Random Sampling Methods (College Board AP® Statistics)

A systematic sample is one in which you jump periodically (systematically) in regular intervals to select elements from the population to be in your sample

A systematic sample of size from a population of size uses intervals of size

• If this is a decimal, round to the nearest whole number

Randomly select a starting element in the first interval

• Then jump by the interval amount to get the other elements

  • If necessary, wrap back around the list again to keep going

No, a systematic sample is not a type of simple random sample (SRS)

• In an SRS, every possible sample of size is equally likely

• But in a systematic sample, two elements next to each other would never be selected

However, a systematic sample is still a type of 'random sample', as selecting the first element is done randomly

In a stratified sample, the population is first divided into distinct groups called strata (singular: stratum)

• The strata cover the whole population but are non-overlapping

  • The same element cannot appear in multiple strata

Elements are chosen for the sample using simple random sampling from each strata in the population

• This can either be done proportionately:

  • where the number of elements from each stratum in the sample correspond to the proportion of the population in that stratum

  • A possible formula is shown below

  

• Or it can be done disproportionately:

  • where the number of elements from each stratum does not need to correspond to the proportions in the population

  • e.g. randomly selecting 20 elements from each strata

If the population can be split into obvious non-overlapping groups then a stratified sample will always be more representative of the population structure than a simple random sample

• Elements from every group are guaranteed to be included in the sample

• This means a sample cannot all come from, say, the group with the lowest values

  • A stratified sample is more balanced across the groups

  • This makes any estimates of population parameters less extreme (more precise)

  • So there is less variability in estimates of population parameters

However, if the sample size is very small, it may not be worth the time to split into groups

• A simple random sample would be quicker

In a cluster sample, the population is first divided into groups called clusters

• Within each cluster there should be a good spread of different elements

  • Different clusters should have similar overall compositions

Then a simple random sample is used to randomly select one or more clusters out of the possible clusters

• All elements in the selected clusters are then used in the sample

Cluster sampling means using all elements from some randomly selected groups in the population

Whereas stratified sampling means using some elements from all groups in the population

Stratified samples are more representative of the population than cluster samples

• But cluster samples can be faster and cheaper than stratified samples

The groupings used for cluster samples should have a good mix of different elements (heterogeneity)

• whereas the groupings used for stratified samples should contain similar type of elements (homogeneity)