Population & Sampling (Edexcel GCSE Statistics)

Population & Sample Types

What are populations, samples and sampling frames?

• The population refers to the whole set of things which you are interested in

  • e.g. if a vet wanted to know how long a typical French bulldog sleeps for in a day

    • then the population would be all the French bulldogs in the world

  • Be careful - the word 'population' can mean different things in different contexts

    • e.g. 'the population of the UK' is usually used to refer to everyone in the UK

    • But if you're studying UK dentists then the 'population' for your study would be restricted to all the dentists in the UK

• A sample refers to a subset of the population which is used to collect data from

  • e.g. out of all the French bulldogs in the world (the population)

    • a vet might take a sample of French bulldogs from different cities and record how long they sleep in a day

• A sampling frame (or sample frame) is a list of all members of the population

  • For example, a list of employees’ names within a company

  • Not every population will have an easily-accessible sampling frame

What's the difference between a census and a sample?

• A census collects data about all the members of a population

  • e.g. the government in the UK does a national census every 10 years to collect data about every person living in England at the time

  • The main advantage of a census is that it gives fully accurate results

  • The disadvantages of a census are:

    • It is time consuming and expensive to carry out

    • It can destroy or use up all the members of a population (imagine a company testing every single firework it produces)

• Sampling is used to collect data from a subset of the population

  • The advantages of sampling are:

    • It is quicker and cheaper than a census

    • It leads to less data needing to be analysed

  • The disadvantages of sampling are:

    • It might not represent the population accurately

    • It could introduce bias, if some parts of the population are more represented in the sample than others

What different sampling techniques do I need to know?

Random sampling methods

• Simple random sampling: here every member of the population has an equal probability of being selected for the sample

  • To select a simple random sample of members of the population

    • Uniquely number every member of the population

    • Then randomly select  different numbers using a random number generator (or other form of random selection)

• Stratified sampling: the population is divided into separate groups (called strata) and then a random sample is taken from each group (stratum)

  • The proportion of a sample that belongs to a stratum is equal to the proportion of the population as a whole that belongs to that stratum

    • e.g. if 1/20 of the population belongs to a particular stratum

    • then 1/20 of the sample should come from that stratum

  • A population could be split into strata by age ranges, gender, occupation, etc.

• See the spec points on 'Random Samples' and 'Stratified Samples' for more info on these two methods

Non-random sampling methods
Note: some of these methods include random elements, but the samples as a whole are not random

• Judgement sampling: here you simply use your judgement to choose a sample of the population

  • You should attempt to make sure that the sample is representative of the population as a whole

• Opportunity (convenience) sampling: a sample is formed using available members of the population who fit the study criteria

  • e.g. for a study of UK consumers you could stand on a street corner and interview the first 50 people who walk by

• Cluster sampling: the population is divided into sensible 'clusters' and then a number of clusters are chosen at random to form the sample

  • e.g. a study of UK education might use schools as the clusters

  • then select 50 schools at random and use the people in those schools as the sample

• Systematic sampling: a sample is formed by choosing members of a population at regular intervals using a list (sampling frame)

  • e.g. to select 1/10 of the students in a school as a sample

    • Start with a list of all students

    • Select one student at random as a 'starting point'

    • Then also select every 10th student on the list after that starting point

    • (If necessary, wrap back around to the start of the list when you get to the end)

• Quota sampling: the population is split into groups (like in stratified sampling) and a quota is specified for each group

  • The quota specifies how many members of the population are to be selected from each group

    • This will often be done in the same way as selecting the sizes of the strata in stratified sampling

    • Or other criteria could be used to set the quota for each group

  • Members of the population are selected until each quota is filled

    • If a member does not want to be included then another member is chosen instead

  • The members do not have to be selected randomly

What are the advantages and disadvantages of different sampling techniques?

• In general

  • Most sampling techniques can be improved by taking a larger sample

  • You want to minimise the bias within a sample

    • This occurs when the sample is not representative of the population

    • The best way to do avoid bias (when possible) is to use a random method

  • Sometimes the 'best' method would cost too much or take too much time

    • So you need to choose the 'best method you can afford (or have the time for)'

  • A sample only gives information about the members in the sample

    • A different sample from the same population could lead to different conclusions about the population!

• Simple random sampling:

  • This is the best sampling method for avoiding bias

    • Although it is possible that members of some groups in the population will not be represented in the sample

    • To avoid this stratified sampling can be used instead

  • Most useful when you have a small population or want a small sample

    • e.g. children in a class

  • This cannot be used if it is not possible to number or list all the members of the population

    • e.g. the fish in a lake

• Stratified sampling:

  • This should be used when the population can be split into obvious groups

  • Useful when there are very different groups of members within a population

  • The sample will be representative of the population structure

    • Members of every group (stratum) are guaranteed to be included in the sample

  • The members selected from each stratum are chosen randomly

    • This helps to avoid bias

  • This cannot be used

    • if the population cannot be split into groups

    • or if the groups overlap

• Systematic sampling:

  • This is useful when you want a sample from a large population

  • You need access to a sampling frame (list of the population)

    • If the order of the sampling frame is random then the sample will also be random

  • This cannot be used if it is not possible to number or list all the members of the population

    • e.g. penguins in Antarctica

  • Be careful of periodic (i.e. regularly recurring) patterns in the sampling frame

    • e.g. a list of names where the names are grouped by 5-person teams with the team captain appearing first

    • If you selected every 5th name in the list you would end up with either all captains or no captains in your sample

• Quota sampling:

  • This is useful when a small sample is needed to be representative of the population structure

  • Useful when collecting data by asking people who walk past you in a public place or when a sampling frame is not available

    • Just keep asking people until the quota is filled for each group

  • This can introduce bias as some members of the population might choose not to be included in the sample

• Cluster sampling:

  • This will usually require less time and be less expensive than simple random sampling or stratified sampling

    • e.g. if your clusters are schools, you will only need to collect data from the people in some of those schools

    • instead of having to collect data from a few people in every school in the country

  • However the clusters may not be representative of the population structure as a whole

    • This can make the sample biased

• Opportunity (convenience) sampling:

  • This should be used when a sample is needed quickly

  • Useful when a list of the population is not possible

  • But the sample is unlikely to be representative of the population structure

    • This can make the sample biased

• Judgement sampling:

  • This can be used when a sample is needed quickly

  • The person choosing the sample should try to make it representative of the population

    • But intentionally or unintentionally the sample can end up being biased

    • Therefore this is rarely a preferred method

Random Samples

What do I need to know about random sampling?

• In a simple random sample every member of the population has an equal probability of being selected for the sample

  • This means that the sample selection is fair and unbiased

    • Therefore the sample is likely to be representative of the population

  • To minimise bias this will usually be the best method

    • But it can also be expensive and time-consuming

    • And some groups in the population may end up not being represented in the sample

• Some other sampling methods also include random selection

  • In stratified sampling, the members of the population chosen from each stratum (group) are chosen randomly

    • A 'simple random sample' is taken from each stratum

    • This leads to relatively unbiased samples that also reflect the population structure

  • In cluster sampling, the clusters to include in the sample are selected randomly

    • This can give good results if the clusters are representative of the population as a whole

  • In systematic sampling the 'starting point' member in the population list is chosen randomly

    • This is not considered a 'random sample' unless the ordering of the list is also random

How is a random sample selected?

• To take a simple random sample you need to have access to a list of all members of the population (i.e. a sampling frame)

  • Every member in the sampling frame must be assigned a number

    • Usually this will mean starting at 1

    • and numbering the rest of the list in order: 2, 3, 4, etc.

• To select a random sample of members of the population

  • random numbers must be generated

  • The members in the list with those numbers are then selected for the sample
    

• You should be familiar with the different options for choosing random numbers

• Random numbers can be selected from a random number table

  • The numbers in the table may have more digits than you need

    • e.g. you want 2-digit numbers but the table shows
      469066      155387      172419      953505

  • In this case you can break the table numbers into smaller numbers

    • So here read the numbers in the table as
      46    90    66      15    53    87      17    24    19      95    35    05

    • '05' is just a two-digit way of writing '5'

• You could use a random number generator on a calculator

  • This may give you random 3-digit decimals between 0 and 1

    • e.g. 0.541, 0.414, 0.929

    • These can be multiplied by 1000 to give you integer answers: 541, 414, 929

  • Or you may be able to ask for random integers between two values

    • e.g. a random integer between 1 and 6

    • This would be just like rolling a fair 6-sided dice

• Apps on a computer or online can also generate random numbers

  • These apps usually let you specify what you want

    • how many numbers

    • between what values

• You can select random numbers by rolling dice

  • A fair 10-sided dice can give values between 0 and 9

  • Use two 10-sided dice (or roll one dice twice) to get numbers between 0 and 99

    • i.e., one dice for the tens (10, 20, 30, ...) and one dice for the units (1, 2, 3, ...)

  • Or use three 10-sided dice to get numbers between 0 and 999

    • i.e., one for hundreds, one for tens, and one for units

• You could also put all the numbers into a hat (or bag, etc.)

  • And draw numbers out at random

    • This is less easy to do with a lot of numbers!

  • Similarly, numbers might be drawn at random from a deck of cards with the numbers written on them

• You should know how to deal with problems that occur when choosing random numbers

• You may get a random number that does not match any of the items in your list

  • e.g. if you have 80 items in a list numbered 1 to 80

    • but get the random number 93

  • If this happens, simply ignore any numbers that don't match

    • and keep generating random numbers until you have enough that do match items in the list

• You may get a random number that occurs more than once

  • In this case keep the first version of the number

    • and ignore any repeated versions

  • Keep generating random numbers until you have enough unique numbers

    • i.e. ones that only occur one time each

Stratified Samples

How is a stratified sample selected?

• To take a stratified sample, the population must first be divided into a number of groups (strata)

  • Every member of the population must belong to one group

  • No member of the population can belong to more than one group

    • i.e. the groups cannot overlap

  • The strata could be based on age ranges, gender, occupation, etc.

• The number of members chosen from each stratum corresponds to the proportion of the population that belongs to that stratum

  • e.g. if 1/20 of the population belongs to a particular stratum

    • then 1/20 of the sample will be chosen from that stratum

• To find the number to be chosen from each stratum use the formula:

  • 

    • i.e. divide the size of the stratum by the size of the population

    • then multiply by the size of the sample

    • the size of the stratum just means how many total members are in the stratum

• Once you know how many members to choose from each stratum

  • those members should be chosen randomly from all the members of the stratum

  • i.e. take a 'simple random sample' of the correct size from each stratum

How do I choose a stratified sample based on more than one category?

• It is possible that the strata to be used will be based on more than one category

• For example the 900 people working for a large company

  • could be divided into managers and employees

  • but could also be divided according to whether they usually walk or bike to work, drive to work, or use public transport

• In a case like this each stratum for a sample could be based on two categories

  • e.g. 'employees who drive to work', 'managers who use public transport', etc.

• Once the strata have been decided, find the number in each stratum in the usual way

  • e.g. if you want a total sample of 100 people (out of the 900 people in the table)

    • the number of employees who walk or bike in the sample would be

      

    • the number of managers who walk or bike in the sample would be

      

    • etc.