Introduction
Businesses, organisations and governments all gather data and conduct surveys to help them make decisions about what people want. The Australian Census, which is conducted by the Australian Bureau of Statistics, is an example of a large-scale data collection. Every Australian citizen is required to fill in a survey so we get a picture of the characteristics of the Australian population. In a census, every member of a population is questioned. In maths, a population does not necessarily refer to the population of a country. It just means every member of a group. It may be a school's population, a sports club's population and so on.
Ideas
Sampling techniques
In a census, every member of a population is questioned. In an unbiased sample, a representative proportion of the population is surveyed.
If you can survey every member of a population, it is the best way to gather information. However, sometimes it is impractical or way too expensive. So sometimes it's better to take a sample that is representative of the wider population.
The most important thing when taking a sample is that it is representative of the population. In other words, we want to try and ensure there is no bias that may affect our results. There are different ways to collect a sample. We'll go through some of them now.
Random sampling
An example of random sampling is numbers being drawn out in the lottery. Every number has an equal probability of being chosen. Each individual is chosen at random (by chance). In other words, each individual has the same probability of being chosen.
Stratified sampling
Stratification is the process of dividing a group into subgroups with the same characteristics before we draw our random sample. Then we look at the size of each subgroup as a fraction of the total population. The number of items from each subgroup that are included in the sample should be in the same ratio as the amount they represent of the total population.
Systematic sampling
To use systematic sampling, a starting point is chosen at random, and then items are chosen at regular intervals. Such as selecting every nth item from a list. For example, we may call every tenth business in the phone book or select every fifth bottle from a production line.
Examples
Example 1
In a group of 360 students, 90 are primary students and 270 are secondary students. A stratified sample of 120 is to be selected from the group based on year level.
How many primary students should be selected?
Example 2
The local mayor wants to determine how people in her town feel about the new construction project. Determine the type of sampling each of the following scenarios describe.
Selecting every 50th name from an alphabetical list of residents.
Giving each resident a random number between 1 and 10 and then selecting everyone with the number 3.
Selecting 10\% of the residents from each suburb.
Example 3
Irene is interested in which students from her school catch public transport. Select whether the following sampling methods are likely to be biased or not.
Selecting every 10th person on the bus she catches.
Selecting every 10th person on the student list.
Selecting the first 50 students that arrive in the morning.
Selecting by having a computer randomly choose student numbers.
In a census, every member of a population is questioned. In an unbiased sample, a representative proportion of the population is surveyed.
The most important thing when taking a sample is that it is representative of the population. Different sampling techniques aim to obtain a representative sample, but some may be more practical to carry out in different scenarios.
Sampling methods:
Random - selects people through a purely chance selection
Systematic - selects people at regular intervals on an ordered list
Stratified - selects a proportional amount of people from the different strata in a population
In general, we can calculate the number required for a given subgroup in a stratified sample using: \dfrac{\text{Number in subgroup}}{\text{Total number in population}}\times \text{sample size}