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10.04 Samples from a population

Lesson

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.

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?

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 90Find \dfrac{1}{3} of primary students
\displaystyle =\displaystyle 30Evaluate
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 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.

a

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.

b

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.

c

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.

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.

a

Selecting every 10th person on the bus she catches.

Worked Solution
Create a strategy

Use the fact that an unbiased sampling method means everyone in the population has an equal chance of being selected.

Apply the idea

Not every student has an equal chance to be selected because the sample is limited only to the bus she catches. The sampling method is likely to be biased.

b

Selecting every 10th person on the student list.

Worked Solution
Apply the idea

Every student has an equal chance to be selected because she is doing a systematic sampling choosing every 10th on the student list at her school. The sampling method is not biased.

c

Selecting the first 50 students that arrive in the morning.

Worked Solution
Apply the idea

Every student does not have an equal chance to be selected because Irene gets only the earliest students in the morning. Students that arrive later will not be selected. The sampling method is likely to be biased.

d

Selecting by having a computer randomly choose student numbers.

Worked Solution
Apply the idea

Every student has an equal chance to be selected through a computer with their student numbers. The sampling method is not biased.

Idea summary

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}

Outcomes

VCMSP323

Investigate reports of surveys in digital media and elsewhere for information on how data were obtained to estimate population means and medians.

VCMSP324

Identify everyday questions and issues involving at least one numerical and at least one categorical variable, and collect data directly from secondary sources.

VCMSP326

Compare data displays using mean, median and range to describe and interpret numerical data sets in terms of location (centre) and spread.

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