Systematic sampling

Here you will learn about systematic sampling, including what systematic sampling is, how systematic sampling works, and the advantages and disadvantages of systematic sampling.

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

What is systematic sampling?

Systematic sampling is a type of probability sampling that selects items of data at regular intervals from a population.

Every data entry for the population must be given in a list (a sampling frame) so that they have an equal chance (equal probability) of being selected.

You select the first item of data using a random number generator and then select the rest at regular intervals (fixed intervals).

The sampling interval

To calculate the interval required to select the sample data, you calculate the size of the population divided by the size of the sample.

\text{Sampling interval}=\cfrac{\text{Population size}}{\text{Sample size}}

For example,

If the population size is 1200 and the desired sample size is 400 items of data, you divide 1200 by 400 to get an interval of 3. This means that every 3 rd item of data in the ordered list is selected for the sample.

For example,

A factory that manufactures cars must regularly assess the quality of production. In one month, 5 \% of cars are selected using a systematic sample to be rigorously tested for quality purposes. The first car is chosen at random then every 10 th car that follows.

This systematic sample helps the company to ensure the quality of their car manufacture is maintained. Testing each vehicle would be costly and take too much time.

Advantages and disadvantages of systematic sampling

Following a systematic sampling methodology has advantages and disadvantages:

What is systematic sampling?

Common Core State Standards

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

• Grade 7 – Statistics and 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 and Probability (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 and Probability (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 systematic sample

In order to take a systematic sample:

1. Order the population and give each data entry a unique reference number.
2. Calculate the number of items of data in the sample.
3. Calculate the interval.
4. Use a random number generator to select the first item of data.
5. Select the remaining items of data following the given sequence.

Systematic sampling examples

Example 1: systematic sampling – production line

A company produces biscuits at 100 per minute. A machine checks the weight of 10 \% of the biscuits.

The biscuits pass through the machine one at a time. Use systematic random sampling to select the biscuits for the sample over 3 minutes.

1. Order the population and give each data entry a unique reference number.

As each biscuit passes through the machine one at a time, you can assume that the first biscuit is number 1, the second is biscuit number 2, etc.

2Calculate the number of items of data in the sample.

In 3 minutes, there will be 3 \times 100=300 biscuits. As the company checks 10 \% of the biscuits, you need a sample of:

\cfrac{300}{100}\times{10}=30 biscuits.

3Calculate the interval.

As you need 30 biscuits, and you are using a systematic sample, you need to choose the biscuits using a sequence. You determine the interval in the sequence by dividing the sample size by the population size:

So you need to pick every 10 th item in the ordered data set.

4Use a random number generator to select the first item of data.

As you need to pick every 10 th term, the first number in the sample (starting point of the sequence) must be randomly chosen from the first 10 terms.

Using a random number generator, you get the number 6, so you choose the first item of data in the sample to be the 6 th biscuit.

Below you have used a table to show how the sequence develops*:

*This table only contains the first 39 biscuits of the 300 in the population.

5Select the remaining items of data following the given sequence.

As you are selecting every 10 th item, you can select the following biscuits from the data:

The sample will therefore contain 30 biscuits with the following numbers:

216, \, 226, \, 236, \, 246, \, 256, \, 266, \, 276, \, 286, and 296.

Note: These numbers are in the sequence 10n-4.

Example 2: systematic sampling – sample as percentage of population

Luke is looking at the beats per minute of tracks in his music player. He has 1,200 tracks. He decides to take a systematic sample of 25 \% of his tracks. Determine the tracks that should be chosen.

Use a random number generator to select the first item of data.

As you need to pick every 4 th term, the first number in the sample must be randomly chosen from the first 4 terms.

Using a random number generator, you get the number 1, so you choose the first item of data in the sample to be the 1 st track.

Below you have used a table to show how the sequence develops*:

*This table only contains the first 39 tracks of the 1,200 in the population.

Select the remaining items of data following the given sequence.

As you are selecting every 4 th item, you can select the following tracks from the data:

The sample will therefore contain 300 tracks that belong to the sequence 4n-3.

Example 3: systematic sampling – small sample size

A traffic management company is researching the proportion of lorries that use a single carriageway between 8\text{:}00 am and 9\text{:}00 am. A traffic camera records the details of every vehicle and produces a list of data in the order of the time that the vehicle passes the camera.

960 vehicles are recorded within the hour on one day. The company uses a systematic sample to select a random sample of 5 \% of the data for their research. Determine which vehicles will be in the sample.

Use a random number generator to select the first item of data.

As you need to pick every 20 th term, the first number in the sample must be randomly chosen from the first 20 terms.

Using a random number generator, you get the number 17, so you choose the first item of data in the sample to be the 17 th vehicle.

Below you have used a table to show how the sequence develops*:

*This table only contains the first 59 vehicles of the 960 in the population.

Select the remaining items of data following the given sequence.

As you are selecting every 20 th item, you can select the following vehicles from the data:

The sample will therefore contain 48 vehicles that belong to the sequence 20n-3.

Example 4: systematic sampling – large population

A local council is researching the distribution of voters in 15,000 homes. They take a systematic sample of 20 \% of homes, listed in order of their zip code and house number. Determine which homes will be asked to participate in the survey.

Use a random number generator to select the first item of data.

As you need to pick every 50 th term, the first number in the sample must be randomly chosen from the first 50 terms.

Using a random number generator, you get the number 8, so you choose the first item of data in the sample to be the 8 th home.

Below you have used a table to show how the sequence develops*:

*This table only contains the first 59 homes of the 15,000 in the population.

Select the remaining items of data following the given sequence.

As you are selecting every 50 th item, you can select the following homes from the data:

The sample will therefore contain 300 homes that belong to the sequence 50n-42.

Example 5: systematic sampling – small sample

An online clothing company is researching the average customer spend over the previous month. There were 12,664 orders purchased, and each order has a unique reference number.

The company takes a systematic sample of 2 \% of orders. Determine which orders will be chosen for the sample.

Example 6: systematic sampling – deciding the population size

A café is carrying out some market research. Out of 1240 customers that entered the café during a weekend, 950 allowed the café to email them a questionnaire.

The café takes a systematic sample size of 12 \% of those who received the questionnaire. Determine which customers will be part of the sample.

Teaching tips for systematic sampling

• Make sure students understand the systematic sampling definition and how it differs from other methods of sampling like simple random sampling and stratified sampling. This understanding will help them grasp when and why each method is used in research and data collection.

• Provide worked examples showing different scenarios, such as different population sizes and sampling intervals.

• Discuss real-world applications where systematic sampling is used, such as in market research or quality control in manufacturing.

• Teach students strategies to minimize sampling errors, such as ensuring a random starting point, verifying the completeness of the population list, and considering alternative sampling methods when systematic biases are suspected.

Easy mistakes to make

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

• Ignoring random start
  Forgetting to randomly select the starting point can introduce bias if there’s any hidden pattern in the list.

• Data not in order
  When you are finding the median value in a set of data, the data must be in order, otherwise the number being picked is not the median, it’s just the middle number in a random list. This is the same for a systematic sample.
  
  Every item of data is structured in an order from a sampling frame (age, zip code, alphabetical order etc), and then your sample is taken.

• Incorrectly calculating the interval for the sample
  Let’s assume you have 1,000 items of data. 5 \% of 1,000 is 50 so you need 50 items of data.
  
  Those 50 items of data must be spread equally across all of the ordered data, so by dividing the number in the population by the sample size, you find the interval between each item of data.
  
  Here you would have \cfrac{1000}{50}=20. You would choose every 20 th item in the list of 1,000 items to get a sample of 50.

Related sampling methods lessons

• Sampling methods
• Stratified random sampling
• Random sampling

The next lessons are

• Probability
• Compound events
• Probability distribution

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