Random Sampling - Math Steps, Examples & Questions

High Impact Tutoring Built By Math Experts

Personalized standards-aligned one-on-one math tutoring for schools and districts

Request a demo

In order to access this I need to be confident with:

Calculator skills

Math resources Statistics and probability Sampling methods

Random sampling

Here you 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.

Students will first learn about random sampling as part of statistics and probability in $7$ th grade.

What is random sampling?

Random sampling is a type of sampling method. It is also called simple random sampling.

To take a random sample, you 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.

You 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.

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 set 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 including quota sampling, convenience sampling (non-random sampling), non-probability sampling, and snowball sampling.

[FREE] Probability Worksheet (Grade 7 to 12)

Teaching random sampling? Use this quiz to check your grade 7 to 12 students’ understanding of probability. 15+ questions with answers covering a range of probability topics to identify areas of strength and support!

DOWNLOAD FREE

[FREE] Probability Worksheet (Grade 7 to 12)

DOWNLOAD FREE

Advantages and disadvantages of random sampling

Following a random sampling methodology has advantages and disadvantages:

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 generalized results for the population.

What is random sampling?

Common Core State Standards

How does this relate to $7$ th grade math and high school math?

Grade 7 – Statistics & Probability (7.SP.A.1)
Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.

High School – Statistics & Probability – Making Inferences & Justifying Conclusions (HS.S.IC.B.3)
Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

High School – Statistics & Probability – Making Inferences & Justifying Conclusions (HS.S.IC.B.4)
Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for 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.

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.

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

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

$\bf{1}, \, \bf{3}, \, \times,$ shift, . , =

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

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.

Show step

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

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

Show step

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

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

Show step

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

$\bf{2}, \, \bf{6}, \, \times,$ shift, . , =

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

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.

Show step

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

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

Show step

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

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

Show step

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

$\bf{2}, \, \bf{0}, \, \times,$ shift, . , =

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

As you need to obtain $30 \%$ of the students, you need to calculate $30 \%$ of $20.$

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

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.

Show step

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

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

Show step

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.

Show step

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

Example 5: student survey

A school wants to investigate what students in the new $7$ th grade class eat for lunch. The school is expecting $250$ new students in $7$ th grade. They wish to take a sample of $10 \%$ of the students.

Each student is ordered alphabetically by their last name 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.

Show step

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

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

Show step

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

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

Show step

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

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

$\bf{2}, \, \bf{5}, \, \bf{0}, \, \times,$ shift, . , =

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

The students involved in the investigation are:

Example 6: production line

A production line produces $1,000$ 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.

Show step

As there are $1,000$ tubes per hour and $8$ hours in one day, there are $8,000$ potential items of data. Theoretically, you could list all $8,000$ items but this would be unnecessary and very time-consuming.

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

Show step

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

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

Show step

As you need a sample of $0.1 \%,$ you need to calculate $0.1 \%$ of $8,000\text{:}$

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

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

$\bf{1000}, \, \times,$ shift, . , =

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

The tubes taken off of the production line for quality control testing are numbers:

$275, \, 278, \, 383, \, 394, \, 662, \, 801, \, 908,$ and $979.$

Teaching tips for random sampling

Start by explaining how random sampling selects a smaller group (sample) from a larger population to ensure every individual has an equal chance of being chosen, making it one of the simplest probability sampling methods.

Clearly define the target population and its importance in ensuring the sample represents the larger group accurately.

Demonstrate the lottery method by assigning numbers to a population (for example, students in class) and drawing numbers randomly to illustrate sample selection.

Explain how the population size affects sample selection and how a larger population requires careful planning to avoid bias.

Teach students to use Excel to generate random numbers or select a random sample, reinforcing the connection between technology and sampling.

Use examples like polling or market research to show how organizations collect data from a sample to understand a larger population.

Easy mistakes to make

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

Using the $\textbf{RAN}\#$ 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, 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 $100$ th person who crosses the line.

Use a random number generator to select $10$ random numbers between $1$ and $1,000.$

Use a random number generator to select the order of all $1,000$ runners in the population.

As each runner has a unique race number, these numbers can be placed into a random number generator to minimize 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 $1,500$ 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 $1,500.$

For a random sample of $25 \%$ of the words in the article, you need to calculate $25 \%$ of the population. Here, $25 \%$ of $1,500$ is $375$ words.

As the location of each word is numbered (example, the first word in this paragraph $“For”$ would be labeled $1, \, “a”$ would be labeled $2$ , etc), you can use a random number generator to select $375$ numbers out of a possible $1,500,$ and therefore select the $375$ words for the sample.

Note: no repeated values allowed.

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, you 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.

Note: no repeated values allowed.

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 $(6\times{4}=24).$ A $12$ -hour period would have a total of $48$ intervals $(12\times{4}=48).$

Labeling each subsequent interval with a unique reference number, you 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, you 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.

Note: no duplicate values.

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 you assign each hour a unique reference number (hour $1$ is Monday $00\text{:}00,$ hour $2$ is Monday $01\text{:}00$ etc), then you 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 FAQs

What is random sampling?

Random sampling is a method of selecting a subset (sample) from a larger population in which every individual has an equal chance of being chosen. This ensures that the sample is unbiased and representative of the population, making it a key technique in probability sampling methods.

What is the difference between random sampling and stratified random sampling?

Random sampling selects individuals randomly from the total population. Stratified random sampling divides the population into groups and randomly selects from each group.

What is the difference between random sampling and systematic random sampling?

Random sampling selects individuals purely by chance. Systematic random sampling selects individuals at regular intervals from a randomly chosen starting point.

What are some other sampling methods?

Some other sampling methods are systematic sampling, stratified sampling, cluster sampling, convenience sampling, quota sampling, snowball sampling, and purposive sampling.

The next lessons are

Still stuck?

At Third Space Learning, we specialize in helping teachers and school leaders to provide personalized math support for more of their students through high-quality, online one-on-one math tutoring delivered by subject experts.

Each week, our tutors support thousands of students who are at risk of not meeting their grade-level expectations, and help accelerate their progress and boost their confidence.