# 11.04 Sampling

Lesson

## Census vs Sample

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.

Remember!

In a census, every member of a population is surveyed. 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.

## 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. 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

Think of a pack of jelly beans. There are lots of different colours in the pack aren't there? Instead of considering them as a whole group of jellybeans, we could divide them up by colour into subgroups.

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.

For example, say we decide to survey $50$50 students to find out what types of music the students at our high school liked best. It is likely that Year $7$7 students may have a different taste in music to Year $12$12 students.

Here is a list of how many students are in each year and how we would calculate the number of students from each year we would need to survey to create a stratified sample:

School Year Number of Students Proportional Number for Sample
$7$7 $200$200 $\frac{200}{1000}\times50=10$2001000×50=10
$8$8 $180$180 $\frac{180}{1000}\times50=9$1801000×50=9
$9$9 $200$200 $\frac{200}{1000}\times50=10$2001000×50=10
$10$10 $140$140 $\frac{140}{1000}\times50=7$1401000×50=7
$11$11 $100$100 $\frac{100}{1000}\times50=5$1001000×50=5
$12$12 $180$180 $\frac{180}{1000}\times50=9$1801000×50=9
Total $1000$1000 $50$50

Remember!

No individual should fit into more than one subgroup, and no group of the total population should be excluded.

### Systematic sampling

If we use systematic sampling, we are basically picking every $n$nth item. From the sample, a starting point is chosen at random, and items are chosen at regular intervals. For example, we may choose every fifth name from a list or call every tenth business in the phone book.

The image to the left shows every $3$3rd person being picked.

#### Practice questions

##### Question 1

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

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

It is cheaper to conduct.

A

There will be no sampling bias.

B

It's more accurate.

C

It takes less time.

D

It is cheaper to conduct.

A

There will be no sampling bias.

B

It's more accurate.

C

It takes less time.

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

It's less accurate.

A

It takes more time.

B

It is more expensive to conduct.

C

There can be sampling bias.

D

It's less accurate.

A

It takes more time.

B

It is more expensive to conduct.

C

There can be sampling bias.

D
##### Question 2

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

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

Census

A

Sample

B

Census

A

Sample

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

Census

A

Sample

B

Census

A

Sample

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

Census

A

Sample

B

Census

A

Sample

B
##### Question 3

The local mayor wants to determine how people in her town feel about the new construction project. Select which type of sampling each scenario uses.

1. Selecting every $50$50th name from an alphabetical list of residents.

Stratified sampling

A

Systematic sampling

B

Convenience sampling

C

Simple random sampling

D

Stratified sampling

A

Systematic sampling

B

Convenience sampling

C

Simple random sampling

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

Stratified sampling

A

Systematic sampling

B

Convenience sampling

C

Simple random sampling

D

Stratified sampling

A

Systematic sampling

B

Convenience sampling

C

Simple random sampling

D
3. Selecting $10%$10% of the residents from each suburb.

Stratified sampling

A

Systematic sampling

B

Convenience sampling

C

Simple random sampling

D

Stratified sampling

A

Systematic sampling

B

Convenience sampling

C

Simple random sampling

D

### Outcomes

#### MA4-20SP

analyses single sets of data using measures of location, and range