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. This lesson will explore techniques for collecting data as well as the practicalities and implications of obtaining data through sampling.

Census vs sample

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.

Collecting data from every member of a population is the most accurate way of gathering information, but it is not always the most practical and can be very expensive or time consuming. Typically, a sample survey is instead done on a subset of the population to make it quicker and less expensive.

Examples

Example 1

Hannah has chosen to collect information using a sample instead of a census.

What are the advantages to doing a sample? Select all that apply.

It is cheaper to conduct.

There will be no sampling bias.

It's more accurate.

It takes less time.

Worked Solution

Create a strategy

Use the fact that samples require asking less people than a census.

Apply the idea

Since a sample requires surveying less people, it would be cheaper to conduct and it would take less time.

The correct options are A and D.

What are the disadvantages to doing a sample? Select all that apply.

It's less accurate.

It takes more time.

It's more expensive to conduct.

There can be sampling bias.

Worked Solution

Create a strategy

Use the fact that samples do not survey the whole population being considered.

Apply the idea

Because the whole population is not being surveyed, the results of a sample would be less accurate and there could be sampling bias depending on how the sample is collected.

The correct options are A and D.

Example 2

For each of the following, determine whether they are a census or a sample.

Lucy has asked everyone in her office what snacks should be provided in the office.

Worked Solution

Create a strategy

Determine the population of interest for each scenario. Then decide if everyone in the population has been surveyed or only a smaller subset of the population has been surveyed.

Apply the idea

The population in this example are the people in Lucy's office. Lucy is receiving information about every person in this population, so she has conducted a census.

James asks a few of his friends how they did in the test to see if he is above average in his class.

Worked Solution

Apply the idea

The population in this example are all the students in James's class. James only asks a few friends, so he has conducted a sample survey.

Joanne finds the height of the entire class to try to find the average height of 15 year old students in Australia.

Worked Solution

Apply the idea

The population in this example are all 15 year old students in Australia. Joanne only collects information from her class, so she has conducted a sample survey.

Idea summary

In a census, every member of a population is surveyed. In an unbiased sample, a representative proportion of the population is surveyed.

Sampling techniques

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. Let's take a closer look at 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 3

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?

Worked Solution

Create a strategy

Find the proportion of primary students that is equal to the proportion of total students selected.

Apply the idea

The proportion of total students selected is \dfrac{120}{360}=\dfrac{1}{3}. So \dfrac{1}{3} of the primary students should be selected.

\displaystyle \text{Selected primary}	\displaystyle =	\displaystyle \dfrac{1}{3} \times 90	Find \dfrac{1}{3} of primary students
	\displaystyle =	\displaystyle 30	Evaluate

Reflect and check

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}

For a stratified sample no individual should fit into more than one subgroup, and no group of the total population should be excluded.

Example 4

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.

Worked Solution

Apply the idea

The scenario describes selecting people at intervals of 50 from a list - this is systematic sampling.

Giving each resident a random number between 1 and 10 and then selecting everyone with the number 3.

Worked Solution

Create a strategy

Compare each scenario to the following descriptions of 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

Apply the idea

The scenario describes allocating residents random numbers, each resident has an equal chance of being selected - this is random sampling.

Selecting 10\% of the residents from each suburb.

Worked Solution

Apply the idea

The scenario describes first grouping the residents by suburb and then taking proportionally from each group - this is stratified sampling.

Idea summary

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

Outcomes

ACMSP284

Investigate techniques for collecting data, including census, sampling and observation

ACMSP206

Explore the practicalities and implications of obtaining data through sampling using a variety of investigative processes

ACMSP293

Explore the variation of means and proportions of random samples drawn from the same population

10.04 Sampling

Introduction

Ideas

Census vs sample

Examples

Example 1

Example 2

Sampling techniques

Examples

Example 3

Example 4

Outcomes

ACMSP284

ACMSP206

ACMSP293

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