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Populations, samples and the main sampling methods (simple random, systematic, stratified, quota and opportunity), with their advantages and disadvantages, plus what you need to know about the large data set of weather data. Questions are short: describe a method, give an advantage or disadvantage in context, or use your knowledge of the large data set.
Key facts
Census: data from every member of the population. Sample: data from a selection of them
Sampling frame: a list of all the sampling units in the population
Stratified: number from a stratum \(= \dfrac{\text{stratum size}}{\text{population size}} \times \text{sample size}\)
Systematic: every \(k\)th item, \(k = \dfrac{\text{population size}}{\text{sample size}}\), with a random start among the first \(k\)
Need a sampling frame: simple random, systematic, stratified
Not random, no sampling frame needed: quota, opportunity
Large data set: 5 UK and 3 overseas stations; 1 May to 31 October (184 days); 1987 and 2015
tr = trace of rain (less than 0.05 mm); n/a = not available; 1 knot = 1.15 mph
Notes
Populations and samples
The population is the whole set of items of interest. A census collects data from every member; a sample from a selection of them.
A census should be fully accurate, but it is slow and expensive, gives a lot of data to process, and is impossible when testing destroys the item (e.g. testing how long batteries last).
A sample is quicker and cheaper but may not represent the population. Different samples can lead to different conclusions (natural variation); larger samples usually give more reliable results.
A sampling unit is one member that can be sampled; a sampling frame is a numbered list of the sampling units.
Random sampling methods (all need a sampling frame)
diagram
Simple random: every sample of size \(n\) is equally likely. Number the sampling frame, generate random numbers and ignore repeats. Free of bias, but impractical for a large population.
Systematic: take every \(k\)th item, where \(k = \dfrac{\text{population size}}{\text{sample size}}\), starting at a random item among the first \(k\). Quick for large populations, but biased if the list has a pattern.
Systematic sampling: a random start among the first k, then every kth item
Stratified: split the population into groups (strata), sample each in proportion to its size, and choose randomly within each stratum. It reflects the population's structure, but the strata must be clearly defined.
Stratified sampling: each stratum is sampled in proportion to its size
Non-random sampling methods
Quota: an interviewer fills a quota for each group (e.g. 20 men and 30 women) with people they choose. No sampling frame is needed and it is quick and cheap, but the interviewer's choices can cause bias, and people who refuse are simply replaced.
Opportunity (convenience): use whoever is available at the time and fits the criteria. Easy and cheap, but unlikely to be representative.
The large data set
Met Office weather data from five UK stations (Camborne, Heathrow, Hurn, Leeming, Leuchars) and three overseas stations (Beijing, Jacksonville, Perth), for every day from 1 May to 31 October in 1987 and in 2015.
Variables include daily mean temperature (°C), total rainfall (mm) and total sunshine (hours), mean windspeed and maximum gust (knots), cloud cover (oktas: eighths of the sky), visibility, pressure (hPa) and relative humidity (%). The overseas stations record only temperature, rainfall, pressure and windspeed.
tr means a trace of rain (less than 0.05 mm); n/a means not available. 1 knot = 1.15 mph.
Perth is in the southern hemisphere, so May to October includes its winter.
How to answer each type of question
M = method mark, A = answer mark, B = a mark on its own.
Describe how to take a simple random or systematic sample
2 to 3 marksGrade E
Say how the sampling frame is numbered (e.g. days 1 to 184).
Simple random: generate random numbers in the range, ignoring repeats, until you have enough.
Systematic: find \(k\) = population size ÷ sample size, choose a random start from 1 to \(k\), then take every \(k\)th item.
Example. Priya wants a systematic sample of 23 days from the 184 days of data for Leeming in 2015. Explain how she could take this sample.
Show the model answerHide the model answer
Number the days 1 to 184 in date order and find \(184 \div 23 = 8\)
(B1)
Choose a random starting day from days 1 to 8, e.g. using the random number function on a calculator
(B1)
Take that day and every 8th day after it, e.g. days 5, 13, 21, …, 181
(B1)
Calculate and describe a stratified sample
3 to 4 marksGrade C
Find each stratum's share: stratum size ÷ population size × sample size.
Round sensibly and check that the numbers add up to the sample size.
Describe a simple random sample within each stratum (numbered list, random numbers, ignore repeats).
Example. A sixth form has 180 students in Year 12 and 120 students in Year 13. A sample of 40 students, stratified by year group, is to be taken. (a) Find the number of students to be sampled from each year group. (b) Describe how the Year 12 students in the sample could be chosen. (c) Give one advantage of stratified sampling in this situation.
Show the model answerHide the model answer
(a) Year 12: \(\dfrac{180}{300} \times 40 = 24\) (M1); Year 13: \(\dfrac{120}{300} \times 40 = 16\)
(A1)
(b) Number the Year 12 students 1 to 180 from a list of names, then use random numbers to choose 24 different students, ignoring repeats
(B1)
(c) The sample has the same proportions of Year 12 and Year 13 students as the sixth form, so both year groups are fairly represented
(B1)
Critique a sampling method and suggest an improvement
2 to 4 marksGrade B
Name the method.
Say who is likely to be left out or over-represented, and why, in context.
Suggest a specific improvement, e.g. a simple random sample from a named sampling frame.
Example. A council wants residents' views on a new cycle lane. An officer stands outside one supermarket on a Monday morning and asks the first 50 adults who pass. (a) Name this sampling method. (b) Give two reasons why the sample may not represent the residents. (c) Suggest an improvement.
Show the model answerHide the model answer
(a) Opportunity (convenience) sampling
(B1)
(b) Any two, e.g. people who work on Monday mornings are unlikely to be included (B1); only people near one supermarket are asked, and some of them may not live in the area
(B1)
(c) Use a list of residents, such as the electoral register, as a sampling frame and take a simple random sample from it
(B1)
Use your knowledge of the large data set
1 to 2 marks eachGrade A
Recall the relevant fact: station location, dates, units or abbreviation.
Apply it to the situation: explain the pattern or problem, not just the fact.
Example. (a) In a sample of daily total rainfall from Camborne in 2015, some values are recorded as ‘tr’. State what ‘tr’ means. (b) A student draws graphs of daily mean temperature against date for Perth and for Leuchars, using the 1987 data. Using your knowledge of the large data set, explain why the two graphs have different shapes.
Show the model answerHide the model answer
(a) A trace of rain: less than 0.05 mm fell that day
(B1)
(b) Perth is in the southern hemisphere, so May to October includes its winter: temperatures tend to fall to a minimum in the middle of the period, then rise (B1). Leuchars is in the UK, where the period includes summer, so temperatures tend to rise to a maximum in the middle of the period, then fall
(B1)
Don’t lose marks
Describing simple random sampling without saying how the items are numbered, how the random numbers are generated, or that repeats are ignored.
Forgetting the random start in systematic sampling: ‘take every 8th day’ on its own is not enough.
Calling opportunity or quota sampling ‘random’.
Giving a generic answer such as ‘it is biased’ without saying who is left out and why.
Rounding stratum sizes so that they no longer add up to the sample size.
More tips
Memory tricks
Only the random methods (simple random, systematic, stratified) need a sampling frame; quota and opportunity sampling do not.
Stratified and quota sampling both split the population into groups: stratified then chooses randomly within each group, quota lets the interviewer choose.
Check a stratified sample: the numbers from the strata must add up to the total sample size.
The large data set has 184 days per station per year (31 + 30 + 31 + 31 + 30 + 31), which is handy for systematic sampling questions.
Exam technique
Advantages and disadvantages must fit this situation: use the words of the question (the shoppers, the days, the students).
You don't need to remember values from the large data set, but you do need to know its structure: stations, dates, years, variables, units and abbreviations.
If asked why two samples lead to different conclusions, mention natural variation between samples (and a small sample size).
What each grade needs
What you need to be able to do, from the first marks up to the top grade.
Grade E
Define population, sample, census and sampling frameA census measures every member of the population; a sample is a selection, chosen from a sampling frame (a list of the sampling units).
Grade E
Describe how to take a simple random sampleNumber every item in the sampling frame, then use random numbers to choose the required number of different items.
Grade D
Describe systematic, quota and opportunity samplingSay how each method is carried out and whether it needs a sampling frame.
Grade C
Calculate the numbers for a stratified sampleSample each stratum in proportion to its size, then choose randomly within each stratum.
Grade B
Give advantages and disadvantages in contextLink each point to the situation, e.g. no list of shoppers exists, so a quota sample is practical.
Grade A
Critique a sampling method and suggest improvementsIdentify who is left out or over-represented, and why, and propose a random method that fixes it.
Grade A*
Reason with knowledge of the large data setUse facts about the stations, dates, variables and abbreviations to explain patterns or problems in a sample.
Quick recall
Cover the answers and test yourself. The app has these as flashcards that come back just before you'd forget them.
A leisure centre manager wants to find out how often adults in a town exercise. The manager asks the first 50 adults who enter the leisure centre on a Monday morning. State the name of this sampling method.
Opportunity (convenience) sampling
Sample questions
Written for this site in the style of Edexcel exam questions. They are not taken from real past papers.
Question 1Easy6 marks
A leisure centre manager wants to find out how often adults in a town exercise. The manager asks the first 50 adults who enter the leisure centre on a Monday morning.
(a) State the name of this sampling method.[1]
(b) Give two reasons why this sample is likely to be biased.[2]
(c) The town has 12 000 adults, of whom 7200 are under 50 years old. The manager decides instead to take a sample of 60 adults, stratified by age group (under 50, and 50 or over). Find how many of the sample should be under 50, and describe how the manager could choose them.[3]
Show the answer and mark scheme
(a)Answer: Opportunity (convenience) sampling
B1 for opportunity (or convenience) sampling
Worked solution: The manager uses the people who happen to be available: this is opportunity (convenience) sampling.
(b)Answer: People entering a leisure centre are likely to exercise more than a typical adult; people at work on a Monday morning are excluded (so some age or employment groups are under-represented).
B1 for the sample consists of leisure-centre users, who probably exercise more often than adults in general
B1 for a second reason, e.g. a Monday morning excludes most people who work (the sample over-represents retired or unemployed people), or everyone was sampled at one time and place
Worked solution: Adults entering a leisure centre are more likely than average to exercise regularly, and on a Monday morning most working adults are at work, so the sample is not representative of all adults in the town.
(c)Answer: 36; number a list of the adults under 50 (e.g. from the electoral register) and choose 36 using random numbers.
M1 for \(\frac{7200}{12\,000} \times 60\)
A1 for 36
B1 for a description of simple random sampling within the stratum: obtain a list (sampling frame) of adults under 50, number them, and use random numbers to select 36 (ignoring repeats)
Worked solution: \(\frac{7200}{12\,000} \times 60 = 36\) of the sample should be under 50. Using a list of all adults under 50 in the town (such as the electoral register with dates of birth), number them 1 to 7200 and use a random number generator to choose 36 different numbers.
Question 2Medium4 marks
A student is using the large data set to investigate the daily mean temperature at Hurn in 2015. The student wants a systematic sample of 8 days from the 184 days from 1 May to 31 October 2015.
(a) Explain how the student could take this systematic sample.[2]
(b) The randomly chosen first day is day 5 (5 May). Find the date of the 7th day in the sample.[2]
Show the answer and mark scheme
(a)Answer: Number the days from 1 to 184 in date order. Since \(184 \div 8 \approx 23\), choose a random day from the first 23 days, then take every 23rd day after it.
B1 for numbering the days 1 to 184 (or listing them in date order) and using the interval 23 (from \(184 \div 8\))
B1 for choosing the first day at random from days 1 to 23 and then every 23rd day after it
(b)Answer: 20 September 2015
M1 for \(5 + 6 \times 23 = 143\) (day number of the 7th day)
A1 for 20 September
Worked solution: Day \(5 + 6 \times 23 = 143\). May has 31 days, June has 30, July has 31, August has 31, so day 143 is 20 September.
Question 3Hard5 marks
The students at a secondary school are divided into three strata by key stage. The table shows the number of students in each stratum. A stratified sample of 73 students is taken, and it contains 28 students from the Key stage 3 stratum.
[object Object]
(a) Find the value of \(x\).[3]
(b) Find the number of students in the sample from each of the other two strata.[2]
Show the answer and mark scheme
(a)Answer: \(x = 40\)
M1 for \(\dfrac{73(3x + 20)}{7x + 85} = 28\) or equivalent
dM1 for solving a correct linear equation in \(x\)
A1 for \(x = 40\)
Worked solution: The population is \((3x + 20) + (4x - 40) + 105 = 7x + 85\), so \(73(3x + 20) = 28(7x + 85)\), giving \(23x = 920\) and \(x = 40\).
(b)Answer: Key stage 4: 24, Sixth form: 21
M1 for \(\dfrac{73}{365} \times 120\) or \(\dfrac{73}{365} \times 105\) (or \(73 - 28\) split correctly)
A1 for 24 and 21
Worked solution: The strata contain 140, 120 and 105 students (total 365), so the sampling fraction is \(\dfrac{73}{365} = \dfrac{1}{5}\), giving 24 and 21.
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