Lesson video

Hi, everyone.

My name is Ms. Ku, and I'm really excited to be learning with you today because today we're looking at histograms under the unit of graphical representations of data.

Hope you enjoy the lesson.

Let's make a start.

Hi, everyone and welcome to this lesson on moving between tables and histograms under the unit Graphical representations of data: cumulative frequency and histograms. And by the end of the lesson, you'll be able to construct a table from a histogram.

Let's have a look at some keywords.

We'll look at continuous data, a histogram, and frequency density.

Continuous data can take any value within a range.

Height, mass, temperature are just examples of continuous data.

Can you think of any more different types of continuous data? A histogram is a diagram consisting of rectangles whose area is proportional to the frequency in each class and whose width is equal to the class interval.

Lastly, frequency density.

Frequency density is proportional to the frequency per unit for the data in each class.

Often the multiplier is one, meaning that frequency density is equal to frequency divided by class width.

Today's lesson will be broken into two parts.

We'll look at constructing a table from a histogram and then incomplete tables and histograms. So let's make a start constructing a table from a histogram.

Now Laura says, "Frequency density allows me to see which classes have the largest frequency per unit, and this makes comparing the classes easier to do." Laura also says, "But what if I want to know the frequencies for each class? How can I see that from a histogram that uses frequency density?" Well, this data shows waiting times at a doctor's office.

We have a completed histogram showing all these wonderful bars, all these wonderful class widths, and the histogram is plotted against a frequency density.

We have a frequency table, but we're missing those frequencies.

So what we need to do is use our histogram to complete our frequency table.

So let's find the area of each bar.

As we know, the area of each bar represents the frequency for that class width.

Here we have our time interval between zero and six minutes.

Well six subtract zero is six, so I've indicated that width of our bar to be six.

Next, we know the frequency density of this bar is two.

In other words, the height of this bar is two.

So look what we've got.

We've got our rectangle, we've got our length, and we have our width.

So how do we find the area? Well, it's simply six multiplied by two, which gives me an area of 12, so that means the frequency is 12.

Let's move on to the next one.

Here we have a time interval between six and 12, so let's subtract six from 12, giving me a width of six.

Now we know the frequency density is three, so we have a length and width of our bar.

That means we can work out the area of our bar.

The area is 18, six multiplied by three, which is 18 so that means the frequency is 18 for our class six less than t, less than or equal to 12 minutes.

Next, let's have a look at the next bar.

Here we have our time interval between 12 and 20 minutes so that means we have a class width of eight.

20 subtract 12 is eight.

Now we have the width of our bar.

Let's identify the height.

Well, the height is the frequency density, so we know that's four.

From our wonderful rectangle, we can work out the area.

Eight multiplied by my four gives me 32 so the area of this wonderful rectangle is 32, so that means the frequency is 32.

Lastly, we have our time interval between 20 and 24 minutes.

This means we have a width of four because 24 subtract 20 is four.

Now we know the width is four.

Let's have a look at that height.

Well, the height is three because the frequency density is three, so that means we can work out the area.

The area is four times three to 12, so that means our frequency is 12.

So what you can see here is we've identified the frequencies from our completed histogram where our histogram had unequal bar widths and we were able to use our frequency density to work out our frequency and complete our frequency table.

Now it's time for a check.

Here's a table showing completion times of a 10K race, and I want you to draw a frequency table from this histogram.

Pay careful attention of your classes and use that right notation as well.

Take your time and press pause.

Well done.

Let's see how you got on.

Well, hopefully you've identified these correct classes.

Massive well done if you did.

Our first class is 40 less than t, less than or equal to 50.

The next is 50 less than t, less than or equal to 60.

The next is 60 less than t, less than or equal to 80.

And lastly, 80 less than t, less than or equal to 100.

Massive well done if you got this.

So now let's calculate that frequency.

Well, remember the frequency is found by looking at the area of each bar.

10 multiplied by one is 10.

We know the width is 10 because it's the difference between 50 and 40.

And we know the height of our bar is one as the frequency density is one.

10 times one is 10.

Next, we know the width is 10 because 60 subtract 50 is 10, and the height of this bar is the frequency density, and the frequency density is nine.

So 10 multiplied by nine is 90.

Next, we have the width of this bar is 20 because 80 subtract 60 is 20.

Multiplying by the height of this bar, which is four, we have 20 multiplied by four, which gives me a frequency of 80.

Last, we have a width of 20 because 100 subtract 80 is 20 and then we're multiplying it by that frequency density, which is two, giving me a frequency of 40.

Really well done if you got this.

Great work, everybody.

Now it's time for your task.

Here are the waiting times for a ride in a theme park.

I want you to complete the table using the histogram.

See if you can give it a go.

Press pause if you need more time.

Well done.

Let's move on to question two.

Question two says, "Complete the table." Here we have different lengths of pieces of string, and I want you to use the histogram to complete our frequency table.

See if you can give it a go.

Press pause if you need more time.

Well done.

Let's move on to question three.

A chocolate factory records the number of faulty chocolates and has grouped them according to mass.

Alex says, "Most of the faulty chocolates are in the range of 22 grammes to 30 grammes, as that bar has the highest frequency density." Is he correct? And I want you to explain.

See if you can give it a go.

Press pause if you do more time.

Great work.

So let's move on to these answers.

For question one, you should have had a frequency of nine, four, nine, seven, and one.

Well done if you've got this.

Press pause if you need more time to copy it down.

For question two, great question, completing the frequency table is done by calculating the area of each bar.

Here's my working out.

For the first bar, 10 multiplied by 0.

4 gives a frequency of four.

Five multiplied by 1.

6 gives a frequency of eight.

Five multiplied by 2.

4 gives a frequency of 12.

10 multiplied by 1.

6 gives a frequency of 16, and five multiplied by 1.

8 gives a frequency of nine.

Really well done if you got this.

Press pause if you need more time to copy it down.

And lastly, we have, Alex says that most of the faulty chocolates are in the range of 22 grammes to 30 grammes as that bar has the highest frequency density.

Is he correct? Well, he's not correct, as 90 chocolates are defective between 10 and 22 grammes, and there are 70 defective chocolates between 22 and 30 grammes.

In other words, we've calculated the frequency using the area of the bar and identified where most of our faulty chocolates lie.

Really, really well done if you got this.

Press pause if you need more time to copy it down.

Great work, everybody.

So now it's time to move on to the second part of our lesson, incomplete tables and histograms. Now, differing class widths affect how we should interpret the data, and in order to make valid conclusions, it can be useful to have both a histogram and a frequency table.

So, here we have an incomplete histogram and incomplete table.

How do we know the area of each bar is the frequency? Now the reason why I'm asking that question is because of our definition of frequency density.

Frequency density is proportional to the frequency per unit for the data in each class.

Often the multiplier is one, meaning that the frequency density is frequency divided by class width, but we need to verify that the multiplier is one.

So we can use this formula and calculate the frequency using the area of our bar.

And to do this, we're going to compare the frequency of our histogram to the frequency in our frequency table.

Well, using the histogram, I'm going to look at our first bar here.

We know the area is 15, as it's the difference between 20 and five, multiplied by two as it's our frequency density.

This means we have a frequency of 30.

This means since the area of the bar is equal to the frequency for the same class, we know our multiplier is one.

Now we can move on, and the final frequency can be found by calculating the area of this bar.

Well, we know the class width this five because 35 subtract our 30 is five.

And we know our frequency density is the height of our bar, which is 13.

So that means we know our frequency is 65.

We have enough information now to fill in that missing bar in our histogram.

We know that missing bar is from the class 20 less than t, less than or equal to 30.

So we have a class width of 10, but let's work out that frequency density.

Frequency density is our frequency divided by our class width, so it's seven.

So that means I've simply added my bar, which has a class between 20 and 30 and a frequency density of seven.

Another useful strategy is to add extra columns for class width and frequency density.

And this allows you to check what is already plotted.

So you can see I've added the extra column of class width, identifying 15, 10, and five, and I've identified the extra column of our frequency density, two, seven, and 13.

This is a really nice strategy as it is a good checking mechanism and/or way of plotting the information from our frequency table into our histogram and vice versa.

Now it's time for a check.

Here, I want you to fill in the missing information about the lengths of pieces of string from a box.

See if you can give it a go.

Press pause if you need more time.

Well done.

Let's see how you got on.

Well, for me, I'm gonna add on those extra columns.

I really like showing all the working out so you can see how we've identified our frequency density and how we've identified our frequency.

Identifying our class width, we have five, five, four, and two.

And then I've worked out the frequency density to be 2.

4, three, five, and two.

From this, you can see how I've simply filled this information into my table.

Well done if you've got this.

Sometimes we're given even less information.

So how can we calculate the scale for the vertical axes given this partially completed histogram and a partially completed frequency table? Well, to do this, let's consider what we already know.

We know the class widths for sure.

So all I'm gonna do is add on that extra column again and identify my class widths.

Between zero and one, I have a class width of one.

Between one and two, I have a class width of one.

Between two and four, I have a class width of two, and between four and 12, I have a class width of eight.

Now I can calculate some of our frequency densities.

Seven divided by one is seven, four divided by one is four.

12 divided by two is six, and that's pretty much all I can fill in for now.

Now let's have a look at this bar.

This class has a frequency density of six.

We know this because we just worked it out before.

We also know the height of this bar should be six.

So all I'm going to do now is plot this six on my Y axis as the height of this bar should have a frequency density of six.

Now we can complete the scale on the vertical axes.

Remember, we know scales increase equally, so I know it goes up in two, four, and then six.

Now I've got enough information to complete the histogram and identify any missing frequency densities.

Well, I know for my first two bars, between the lengths of zero and one centimetres, it should have a frequency density of seven.

The next class is between one and two centimetres, and it should have a frequency density of four.

And let's extract our information from our histogram and pop it into our frequency table.

We know the class width between four and 12 centimetres has a frequency density of 0.

5, so that means we can work out our frequency to be four.

Really well done if you got this.

We've now completed both our histogram and our table.

Well done, everybody.

So now it's time for your check.

I want you to fill in the table and histogram of these race times.

Here we have a partially completed table and a partially completed histogram.

Use the squares in your book if needed.

Press pause as you'll need more time.

Well done.

Let's see how you got on.

Well, first things first, I'm gonna add on these extra columns of class width and frequency density.

By working this out, I've identified this scale and I've plotted my remaining bars and identified my frequencies.

Well done if you've got this.

Great work, everybody.

So now it's time for your task.

I want you to fill in the table and histogram of these race times.

Press pause as you'll need more time.

Great work.

Let's move on to question two.

Here are some lengths of haircuts in a salon, and I want to fill in the table.

This is a tough question.

As you'll notice, we are missing that y axis.

Add on any extra rows if it helps.

See if you can give it a go.

Press pause as you need more time.

Great work.

Let's move on to these answers.

Well, here we should have this completed histogram, and the missing frequency was five.

Press pause if you need more time to copy it down.

For this one, we should have these missing frequencies and this completed histogram.

Absolutely well done if you got this one right.

It was really tough.

Press pause if you need more time to copy it down.

Superb work, everybody.

So in summary, histograms are graphical representations of continuous data, which allow us to see a quick and easy snapshot of the distribution of the data.

And when the class widths are equal, we can use the frequency for the vertical axis, but when the class width are unequal, we can use frequency density for that vertical axis.

Massive well done, everybody.

It was tough today.

Really impressed with you.

Great work.