frequency tables.

And summary statistics, such as the mean, mode, median, and range, can be calculated from frequency tables.