# Sampling

## Topic Overview

In mathematics, a **'sample'** is a selection of values which have been taken from a larger group of data. By **'taking a sample'** of a set of data, you can reach conclusions about the larger group of data as a whole. There are several different types of sampling:

A **'random sample'** is when you randomly select a series of values from a larger group of data. These values will be chosen completely by chance with no clear reason or predictability. For example, if you were taking a random sample of the population you would choose values without paying any particular attention to categories such as age group or gender.

Alternatively, there is also a method known as **'stratified sampling'**. The term **'stratified'** comes from the Latin 'strata' which means 'layer'. Consequently, a stratified sample consists of several different layers of data. For example, if you were taking a stratified sample of the population you could select various layers of samples such as samples from different age groups. **With stratified sampling, the sample size for each layer is proportional to the size of the 'layer'.**

During your GCSE maths exam, you will encounter various different types of data. There are two main types of data; **'qualitative'** and **'quantitative'**.

**Qualitative data** is the term used to denote information which is descriptive. For example; whether someone has brown hair or blonde hair.

**Quantitative data** is the term used to denote information which is numerical. For example; 10 girls have brown hair and 12 girls have blonde hair.

When solving questions related to samples of quantitative data, you will encounter two different types of variable. These are known as **'discrete'** and **'continuous'** measures.

**A discrete variable is a type of measure which can only take certain values from a finite set**. For example, the amount of boys in a classroom is a discrete variable because it must be a whole number- you cannot have a fraction of a student!

Alternatively, **a continuous variable is a type of measure which can take any value within a range**. For example, the height of various boys in a classroom is a continuous variable because the values could be of varying amounts within the range of the students' heights.

## Key Concepts

In the new linear GCSE Maths paper, you will be required to solve various mathematical problems related to sampling. The specific questions you will be expected to answer will vary depending upon which examination board with which you are registered, but as a rule you will be required to:

- Understand the difference between random and stratified sampling
- Understand the difference between discrete and continuous variables
- Identify possible sources of bias
- Recognise that increasing the sample size will generally lead to better estimates of probability and population characteristics

Listed below are a series of summaries and worked examples to help you solidify your knowledge about various different types of sampling.

## Worked Examples

**1 - Bias and fairness**

During your GCSE maths exam, you will be required to take samples of various sets of data. When doing so, it is important to identify the accuracy of your sample. In order to do this correctly, you will need to understand the concepts of **'bias'** and **'fairness'**:

In mathematics, the term **'bias'** refers to a systematic error which causes the values of a sample to be inclined towards a certain group. As a result, the sample will produce an unfair verdict. For example;

*"Do you agree that Manchester United is the best football team?"*

This is a biased question because it pressures participants to agree with the person who is conducting the sample. It also does not provide an opportunity for participants to state the team which they believe is the best.

A **fair** sample question would be a question which enables participants to provide an accurate answer. For example:

*"Which football team do you think is the best?"*

This provides participants with the opportunity to provide multiple answers which represent their true opinions and which are not biased towards a particular category.

**2 - Tallying and grouping data**

When carrying out samples, you will be required to record large amounts of data. When doing so, it is helpful to use a **'tallying'** system in order to record your data more efficiently.

**Tallying is a method of recording values into groups of five**. Your values can be grouped as follows:

**Example**

(a) 52 pupils were asked to state their favourite colour. 12 students chose 'blue', 15 chose 'red', 7 chose 'yellow', 8 chose 'green' and 10 chose 'purple'. Represent this data in a tally chart and bar graph.

**Solution**

(a) When collecting large groups of data, it is useful to draw a 'tally chart'. This enables you to place a tally next to each value so that all of your records are grouped together as you go along:

Once you have collected all of your data, you can use the frequency column in order to display your sample as a chart:

**3 - Random sampling**

During your GCSE maths exam, you may be required to take a random sample of a set of data. When taking a random sample, this means that all of the data within the sample has an equal chance of being selected:

**Example**

(a) Out of 250 pupils in a school, 50 are asked what their favourite colour is. How should the sample of 50 pupils be chosen?

**Solution**

(a) In this case, it is not appropriate to ask every single pupil in your school what their favourite colour is. Therefore, in order to produce a fair sample of the favourite colours of pupils in your school, you should ask a random sample of people. In this example, your random sample size is 50.

In order for your sample to be completely random, you should assign each student a number ; from 001 all the way up to 250. You can then use a calculator to generate random numbers. Three-digit random numbers can then be used to choose 50 pupils.

**4 - Stratified sampling**

Stratified sampling is the process through which layers are used to take samples of various data. In order to carry out stratified sampling, you must calculate the sample size for each layer by using the following rule:

**Sample size for each layer = (size of whole sample ÷ size of population) × size of layer**

By using stratified sampling, you can produce a more accurate sample of a population:

**Example**

(a) In a school of 1000 pupils, 50 were asked what their favourite type of music was.

There are 180 pupils in Year 7, 200 in Year 8, 240 in Year 9, 220 in Year 10 and 160 in Year 11.

How would you create an accurate a sample for this survey?

* Solution*
(
a) In order to create an accurate sample for this survey, you would need to ask pupils across all of the different year groups what their favourite type of music was. For instance, students in Year 9 may prefer a certain type of music to students in Year 7. You must ask a broad spectrum of pupils of different ages in order to receive a fair sample of the entire school.

Therefore, you will need to calculate a suitable sample size for each year group. Using the values given in the question, you can use the stratified sampling rule to calculate the sample size for each layer:

**Sample size for each layer = (size of whole sample ÷ size of population) × size of layer**

Sample size for Year 7 = (50/1000) x 180 = 9 Sample size for Year 8 = (50/1000) x 200 = 10 Sample size for Year 9 = (50/1000) x 240 = 12 Sample size for Year 10 = (50/1000) x 220 = 11 Sample size for Year 11 = (50/1000) x 160 = 8

Once you have calculated these sample sizes for each layer, you can take random samples from each Year group.

To do so, you can allocate the numbers 000 to 199 to pupils, and then assign an equal number of random numbers to each pupil. For example;

pupil 000 is allotted the numbers: 000, 200, 400, 600 and 800, pupil 001 is allotted the numbers: 001, 201, 401, 601 and 801, all the way up to pupil 199 who is allotted the numbers: 199, 399, 599, 799, 999.

During your GCSE maths exam, you may also be asked to use the rule of stratified sampling to calculate how many values are within a particular sample layer:

**Example**

(b) 20 pupils in Year 9 are questioned for a survey. The total sample size of the survey is 100. There are 500 pupils in the school. Calculate how many pupils are in Year 9.

**Solution**

(b) Using the rules of stratified sampling, you can calculate how many pupils are in Year 9:

From the question, you know that there are 20 pupils in Year 9.

This can be represented as 20/100 of the total sample, which can be simplified to 1/5.

There are 500 pupils in total. Therefore, the amount of Year 9 pupils = 1/5 x 500

**The amount of Year 9 pupils = 100**

## Exam Tips

- Remember that a
**discrete variable can only take certain values from a finite set**and that a**continuous variable can take any value within a range.** - Memorise the stratified sampling rule that:
**Sample size for each layer = (size of whole sample ÷ size of population) × size of layer.** - When carrying out a sample, ensure that you have created a survey which is fair without bias towards a particular variable or sample layer.
- When tallying your data,
**double check that the total number of tallies which you have recorded is equal to the total sample size.**

## Topic Summary

Sampling is a very useful mathematical concept which can be used to accurately record and evaluate data. If you have a comprehensive understanding of how to carry out surveys without bias, you will be able to score maximum marks for this topic in your exam. What's more, the topic of sampling is also incredibly useful if you are revising for GCSE science topics which concern carrying out and evaluating surveys. Ultimately, you will need to be able to recognise the difference between:

- random and stratified sampling
- discrete and continuous variables
- qualitative and quantitative data

Therefore, you should attempt as many past paper questions as possible in order to improve your understanding of the different types of sampling as well as the different types of data. With practice and perseverance, you will soon be able to tackle all sampling questions with ease!

## Related Topics

- Probability Diagrams
- Charts and Tables
- Average and Spread
- Conditional Probability