Random sampling

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GCSE Maths Statistics Types Of Sampling

Random Sampling

Here we will learn about random sampling, including what random sampling is, how to take a random sample of data, and the advantages and disadvantages of this sampling method.

There are also random sampling worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What is random sampling?

Random sampling is a type of sampling method.

To take a random sample, we list each individual member of the population, assign a unique number to each member, and use a random number generator or a random number table to select the number of pieces of data required for the sample size.

We use simple random sampling to choose the individual items of data within the population.

Each member of the sample has an equal chance of being selected, reducing bias and sampling error.

Sampling method	Description	Example
Random sampling (for simple random sampling)	Gathering a representative sample from a population where each member in the population has an equal chance of being selected.	Using a random number generator to select students in a class to complete a task.

Random sampling is also used for other sampling techniques such as stratified sampling.

Stratified sampling requires another sampling method such as a simple random sample to generate a random selection of data values once the data is divided into subgroups (or subsets). This means that each item of data has an equal probability of being chosen and each subgroup within the sample is represented proportionally to the whole population.

Other types of random sampling methods include: cluster sampling, stratified sampling, and systematic sampling.

There are also other types of sampling methods that do not require simple random sampling include: quota sampling, convenience sampling (non-random sampling), non-probability sampling, and snowball sampling.

Advantages and disadvantages of random sampling

Following a random sampling methodology has advantages and disadvantages:

Advantages	Disadvantages
Results can be generalised for a population It is more time efficient than asking the entire population. Reduced bias.	Expensive. Time consuming. Not always possible if there is no sampling frame or list to sample from.

Sampling error

If every member of the population is in the sample, there is no sampling error. As the sample gets smaller, or the methodology has introduced a selection bias, the sampling error becomes more significant as this means that the sample may not be representative of the population.

The more random a sample is, the smaller the sampling error.

What happens next?

Once a random sample is chosen, the next step is data collection where respondents offer data to fulfill the requirements of the questionnaire or survey (for example). The collected values within the sample then go through data analysis to find generalised results for the population.

What is random sampling?

How to take a random sample

In order to take a random sample:

List every member of the population.
Associate each member of the population with a unique reference number.
Use a random number generator to select the number of data points in the sample.

How to take a random sample

Types of sampling methods worksheet (includes random sampling)

Get your free random sampling worksheet of 20+ types of sampling methods questions and answers. Includes reasoning and applied questions.

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Types of sampling methods worksheet (includes random sampling)

Get your free random sampling worksheet of 20+ types of sampling methods questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREE

Random sampling is part of our series of lessons to support revision on types of sampling methods. You may find it helpful to start with the main types of sampling methods lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

Random sampling examples

Example 1: deck of cards

A deck of cards contains $52$ cards divided into $4$ suits. Use a random number generator to collect a sample of $5$ cards from one suit. Do not allow any duplicates.

List every member of the population.

The $13$ cards in one suit are: Ace, $2, 3, 4, 5, 6, 7, 8, 9, 10,$ Jack, Queen, King.

2Associate each member of the population with a unique reference number.

$A$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$J$	$Q$	$K$
$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$	$12$	$13$

3Use a random number generator to select the number of data points in the sample.

Using a Casio FX series calculator, we can generate random numbers between $1$ and $13$ by pressing the key combination:

$1, 3, \times , shift, . , =$

This should show the expression: $13 \times RAN$ # =. Round any decimal to the nearest integer (whole number). Ignore any duplicates as stated in the question.

$13RAN\times$ #=	$5.448$	$6.889$	$6.109$	$0.903$	$11.375$
Rounded	$5$	$7$	$6$	$1$	$11$
Card	$5$	$7$	$6$	$A$	$J$

The $5$ cards in the sample are: Ace, $5, 6, 7,$ Jack.

Example 2: the alphabet

There are $26$ letters in the English alphabet. Use a random sample to obtain a sample of $10$ letters with no duplicates.

List every member of the population.

Here we will list all $26$ letters of the English alphabet.

$A$	$B$	$C$	$D$	$E$	$F$	$G$	$H$	$I$	$J$	$K$	$L$	$M$
$N$	$O$	$P$	$Q$	$R$	$S$	$T$	$U$	$V$	$W$	$X$	$Y$	$Z$

Associate each member of the population with a unique reference number.

$A = 1, B=2,$ etc:

$A$	$B$	$C$	$D$	$E$	$F$	$G$	$H$	$I$	$J$	$K$	$L$	$M$
$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$	$12$	$13$
$N$	$O$	$P$	$Q$	$R$	$S$	$T$	$U$	$V$	$W$	$X$	$Y$	$Z$
$14$	$15$	$16$	$17$	$18$	$19$	$20$	$21$	$22$	$23$	$24$	$25$	$26$

Use a random number generator to select the number of data points in the sample.

Using a Casio FX series calculator, we can generate random numbers between $1$ and $26$ by pressing the key combination:

$2, 6, \times, shift, . , =$

This should show the expression: $26 \times RAN$ #=. Round any decimal to the nearest integer (whole number). Ignore any duplicates as stated in the question.

$26RAN\times$ #=	$6.301$	$23.244$	$5.452$	$15.814$	$12.476$
Rounded	$6$	$23$	$5$	$16$	$12$
Letter	$F$	$W$	$E$	$P$	$L$

$26RAN\times$ #=	$13.332$	$1.375$	$25.487$	$13.913$	$9.684$
Rounded	$13$	$1$	$25$	$14$	$10$
Letter	$M$	$A$	$Y$	$N$	$J$

The $10$ letters in the sample are: $F, W, E, P, L, M, A, Y, N, J.$

Example 3: questionnaire

A class of $20$ students completed a questionnaire. Of the $20$ students a random sample of $30%$ will be given a prize. Each student is given a number between $1$ and $20$ determined by their position in the register. Determine which students will be given a prize. Each student can only receive one prize.

List every member of the population.

As each student has been given a number between $1$ and $20$ , we do not need to know their names in order to choose the sample. We therefore only need to reference the numbers between $1$ and $20$ .

$1$	$2$	$3$	$4$	$5$
$6$	$7$	$8$	$9$	$10$
$11$	$12$	$13$	$14$	$15$
$16$	$17$	$18$	$19$	$20$

Associate each member of the population with a unique reference number.

As each member of the population is numbered between $1$ and $20$ , we can use this number.

Use a random number generator to select the number of data points in the sample.

Using a Casio FX series calculator, we can generate random numbers between $1$ and $20$ by pressing the key combination:

$2, 0, \times, shift, . , =$

This should show the expression: $20 \times RAN$ #=. Round any decimal to the nearest integer (whole number). Ignore any duplicates as stated in the question.

As we need to obtain $30%$ of the students, we need to calculate $30%$ of $20$

$20\div{100}\times{30}=6\text{ students}$ .

$20RAN\times$ #=	$2.201$	$5.194$	$1.730$	$7.171$	$19.205$	$3.297$	$9.454$
Rounded	$2$	$5$	$2$	$7$	$19$	$3$	$9$
Number	$2$	$5$	Dup.	$7$	$19$	$3$	$9$

The $6$ students in the sample who win a prize are: $2, 3, 5, 7, 9,$ and $19$ .

Example 4: counters

There are $36$ counters in a bag. Each counter is uniquely numbered with a two digit number that can be determined by rolling two fair, six sided dice. Counters are chosen by the number rolled on the first dice being the first digit, and the number rolled on the second dice being the second digit. A sample of $12$ counters is chosen from the bag. Each counter is replaced and so can be chosen twice. Determine the numbers on the $12$ counters.

List every member of the population.

As each dice has the numbers $1$ to $6$ , we have the following sample space:

Associate each member of the population with a unique reference number.

Each member of the population (each counter) has a unique reference number

Use a random number generator to select the number of data points in the sample.

This time, as each dice is fair, we can use the dice to generate the $12$ counters from the bag. Rolling the dice together $12$ times, and allowing repetitions, we obtain the following random sample:

$35$	$42$	$13$	$31$	$51$	$32$
$11$	$16$	$34$	$35$	$15$	$21$

Example 5: student survey

A school wants to investigate what students in the new Year $7$ cohort eat for lunch. The school is expecting $250$ new students in Year $7$ . They wish to take a sample of $10%$ of the students. Each student is ordered alphabetically by their surname and given a unique number between $1$ and $250$ . Use a random number generator to determine the students who will participate in the investigation.

List every member of the population.

As there are $250$ students in the population, we would use a list to determine each unique reference number. For this example, we will visualise the list without writing down all $250$ numbers!

Associate each member of the population with a unique reference number.

Each member of the population has a unique reference number as stated in the question and so we can move on to step $3$ .

Use a random number generator to select the number of data points in the sample.

As we need a sample of $10%$ , we need to obtain a sample of $25$ students as $10%$ of $250$ is equal to $25$ .

Using a Casio FX series calculator, we can generate random numbers between $1$ and $250$ by pressing the key combination:

$2, 5, 0, \times, shift, . , =$

This should show the expression: $250 \times RAN$ #=. Round any decimal to the nearest integer (whole number). Ignore any duplicates as stated in the question.

$250RAN\times$ #=	$8.75$	$62$	$102$	$137.75$	$247.75$
Number	$9$	$62$	$102$	$138$	$248$

$250RAN\times$ #=	$45$	$46$	$130.25$	$240$	$51$
Number	$45$	$46$	$130$	$240$	$51$

$250RAN\times$ #=	$187.5$	$75.25$	$178$	$33.75$	$72.25$
Number	$188$	$75$	$178$	$34$	$72$

$250RAN\times$ #=	$215.75$	$82$	$231.25$	$113.25$	$116$
Number	$216$	$82$	$231$	$113$	$116$

$250RAN\times$ #=	$191.75$	$47$	$241.5$	$190$	$219.75$
Number	$192$	$47$	$242$	$190$	$220$

The students involved in the investigation are:

$9$	$51$	$102$	$178$	$220$
$34$	$62$	$113$	$188$	$231$
$45$	$72$	$116$	$190$	$240$
$46$	$75$	$130$	$192$	$242$
$47$	$82$	$138$	$216$	$248$

Example 6: production line

A production line produces $1000$ tubes of toothpaste every hour. In one $8$ hour day, $0.1%$ of the tubes must be taken off of the production line for quality control testing. These tubes must be chosen at random to ensure that the product quality remains consistent throughout production. Determine the tubes that are selected for quality control in one day. Do not include duplicated data.

List every member of the population.

As there are $1000$ tubes per hour and $8$ hours in one day, there are $8000$ potential items of data. Theoretically we could list all $8000$ items but this would be unnecessary and very time consuming.

Associate each member of the population with a unique reference number.

Each tube is numbered from $1-8000$ depending on their place in the production line (chronologically).

Use a random number generator to select the number of data points in the sample.

As we need a sample of $0.1%$ , we need to calculate $0.1%$ of $8000$ :

$(8000\div{100})\times{0.1}=8\text{ tubes}$ .

Using a Casio FX series calculator, we can generate random numbers between $1$ and $250$ by pressing the key combination:

$1000, \times, shift, . , =$

This should show the expression: $1000 \times RAN$ #=. Each number will be an integer and so we can use this number to select the item in the population for the sample.

1000RAN×#=

394

278

383

908

979

801

662

275

The tubes taken off of the production line for quality control testing are numbers: $275, 278, 383, 394, 662, 801, 908,$ and $979.$

Common misconceptions

Mixing up a sampling method

Using the incorrect sampling method to select data (such as using systematic sampling or non random sampling)

Using the RAN# button incorrectly

The RAN# button returns a number to $3$ decimal places. Different calculators will have different ways to generate random numbers over an interval and so make sure you know how your calculator does this function.

Incorrect number of items in the sample / duplicates

If a sample asks for a specific number of data values, then the sample must contain this number of data. Be careful as to whether the sample can contain duplicates as some will allow repeated results.

Practice random sampling questions

Take the first $10$ people to cross the finish line.

Select the first runner at random then take every $100^\text{th}$ person who crosses the line.

Use a random number generator to select $10$ random numbers between $1$ and $1000$ .

Use a random number generator to select the order of all $1000$ runners in the population.

As each runner has a unique race number, these numbers can be placed into a random number generator to minimise the bias when selecting runners for the drug test. This would be a suitable way to select a simple random sample.

Use the first $375$ words in the article.

Find the average word length of all $1500$ words.

Select the first word out of the first $4$ words at random, then use every fourth word in the article following this.

Use a random number generator to select $375$ random numbers between $1$ and $1500$ .

For a random sample of $25$ % of the words in the article, we need to calculate $25$ % of the population. Here, $25$ % of $1500$ is $375$ words.

As the location of each word is numbered (e.g. the first word in this paragraph “For” would be labelled $1$ , “a” would be labelled 2, etc.) we can use a random number generator to select $375$ numbers out of a possible $1500$ , and therefore select the $375$ words for the sample.

Assign each weather station a random number then use a random number generator to select $15$ weather stations.

Divide the list into smaller groups according to their height, find the proportion within each subgroup for the sample then select the tallest weather stations in each group.

Select the last $15$ weather stations that appear in the list alphabetically.

Check every weather station in the county.

As the $300$ weather stations can be listed in alphabetical order, we can assign each weather station with a unique reference number. Using a random number generator to select $15$ numbers out of the list of $300$ would successfully select $15$ weather stations. This would be a random sample.

Observe for $15$ minutes every half an hour throughout the night.

Observe for the entire night

Assign each interval a consecutive number between $1$ and $48$ . Use a random number generator to select $24$ intervals. No duplicates.

Assign each interval a consecutive number between $1$ and $48$ . Use a random number generator to select $6$ intervals over the $12$ hours. No duplicates.

There are four, $15$ minute intervals each hour. Over $6$ hours, this would be $24$ intervals (64=24). A $12$ hour period would have a total of $48$ intervals (124=48). Labelling each subsequent interval with a unique reference number, we can put the $48$ intervals into a random number generator to select the $24$ intervals needed for the sample. This would generate a random sample for the research.

Split the deck in half and take the next $13$ cards from the top of the pile.

Turn all of the cards over and pick all of the Spades out of the deck.

Pick every fourth card from the pile.

Assign the top card as number $1$ through to number $52$ . Use a random number generator to choose $13$ random numbers between $1$ and $52$ .

As the deck of cards is face down, each card can be numbered from $1$ to $52$ (the top card in the deck would be number $1$ , the bottom card in the deck would be number $52$ etc). As each card has a unique reference number, we can use a random number generator to select the $13$ cards in the sample. This would mean that each card has an equal chance of being selected for the sample.

Associate each hour with a unique reference number and use a random number generator to determine which hours will be used in the sample.

Observe the CCTV camera continuously for the entire week, recording the total number of pedestrians that cross the junction.

Record the number of pedestrians that cross the junction between $7-9$ am, and $4-6$ pm Monday to Friday.

Divide the number of hours per day by the number of letters in each day (M-O-N-D-A-Y = $6$ ) to determine how many hours of CCTV is recorded overnight.

There are $168$ hours in one week. If we assign each hour a unique reference number (hour $1$ is Monday $00:00$ , hour $2$ is Monday $01:00$ etc), then we would place $168$ numbers into a random number generator, in order for that generator to select the sample size. This would be a random sample as there are no subgroups for the sample to be stratified, and the numbers are not selected using a sequence (a systematic sample).

Random sampling GCSE questions

1. Freddie is investigating the distribution of prime numbers. He knows that there are $25$ prime numbers between $1$ and $100$ . He uses a random number generator to select $20$ numbers between $1$ and $100$ . Below are his results:

3	4	9	15
20	24	23	25
27	34	51	55
58	64	70	75
82	84	92	95

Freddie states “These numbers are not random”.

(a) Why would Freddie think that these numbers are not random?

(b) How could this investigation be improved?

(2 marks)

Show answer

(a)

Only $2/20$ are prime numbers

(1)

He should expect $5$ numbers to be prime as $25%$ of $20$ is $5$

(1)

There are lots of patterns (numbers ending in a $4$ or $5$ )

(1)

(b)

Take a larger sample

(1)

2. (a) There are $26$ letters in the English Alphabet. Use a random sampling method to obtain a sample of $5$ letters with no duplicates. Use the grid below to help you.

(b) Harriet says her sample consists of the letters $“PQRST”$ . Do you think these letters were chosen at random? Explain your answer.

(4 marks)

Show answer

(a)

A	B	C	D	E	F	G	H	I	J	K	L	M
1	2	3	4	5	6	7	8	9	10	11	12	13
N	O	P	Q	R	S	T	U	V	W	X	Y	Z
14	15	16	17	18	19	20	21	22	23	24	25	26

Each letter associated with a unique reference number

(1)

$5$ random numbers / letters selected with no obvious pattern / order

(1)

(b)

Unlikely they are chosen at random (not impossible)

(1)

They are all consecutive letters

(1)

3. Two dice are rolled $1000$ times and their scores are added together. Roger wants to check the validity of the experiment and so he takes a sample of $2%$ of the data. Below are the values in the sample.

7	11	1	8	3	8	6	11	0	6
9	9	6	9	13	9	1	7	2	9

Should the results of the experiment be trusted? Explain your answer.

(2 marks)

Show answer

(1)

$4/20$ items of data are not possible solutions for the sum of two dice

( $1, 0, 13, 1$ not possible for the sum of two dice)

(1)

Learning checklist

You have now learned how to:

Infer properties of populations or distributions from a sample, whilst knowing the limitations of sampling

The next lessons are

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