Understanding Populations and Sampling in Statistics

okay so we were talking about populations and samples what you're going to see in stats is

that we're going to be looking at a lot of different types of sampling and we need to know all of these different

things about it so i've said at the top we'll be investigating different types of sampling what they are

how to carry them out and their advantages and disadvantages and actually at the back of your

booklets on the very back page i've included a blank version of this for you to use to test yourself during

revision because these are a lot of facts that you need to know and it's a bit weird most of the stuff

in maths is about skills but there's some stuff at the beginning of statistics it's like you need to know

these things off by heart and you'll see i've divided this into random sampling and then some non-random

sampling that we have at the bottom so we're just going to concentrate now on looking at these three which are the

three types of random sampling so random sampling and there are three types of random sampling

here i've got a population and i've sort of shown that this population here i've

tried to represent the fact that there are all different kinds of things in this population

i've got reds greens yellows oranges plain ones and the random sampling the sample that i'm taking are these ones

that i've put a box around and these ones that i've put a box around have been selected

randomly so this is the definition of something being randomly sampled where we want each sampling unit each of

these smiley faces in our sampling frame which is possibly our list of things that we've got you

have to have a sampling frame to do this to have an equal chance of being chosen in order to avoid bias

so you're not selecting them yourself which may have bias they're just getting randomly selected for you

people often don't really realize this is the definition they need to have an equal chance

each one has an equal chance of being chosen so this is known as a rand this is known as random sampling and there

are a few different ways of doing this and we're going to look at the three ways that you could go about doing this

so the first one is called a simple random sample and a simple random sampling

so this is all written on your table it's all actually written on the next few pages for you

and then at the end you'll see i've also got it written up here so you don't need to do any of the writing down okay so

i've got them all written in the next few slides for you so for simple random sampling what it is

well simple random something says every sampling unit in the sampling frame has

an equal chance of being selected that's what we've just talked about now how you carry it out in your

sampling frame each item is given an identifying number and then you can use a random

number generator or you can use lottery sampling which is names in a hat so you need to have a sampling frame to

do a random sample you would assign everything a number you'd use a random number generator

that would pull out all of those different numbers for you if a random number comes out twice

you just put the number and you just try another one again and this lottery sampling means names in

a hat you've seen you know in hunger games where they have that giant bowl with all the names in it and they pick

out the tributes that's an example of lottery sampling because they've literally got all of the

names in a in a bowl so you don't necessarily need them to be numbered but they need

to be then identified in a way that you can pull them out of a hat as they normally say so the advantages of

this simple random sample is that it's bias free it's easy and it's

cheap to implement and each number has a known equal chance of being selected because we're using a random number

generator the disadvantages though that it's not suitable when the population size

is large and that a sampling frame is needed you can't do a random sample if you don't have the list of things

you're trying to do the sampling from because how would you assign the numbers to those things

because it's long for this one the population size so it's more because if you have a large

population you probably want it there's not there's going to be too many things on the list

it's going to be too difficult to pick out the ones that you need and often when there's a large

population you may want to do something called stratified sampling where you want to reflect the population a bit

a bit more clearly so for example i'm just going to show you one with a mark scheme so we can see where the

marks actually come from it says there are 64 girls and 56 boys in a school explain briefly how you could take a

random sample of 15 pupils using a simple random sample so this is three marks it's got

actually have quite a bit of description here so the first mark that you get is it says allocate a number

between one and n or equivalent to each pupil so in this case there are 120

pupils so you'd need to allocate a number from 1 to 120 for these pupils then use a random number table or a

computer or a calculator to select 15 different numbers between 1 and 120

notice it has to have the word different in there because if you pick the same random number

you'd be picking the same pupil and then the last thing is it says pupils corresponding to these numbers

becomes the sample i know it kind of all makes sense but it's just making sure you'd get all three marks there

so you get the first mark for allocating an identifier to each sampling unit you'd get a mark for one bias-free

method to select such a number and a mark for explicitly mentioning how that number is actually used

alternatively if you didn't want to do it with a numbering system you could say get get a list of all of the names of

the pupils put them written on pieces of paper into a hat

and select 15 names out of the hat and that would also correspond to that that would be a

lottery sample so the next type of one is called systematic

sampling so systematic sampling says that the required elements are chosen

at regular intervals in an ordered list and i'll demonstrate what that means in just a second so how do you carry it out

you take every case elements where k is the population size divided by the sample size i'll explain what that means

in just a second and you started a random item between 1 and k so let's say that we could talk

about it with this population that i've got here how many how many um sampling units have i got in

this population uh i've got 50 yep because there's 10 across

and there's 5 down the side here so let's say that i wanted to take a sample of 10 okay i'm going to try and take a

sample size of 10. so my sample size little n is going to be 10. my

population size capital n is going to be 50. and it says you're going to take

every k element well this kind of makes sense if you've got you want to take 10 out of 50

you will be taking every well k in this case would be the fifth one because you've got 50

divided by 10 so you'll be taking every fifth one and it says you started a random

item between one and k so you need to start at a random one between

one and five so i'm going to select a number randomly between one and five i'm just going to say two and pretend

i've done it on a random number calculator there and i'm going to imagine that my list is

going one two three four five six etc as we go all the way across here okay

so this tells me you're going to take every fifth one i picked a random number between one and five and i picked

two so i would start off by taking two and i'd go one two three four five one two three four five one two three

four five one two three four and you can see if i'm picking

every fifth one i would have this as my systematic sample i had them in a list i decided i was going to pick every

fifth one because i did the population divided by the sample and i decided that i was going to

randomly start at the second one because i used a random number generator which i use my mind which is not a

random number generator but you could use a calculator and we started it at two and you'll see

how that list went like that so this isn't random in the sense of them just being like pulled out

but because they're not in a particular order we can just take them from that list using a

systematic sample and the reason it's called cis it could have been any number from one

to five so if it was if i'd picked the number four it would have ended up being these ones

and these ones here and if i picked a different sample size it may have looked

like a different kind of pattern it just happened to be this pattern that i'd got here and the reason it's called

a systematic sample is because you should notice there's that word system at the beginning and system means

following a process so the random sample we had before was like just pulling them out of a hat

whereas this time we're following a strict system and the strict system is you pick everyone at a fixed interval

with a fixed gap that we've got here so what are the advantages the advantage is that it's simple

and that it's quick to use and it's suitable for large samples and populations

it's just an easy way of just you're not having to do a random number for every single one

you can just quickly grab them and say i'm just going to pick this this this and this

the disadvantage is that you do need a sampling frame again for this okay you need to have a list

to be able to even select them from and it can introduce bias if the sampling frame is not random

if these were ordered in a particular way i might end up selecting ones that are going to be

biased so you've got to be careful because you want this original list to be in a random order so here is an

example of how it comes up in an exam and it says a telephone directory contains 50 000

names so what is this telephone directory what would we call the telephone directory

without this language that we've got is it the population yeah yeah but i i don't want to know

specifically what is this this is this is the population and the the directory itself

is the sampling frame because it's the list of all the things that we need the population are all of

their names the directory is the sampling frame and the individual names are going to be the

sampling units so it says a researcher wishes to select a systematic sample of a hundred names

from the directory explain in detail how the researchers should obtain such a sample

wouldn't what sorry no the hundred names will be the sample the sampling frame is the is the the

original list of things that you've got yeah so in this case the population is 50 000 so it's 50 000

and we're dividing it by the sample size which is 100 so it's just going to be every 500 names

is what you would want to select and so here they've said randomly select a number between 0 and 499 or between 1

and 500 to find out where you will start and then you select every 500th person so if

you randomly picked the number was 17 you would then pick 517 and then 1017 until you had picked

100 names from that list okay so this is what we call uh systematic sampling and the most

important bit for that extra mark is you need to decide where you're going to randomly start you can't just say pick

every 500th person you also need to instruct them where to start on that as well

so we're going to look at the third type and this is one that i think comes up in gcse and this one kind of makes a bit

more sense here stratified sampling so with stratified sampling

we want to in this example i've got here i want to sample 20 percent of the population

if the population were divided into distinct groups e.g age ranges which are known as

strata hence it being called a stratified sample so a strata is a distinct group what we could do is

we could randomly sample 20 from each of those groups ensuring that each group is equally represented

so in this particular population that i've got here the groups of my sampling units

are different colors okay so i've got these blue units they might not be representing people they might be

representing a product or whatever it might be and so i've got these blue ones orange

ones green ones and red ones and i want to pick 20 of them so you'll notice

out of this group of what's that five times well out of 20 i've picked one two three four of them

at random and then out of the orange group where there's 10 of them 20 of 10 is two so

i've picked two of them out of this group of 15 i've picked three of them to keep it in the same

proportion and out of this last group of five i've picked 20 of them which is just one of

them it's usually done with people this is the stratified sampling is the one that

when we talk about political polling when they will say um you know how do you think people are going to vote in

like the american election that's the one that's done usually by stratified sampling because it's going

to represent the population in the best kind of way so it's still random because within the

strata you're picking something randomly so the type is called stratified sampling what

it is the population is divided into groups which are called strata

and a simple random sample is carried out in each group so a stratified sample also is a simple

random sample inside it it's used when the sample is large and the population naturally

divides into groups often if we were going to do like a survey of the school we would

probably want to survey year sevens year eights or nines or tens etc

and they are the natural strata that we have so how do we carry it out we need the same proportion

which is the sample size divided by the population size sampled from each strata so in this case

the sample size that i think we would have wanted is one two three four five six seven eight nine

ten we needed a sample size of ten out of the population size of 50 so we needed 20 percent from each group

now the advantages of this is that it reflects the population structure which is great because if it reflects

the population structure it's going to guarantee proportional representation of groups within the

population and that really should give you a better picture of of what the population thinks

about something the disadvantages though is that the population must be clearly classified

into distinct strata so you'd need to have groups that you want to put them into

now the selection within each stratum which again is about a group it suffers from the same disadvantages

as simple random sampling so when you're then doing simple random sampling within here

you're still having those disadvantages because you're actually doing a simple random sample

within this one that we have so on the next page i'm going to break down uh quite a big

question here it's seven marks because there's so many steps that you would need to explain to do

and then we'll do a couple of examples ourself and then we'll do some practice so it says here that a school has got 15

classes and a sixth form in each class there are 30 students

in the sixth form there are 150 students there are equal numbers of boys and girls in each class

and there are equal numbers of boys and girls in the sixth form the head teacher wishes to obtain the

opinions of the students about school uniforms explain how the head teacher would take

a stratified sample of size 40. so they've already told us that the sample size which is little

n is 40. the first thing we would do is we would work out the population size the population of this school it

says that there are 15 classes of 30 students so that's why they've done 15 times 30

and they've added 150 for the sixth form because it says the sixth form has got 150 students

so that's why there's a total of 600 so we know that the proportion or the percentage

the proportion from each strata is going to be 40 out of 600 which simplifies to 4 out of 60

or 2 out of 30 or 1 out of 15. so that's really what they're doing in here

they're working out the random sample size they've got the 40 in the 600s and here and multiplying by 30. it's the

same thing as what we're doing so you need to pick two at one person out of every 15

that there is so from the sixth form um let's say how many we would need so for the

classes there are 15 classes with 30 students in and there is half boys and half girls so

how many would we need from each class how many boys from each class and how many girls from each class

we need one boy because there are 15 uh 15 people 15 boys in a class and 15 girls in a class

so we need one boy and one girl from each of the 15 classes and then from the sixth form we're going

to have 40 out of 600 which we've said is 1 out of 15 we need a 15

of the 150 people and the 15th of 150 is 10. so we need 10 from the sixth form and we need five boys and five girls

so let's just check that that's going to be the right amount of people we need one boy and one girl from each of the 15

classes that's 30 people 35 40 people that's going to hit our sample of

of the 40 people that we need so what we've said here then you need to get these last marks

for doing the simple random sample it says label the boys in each class from 1 to 15 and the girls from 1 to 15

use a random number to select one girl and one boy from that list you also need to label the boys and the

girls in the sixth form from 1 to 75 because there's 75 of them and you need to use a random number generator to

select five different boys and five different girls

from that list of 75 people that we've got there so i don't think they're ever gonna do a

seven mark question like this this is from an old exam but it's just worth thinking there's

quite a lot of stages that you have to do there you think about the different strata you

think about the proportions and then you describe how to do the simple random sample

so we're going to just do two examples here and then we're going to practice exercise 1b so this first one says that

their hundred members of a yacht club are listed alphabetically in the club's membership book

the committee wants to select a sample of 12 members to fill in a questionnaire

explain how the committee could use a calculator or random number generator to take a simple random sample of the

members so that's the first one that we had a look at here what would be the first

thing that we might need to do here if we're going to do if we wanted to use a calculator or a random

number generator for these members of the yacht club yeah what would i need to do if i had

these 100 members of a yacht club as a list and i wanted to select 12 of them

great so the first thing i'm going to need to do is i will assign a number

from 1 to 100 to each member and then the next thing that i would do

yep so i will generate 12 random numbers i should say 12 different random numbers 12 different random numbers

and select the corresponding people so that's using a random number generator then it says explain how the

committee could use a lottery sample to take a simple random sample of the members so what did we say the lottery

sample was with names in a hat that hunger games like big bowl of names okay so what you

would do is you would write all 100 names on pieces of paper

and place into a hat doesn't have to be a hat though or a hat or a bowl

and then draw 12 names these are your sample i know this is uh it doesn't really feel

like maths but it's like the build-up to what we'll be doing in stats um well they they definitely will be

different because once you've taken them out the hat you won't be putting them back in again

so the next one we're going to do it says the factory manager wants to find out what his workers think about the

factory canteen facilities the manager decides to give a questionnaire to a sample

of 80 workers so that's my sample size of 80. it is thought that different age groups

will have different opinions there are 75 workers between 18 and 32 140 between 33 and 47

and 85 workers between 48 and 62. write down the name of the method of sampling the manager should use

what method should he use why should he use stratified yeah and it's because actually in this

question it is thought that different age groups will have different opinions

so that's why that's the sentence that tells me it's a stratified sample explain how he could use this method to

select a sample of workers opinions so the proportion that we're going to look for will be the sample size

which is 80 divided by the population size so we need to quickly work out what the population size is here

by adding up all of the different what is it we've got the factory is it 300 yeah 75

1 40 and 85 is 300. so the proportion from each group is going to be 80 out of 30.

80 out of 300 which is 4 out of 15. so from the 18 to 32 year olds

we're going to need to do 4 out of 15 multiplied by 75 which is 20 for the

what's the next group 33 to 47 we need to do 4 15 of 140 which is 37.3

recurring so how many people should we pick so we're going to pick 37.

and then for our last group we've got 48 to 62 so it's going to be 4 15 of 85

which is 22.6 so we're going to pick 27. we're going to round it but what you need to do

is we need to check actually to see if this is all rounded up oh pick 27 23. we just need to check that

these things do add up to the correct sample size because sometimes when you do rounding you might get one more or

one less so we've got 20 37 and 23 yet they do add up to 80 that we've got there

so that's all good now to explain that last part we're gonna then

number the workers in each group and use a number generator to select required numbers from each

group and then after that give the questionnaire

to those selected okay so that is the three types of random sampling these are easy marks to

get but only if you've memorized the stuff you know how to do it so we're gonna have a look at exercise

1b um and it's just going to talk about the advantages and disadvantages of these things so use the stuff that we've

written so far and it's just good practice to see that we can get all of these ones right okay