Lesson video

Hello and good day to you all.

Welcome to today's maths lesson on sampling, with me, Mr. Grattan.

In today's lesson we will look at a very specific type of sampling, the stratified sample.

Pause here to check some of the keywords that we'll be using in today's lesson.

Two important keywords are stratum, which is a group within a population that has distinct properties you care about for your investigation.

And stratified sample, which is a representative sample taken from each stratum within a population.

A type of sample you may be familiar with is a random sample, but is it always appropriate? Let's have a look.

A random sample is a sample where every member of a population has an equal chance of being selected for a sample.

Laura claims, "A random sample is always the best kind of sample, as it is guaranteed to be fair and representative of the population." But Alex disagrees, as it's "Never guaranteed a random sample will be representative of the whole population.

Random chance may lead to very odd samples, especially with a small sample size.

It is possible that the sample could end up biassed and unrepresentative, defeating the purpose of collecting many samples." And here's an example of what Alex means.

In an investigation for how many people at school travelled to school by car each day.

A simple random sample from the population could look like this or this or this.

Which looks representative of the population and which ones don't? Well, this sample represents all of the staff, but only some of the pupils.

Whilst this sample represents no staff at all, but only some of the pupils.

This one however, looks a little bit more proportional to the number of staff and pupils at the school.

I agree with Laura here.

"The school has a lot more pupils than staff, so it makes sense for the sample to have more pupils as well.

But there should still be some staff in the sample as their answers are still important to any investigation about everyone in the school." This sample matches the criteria that we want.

Okay, for this quick check, pause here to match the population to the sample that most represents that population.

For population A, there is an equal number of people in group A and group B.

Therefore, sample C seems most suitable.

However, for population B, group A had three times the amount of people as group B.

Therefore, sample two seems more suitable, as Group A in that sample also had three times the number of people as group B.

A strata is a description for how a population can be split up into groups.

A strata can be many things, depending on the population that you are dealing with.

For example, in this population of a collection of toys, a strata can be the different things on the heads of each of the toys, such as an apple, a hat, or nothing.

We can then split this population into these three groups.

Each of these groups is called a stratum.

This is one stratum, this is another stratum, and this is a third stratum.

Together, all of the groups form the whole population.

The population is split into groups based on the strata of "Things on the heads of the toys." It's definitely worth noting that no toy can be two separate groups.

Each stratum must be completely distinct from all other groups.

A population can be split into different strata.

Again, what strata you choose to split your population into will depend on what is helpful for your investigation.

Pause here to think about or discuss, what could the name or description of the strata be for these groupings of toys on screen? The strata could be the type of toy.

Robots, bears, and penguins are all different types of toys.

And pause here to think about or discuss, what other strata could you split this population into? You could split this population into groups based on colour, size, or fur, or many other correct answers.

Okay, for this check.

A hair salon wants to split the population of the local town into groups based on the strata of their hair colour.

Pause now to write down one stratum based on this strata.

Each stratum will be an example of a hair colour, such as black hair, brown hair, ginger hair, et cetera.

Hair colour is one example of a strata that we could split the population into.

But what other strata would be sensible for a hair salon to group the population into? Pause now to think about this question.

A hair salon could group the population into hair length, beard length, type of haircut wanted, type of hair.

There may be other perfectly correct answers.

Okay, great stuff.

Onto the practise task.

For question one, you'll be given four different populations.

For each population, state which of the three given samples is most representative from the given strata? Pause now for the first two populations.

And pause again here for the second two populations.

And for questions two and three, pause here to give examples of strata and stratum for these two different populations.

Okay, onto the answers.

For question 1A, sample one was correct.

This is because both in the population and in sample one, there were three times as many passengers as there were crew.

For question 1B, sample three was correct.

Whilst there were slightly more home supporters than away supporters in the population, this difference just wasn't big enough to reflect itself in any significant way in the sample.

For part C, sample three was correct.

As in both the population and the sample there were four times as many customers as there were staff members.

And well done If you spotted for D, that option two was correct because both in the sample and population there were nine times as many pupils as there were staff.

And pause here briefly to think about or discuss, how did you identify the proportionality that led to you identifying option two as the correct proportional sample for part D? For question two, examples such as fish and reptiles are other stratum that you could have found.

And for question three, for example, a strata could be the county that you live in.

A stratum would be an example of a county, such as the group of people who live in Devon.

So, we've looked at the idea of grouping a population in terms of strata.

Let's formalise this through a great sampling method, the stratified sampling method.

Here's the same investigation that we saw in the previous cycle.

In a school of 600 pupils and 40 staff, the strata are the roles of the people at the school broken down into two stratum, pupils and staff.

Laura asks, "Is there a way to guarantee that each in a sample has the same proportion as in the population? Is there a way to guarantee that the sample looks like this?" A sample that is proportional to the population because there are 15 times as many pupils as staff.

There definitely is.

A stratified sample is a sample taken from a population broken down into strata.

The proportion of each stratum in the sample is equal to the proportion of that same stratum in the population.

Stratified sampling is also known as proportional sampling because the focus is on ensuring that the sample has a stratum at a frequency proportional to the stratum's frequency in the population.

This population of toys has been grouped into three different stratum, with the strata being the type of toy.

A sample is chosen proportional to the size of each stratum.

In this sample, for every four members in the population, one member is taken for the sample, as you can see by the one robot out of four total robots being chosen for the sample.

Pause here to think about or discuss observations about the number of bears being chosen for the sample out of the population of bears.

A very well done if you spotted that there are nine bears.

However, nine divided by four is 2.

25.

You simply cannot sample one quarter of a bear, so we must round to the nearest integer, this is why we have two bears chosen for our sample.

So, on the left we have our population, and on the right we have our stratified sample.

The sample from each stratum is still randomly chosen.

We try to ensure that the selection from each stratum is as fair as possible to avoid any unwanted bias from creeping in to each sample from each stratum.

All we know is how many from each stratum to choose.

How do we calculate how many people or objects we need to select for each stratum in our sample? Well, a ratio table will help.

Let's go back to our previous example of a population of a school.

For every stratified sampling question, a good bit of advice is to create a row or column that is explicitly for the total of the population or our sample.

For this example, the total population is 600 plus 40, which is 640.

And we want to take a sample of a total of 80 people.

The multiplier from the population to the sample is multiplied by 1/8, or 80 over 640, without simplification.

We can then apply this multiplier of 1/8 to both the pupils and the staff.

600 divided by eight is 75, and 40 divided by eight is five.

In our stratified sample of 80 people, we should randomly choose 75 pupils and five staff.

Let's check that we are correct.

This sample of each stratum should sum to the total sample required.

75 plus five is 80, so our sample is likely valid as a stratified sample.

Okay, onto this check question.

For this investigation about taking a stratified sample of light bulbs in a factory, pause here to figure out what words or numbers go in place of A, B, and C in the ratio table.

Remember to always include a total row, even if the ratio table doesn't initially give it to you.

Sticking with the same investigation, pause here again to figure out what values would go in place of A, B, C, and D in this ratio table.

We can use the multiplier of 1/7, or 170 over 1,190 to get our samples of 100, 50 and 20.

And since 100, 50 and 20 sums to 170, the size of our whole sample, our calculations are likely correct.

There are times when a stratified sample will result in the sample size of a stratum being a decimal.

When this happens, round each sample size to the nearest integer.

For example, a car company wants to take a sample of 600 components that it manufactures.

Let's go through this example to see where the decimals appear.

We have a total of 18,250 components, and 600 of those components are taken for a sample.

The multiply is the output of 600, divided by the current value of 18,250.

Taking that multiplier and multiplying it by 4,200 gives a decimal of 138.

082.

We round this to the in integer 138.

We can do this for the other stratum to give us 35 and 427.

In the majority of cases, the sum of the sample for each stratum should still equal the total sample required, even after rounding.

In this case, the sum of these three stratum in the sample does still equal 600.

Okay, for this check, calculate the sample for each stratum by finding the values of A, B, and C in this ratio table.

Pause now to have a look at this question.

And the answers are 7,300, 42, and 28.

Our calculations are likely correct because 42 plus 28 equals the total sample of 70.

There are times when you are given more information than is required to find the size for sample for one stratum.

The sample size for a stratum can be calculated from the information in a two-way table by taking all of the relevant information and creating your own ratio table.

For example, a stratified sample of 700 pupils of the sciences across two universities will be taken.

How many maths pupils from Oakfield Uni should there be in the sample from this population? As usual, make sure to include total rows and columns.

This is especially important in a two-way table where totals are required on both the final column and row, the information provided.

To construct our ratio table for maths pupils from Oakfield Uni, we only need to look out for the total population and total sample, and the population in the stratum that we care about.

In this case, math students from Oakfield Uni, of which there are 500.

We can then take this relevant information and place it into a ratio table that shows the stratum of maths from Oakfield Uni, and total pupils in both the population and sample.

500 pupils in the population are math pupils from Oakfield Uni, out of a 2,500 pupil population who does one of the sciences.

700 pupils are in the whole sample.

And we can use these three values and proportionality of a ratio table to get the total sample size of the stratum we want from a multiplier of 1/5, giving us 140 maths pupils from Oakfield Uni that are required for our sample.

For this last check, by finding the values of A, B, C, D, and E across both the two-way table and ratio table.

Pause here to find the sample size of hardback horror books.

One hardback horror book should be taken for the sample.

Okay, brilliant work so far.

Onto the final set of practise questions.

Pause here to complete the ratio tables for question one and two to find the stratified sample sizes for these two populations.

And pause here to either complete or construct your own ratio tables to answer these two stratified sampling questions.

And finally, question five.

By modifying the two-way table appropriately, and taking relevant information from the two-way table, construct a ratio table to find out how many Rowanwood customers who shop by delivery will be given the questionnaire for the sample.

Pause now to do this last question.

Great work in all of these challenging stratified sampling questions.

Pause here to check your answers to questions one and two to the ones on screen.

And pause here to check your calculations and answers for questions three and four.

And finally for question five.

71 customers who shop by delivery in Rowanwood will be given the questionnaire.

Pause here to check if your calculations match those on screen.

Once again, great effort in using ratio tables to your advantage to help you answer all of these questions in a lesson where we have considered that some bias samples are actually more effective at representing a population than a random sample.

We've looked at the definitions of strata and stratum, which are ways of grouping a population.

We use strata in order to collect a stratified sample, where each stratum has a sample size in the same proportion as in the population.

Ratio tables are great at calculating a stratified sample, and we can round decimal sample sizes to the nearest integer.

Thank you all for joining me, Mr. Grattan, in today's lesson and for all of the hard work that you have put in.

Until our next math lesson together, take care, and have an amazing rest of your day.