When we form a statistical sample we always need to be careful in what we are doing. There are many different kinds of sampling techniques that can be used. Some of these are more appropriate than others.

Often what we think would be one kind of sample turns out to be another type. This can be seen when comparing two types of random samples. A simple random sample and a systematic random sample are two different types of sampling techniques. However, the difference between these types of samples is subtle and easy to overlook. We will compare systematic random samples with simple random samples.

### Systematic Random vs. Simple Random

To begin with, we will look at the definitions of the two types of samples that we are interested in. Both of these types of samples are random and suppose that everyone in the population is equally likely to be a member of the sample. But, as we will see, not all random samples are the same.

The difference between these types of samples has to do with the other part of the definition of a simple random sample. To be a simple random sample of size *n*, every group of size *n* must be equally likely of being formed.

A systematic random sample relies on some sort of ordering to choose sample members. While the first individual may be chosen by a random method, subsequent members are chosen by means of a predetermined process. The system that we use is not considered to be random, and so some samples that would be formed as a simple random sample cannot be formed as a systematic random sample.

### Example

To see why this is not the case, we will look at an example. We will pretend that there is a movie theater with 1000 seats, all of which are filled. There are 500 rows with 20 seats in each row. The population here is the entire group of 1000 people at the movie. We will compare a simple random sample of ten moviegoers with a systematic random sample of the same size.

- A simple random sample can be formed by using a table of random digits. After numbering the seats 000, 001, 002, through 999, we randomly choose a portion of a table of random digits. The first ten distinct three digit blocks that we read in the table are the seats of the people who will form our sample.
- For a systematic random sample, we can begin by choosing a seat in the theater at random (perhaps this is done by generating a single random number from 000 to 999). Following this random selection, we choose this seat’s occupant as the first member of our sample. The remaining members of the sample are from the seats that are in the nine rows directly behind the first seat (if we run out of rows since our initial seat was in the back of the theater, we start over in the front of the theater and choose seats that line up with our initial seat).

For both types of samples, everyone in the theater is equally likely to be chosen. Although we obtain a set of 10 randomly chosen people in both cases, the sampling methods are different. For a simple random sample, it is possible to have a sample that contains two people who are sitting next to each other. However, by the way that we have constructed our systematic random sample, it is impossible not only to have seat neighbors in the same sample but even to have a sample containing two people from the same row.

### What’s the Difference?

The difference between simple random samples and systematic random samples may seem to be slight, but we need to be careful. In order to correctly use many results in statistics, we need to suppose that the processes used to obtain our data were random and independent. When we use a systematic sample, even if randomness is utilized, we no longer have independence.