In statistics there are many different types of sampling techniques. These techniques are named according to the way in which the sample is obtained. In what follows we will examine a systematic sample and learn more about the orderly process used to acquire this type of sample.

### Definition of a Systematic Sample

A systematic sample is obtained by a very straightforward process:

- Begin with a positive whole number
*k.* - Look at our population and then select the
*k*th element. - Select the 2kth element.
- Continue this process, selecting every kth element.
- We stop this selection process when we have reached the desired number of elements in our sample.

### Examples of Systematic Sampling

We will look at a few examples of how to conduct a systematic sample.

For a population with 60 elements will have a systematic sample of five elements if we select population members 12, 24, 36, 48 and 60. This population has a systematic sample of six elements if we select population members 10, 20, 30, 40, 50, 60.

If we reach the end of our list of elements in the population, then we go back to the beginning of our list. To see an example of this we start with a population of 60 elements and want a systematic sample of six elements. Only this time, we will start at the population member with number 13. By successively adding 10 to each element we have 13, 23, 33, 43, 53 in our sample. We see that 53 + 10 = 63, a number that is greater than our total number of 60 elements in the population. By subtracting 60 we end up with our final sample member of 63 - 60 = 3.

### Determining k

In the above example we have glossed over one detail. How did we know what value of *k* would give us the desired sample size? The determination of the value of *k* turns out to be a straightforward division problem. All that we need to do is to divide the number of elements in the population by the number of elements in the sample.

So to obtain a systematic sample of size six from a population of 60, we choose every 60/6 = 10 individuals for our sample. To obtain a systematic sample of size five from a population of 60, we choose every 60/5 = 12 individuals.

These examples were somewhat contrived as we ended up with numbers that worked together nicely. In practice this is hardly ever the case. It is quite easy to see that if the sample size is not a divisor of the population size, then the number *k* may not be an integer.

### Examples of Systematic Samples

A few examples of systematic samples follow below:

- Calling every 1000th person in the phone book to ask their opinion on a topic.
- Asking every university student with ID number ending in 11 to fill out a survey.
- Stopping every 20th person on the way out of a restaurant to ask them to rate their meal.

### Systematic Random Samples

From the above examples, we see that systematic samples do not necessarily need to be random. A systematic sample that is also random is referred to as a systematic random sample. This type of random sample can sometimes be substituted for a simple random sample. When we make this substitution we must be certain that the method we use for our sample does not introduce any bias.