1. Dr J Frost (jfrost@tiffin.kingston.sch.uk) www.drfrostmaths.com
GCSE: Sampling
Dr J Frost
Objective from GCSE9-1 specification:
Last modified: 22nd December 2018

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3. Starter
Naomi wants to find out how often people go to the cinema. She uses this question on a questionnaire.
“How many times do you go to the cinema?”
 Not very often  Sometimes
A lot
Identify three things wrong with the question. ?
There is no time frame in the question. Does she mean per week, per month?
The options are not exhaustive. What if they don’t go to the cinema at all?
The options are subjective (even if there was a time frame) – ‘sometimes’ means different things to different people. ? ?
Teacher note: ‘Questionnaires’ has been removed from the new GCSE. This starter question just aims to identify some problems potentially encountered with sampling.

4. Populations vs Samples!
A population is: the whole set of items that are of interest.
A sample is: some group within the population intended to represent the population. ? ?
You’re probably used to a ‘population’ meaning all humans/animals within a country/ecosystem. But a population could be “all the lightbulbs in a factory” or “all the cars in the UK”.

5. Populations vs Samples
We could collect data either from a sample, or from the entire population.
Data collected from the entire population is known as a census. ?
Advantages
Disadvantages
Census
Should give completely accurate result.
Time consuming and expensive.
Can not be used when testing involves destruction.
Large volume of data to process.
Sample
Cheaper.
Quicker.
Less data to process.
Data may not be accurate.
Data may not be large enough to represent small sub-groups. ? ? ? ?
Example: A supermarket wants to test a delivery of avocados for ripeness by cutting them in half.
Suggest a reason why the supermarket should not test all the avocados in the delivery.
The supermarket tests a sample of 5 avocados and finds that 4 of them are ripe. They estimate that 80% of the avocados in the deliver are ripe. Suggest one way that the supermarket could improve their estimate. ? a
Testing the avocados destroys them (and thus can’t be sold).
Use a larger sample size (as this would be better estimate of the proportion of ripe avocados). b ?

6. How to we select a sample?
We have a population of 100 students and want to pick 20 of them to determine the favourite books. Can you think of some ways we pick these 20?
Simple Random Sample
Each thing in sample is equally likely to be selected. Could use names out of a hat or random number generator to select these items in a way that is bias-free.
Stratified Sampling
This is when we divide the population into groups, e.g. ethnicity or year group.
We might want to fairly represent each year group. Since we’re sampling 20 out of 100, i.e. a fifth of the students, we might sample a fifth of Year 7, a fifth of Year 8, and so on. ?
? Note: No longer in GCSE syllabus
Systematic Sampling
Get a list of the 100 students, pick a random one from the first 10, and the systematically pick every 10th student after that, e.g. the 4th, 14th, 24th, 34th, …
Non-Random Sampling
Stand at the entrance of the canteen and survey the first 20 students that pass you. Lots of potential for bias, e.g. one year group might have been let out for lunch early, so sample wouldn’t fairly represent the school.
? Note: Not in GCSE syllabus ?

7. Exam Questions on Random Sampling
“Describe a random sample”
a sample where each thing in the population is equally likely to be chosen. ?
Fro Tip: If you are asked in an exam what a random sample is, the key phrase they’re looking for is “equal chance”.
“You want to take a random sample a student’s favourite TV programmes at school. Describe how you could achieve a random sample.” ?
Since it is a random sample, you need to ensure each thing is equally likely to be chosen. Mark schemes would expect an approach such as:
Put all student names into a hat and pick them out to decide who to sample.
Use a random number generator where each number represents a student.
“Dave wants to determine the spending habits of the UK population.
He stands outside HSBC bank’s headquarters and asks the first 10 people he sees. Suggest two reasons why his sample will likely be poor.” ?
Sample not random: People working for a bank likely to earn more money.
Sample size too small.

8. Question Bias
How much does the wording of a question change the results?
British Social Attitudes Survey (2002):
“Do you regard yourself as belonging to any particular religion?”
English/Welsh National Census (2001):
“What is your religion?” ?
15.7% - No religion
50.5% - No religion ?

9. Using a sample to estimate frequencies
Whatever proportion of the sample is a particular value should reflect the proportion of the population.
e.g. If a quarter of the students in the sample liked red, then we should similarly get red books for a quarter of the whole school.
Edexcel Mock Set 4 Paper 2 ?
Fraction red in sample: 𝟏𝟖 𝟓𝟎
Red books to buy: 𝟏𝟖 𝟓𝟎 ×𝟑𝟎𝟎𝟎=𝟏𝟎𝟖𝟎 ?
Assuming the same proportion of people in the sample like red than in the whole school. If assumption were not true, we may have too many or too few red books.

10. Test Your Understanding
George is doing a survey on people’s favourite newspaper in Surrey. He asks a sample of 80 people and gets the following results:
If the population of Surrey is 1.2 million, estimate the number of people that read The Post.
Newspaper
The Post
The Shine
The Herald
Can’t read
Readers 25 10 40 5 ?
25 80 × =

11. Capture, Tag and Release
A similar principle of ‘considering proportions’ can also be used to estimate animal populations.
50 of the llamas in a field are captured, given an ear tag and then released. The following day, I capture 20 llamas, and 8 have an ear tag.
Estimate the total number of llamas in the field.
State any assumptions you have made in making your estimate.
? a
(a) of llamas = 50
1 5 of llamas = 25
Total llama population = 25×5=125
(b) We have assumed that there has been no change to the proportion of llamas that have been tagged, e.g. no llamas have died.
8 20 = 2 5 of the second sample of llamas were tagged, so the 50 llamas we initially tagged were approximately ths of the total llamas:
? b

12. Test Your Understanding
I herd 30 Tiffinians into a holding pen, and tag all of them. I then release them into the wild (i.e. the playground).
I then capture 50 Tiffinians and of these, 15 of these have tags. Estimate the number of Tiffinians in the playground. ?
15 50 = of Tiffinians = 30
1 10 of Tiffinians = 10
Total playground population = 10×10=100

13. Exercise 1
I want to estimate how many vegetarians to cater for at a large even of 600 people. I randomly ask 15 people and 2 are vegetarian.
Estimate how many vegetarians to cater for at the event. 𝟔𝟎𝟎× 𝟐 𝟏𝟓 =𝟖𝟎
State any assumption you made. The proportion of vegetarians in the sample reflects those attending the event.
I want to estimate the population of dolphins in Ingall Bay. I capture and tag 20 dolphins before releasing them. I then capture 56 dolphins and 7 have tags. Estimate how many dolphins are in the bay.
𝟕 𝟓𝟔 = 𝟏 𝟖 of dolphins had tag.
Total dolphins =𝟐𝟎×𝟖=𝟏𝟔𝟎
In order to determine a hamster population, 340 hamsters were captured, tagged, and released. Later, 420 hamsters were captured. Of the 420 hamsters, 16 were already marked. Estimate the size of the hamster population?
𝟏𝟔 𝟒𝟐𝟎 = 𝟒 𝟏𝟎𝟓 of hamsters tagged.
𝟒 𝟏𝟎𝟓 of hamsters = 340
𝟏 𝟏𝟎𝟓 of hamsters = 85 hamsters
Total hamsters 𝟖𝟓×𝟏𝟎𝟓=𝟖𝟗𝟐𝟓 1
Melanie wants to find out how often people go to the cinema. She gives a questionnaire to all the women leaving a cinema. Her sample is biased. Give two possible reasons why. Only women were asked/you need to ask men. Only people leaving the cinema were asked/you need to ask people in different places.
I want to find out information about the types of homes that students in a school live in.
I initially sample 5 students. What are the limitations of this sample? Sampled too few people so not likely to adequately represent whole school.
What is a random sample? A selection of items from the population where each thing has an equal chance of being selected.
Explain how you would obtain a random sample of 20 students from a school of 800 students. Use a random number generator to get 20 different numbers from 1 to 800. Select students with these numbers from an ordered list. 3 ? ? ? 2 4 ? ? ? 5 ? ?