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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.

**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**.

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. |

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**.

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.

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.**

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

DOWNLOAD FREEGet 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:

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.

2**Associate 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 |

3**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 13 by pressing the key combination:

1, 3, × , shift, . , =

This should show the expression: 13 × 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.

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, ×, shift, . , =

This should show the expression: 26 × 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.

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, ×, shift, . , =

This should show the expression: 20 × 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 .

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 |

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, ×, shift, . , =

This should show the expression: 250 × 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 |

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, ×, shift, . , =

This should show the expression: 1000 × 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.

**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.

1. 1000 people enter a running race. Each person is given a race number to attach to his/her shirt to track their position in the race. At the end of the race, 10 runners are randomly selected for a drug test. Describe how they should be selected using a simple random sampling method.

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.

2. An article in a newspaper contains 1500 words. The editor would like to find out the reading age of the article by taking a random sample of words and calculating the average word length. The location of each word in the article is numbered from 1-1500 Describe how you would take a random sample of 25% of the words in the article.

Use the first 375 words in the article.

Find the average word length of all 1500 words.

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.

3. There are nearly 300 weather stations across a county. A sample of 15 weather stations are routinely checked every year to ensure the equipment is working optimally. Each weather station is named alphabetically by the road/street name it is located by. Describe how you would take a random sample of 15 weather stations.

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.

4. An astronomer is exploring the night sky using a telescope. He wants to count the number of shooting stars in the sky over a 12 hour period. He decides to observe for 15 minute intervals chosen randomly throughout the night. He wants to observe for at least 6 hours. Describe how he should choose a random sample of data.

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

Observe for the entire night

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.

5. A deck of cards is shuffled and placed face down on a table so that you cannot see what is on each card. Describe how you would use a random sampling method to pick 13 cards.

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.

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.

6. A CCTV camera captures the number of people crossing a busy junction 24 hours a day. A road safety company would like to determine whether a pedestrian crossing should be installed in the area. They decide to take a random sample of hours throughout one week to monitor how many pedestrians cross the junction. Determine how they should take a random sample of times.

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).

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)**

**Or**

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

**(1)**

**Or**

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.

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 |

(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

No

**(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)**

You have now learned how to:

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

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8 sets of free exam practice papers written by maths teachers and examiners for Edexcel, AQA and OCR.

Each set of exam papers contains the three papers that your students will expect to find in their GCSE mathematics exam.