Samples and Populations

Kerby Shedden

September 8, 2020

In most statistical analyses, the sample is the data that we have and can work with.

The sample is a subset of the population, which is usually much larger than the sample (often it is infinite).

Ideally the sample should be representative of the population. We will define the term “representative” more precisely later.

If the sample is not representative of the population, then there is still a possibility of obtaining meaningful results from our analysis of the data, but we need to be aware of the ways in which our sample fails to be representative.

Whether a sample is representative is largely determined by how the sample is collected. “Random samples” (to be discussed later in detail) are often but not always representative. Non-random samples can also be representative, if chosen carefully.

Examples of populations

A population is to some extent imaginary, but some populations are more tangible and easier to work with than others.

If we are studying the adult population of the United States, the population is fairly concrete (if not taken too literally).

Suppose we are conducting an experiment in which we grow plants using different types of fertilizer. We select eight plots of land, and randomly select four of them to use fertilizer A, and the remaining four plots use fertilizer B.

We then grow the plants for one growing season, and measure the yield.

What is the population here? There are several ways to approach this: one possibility is to imagine all the similar plots of land that we could have chosen but did not, then imagine the results that we would have obtained had we used these other plots in our study.

Simple Random Samples

A simple random sample (SRS) from a finite population is one in which every subset of the desired size is equally likely to be selected.

Imagine that our population contains three elements labeled A, B, and C.

A simple random sample of size 2 results from a process in which we obtain the samples {A, B}, {A, C}, and {B, C} one-third of the time each.

Note that in a simple random sample, each element of the population can only be included in the sample one time.

A SRS is often obtained sequentially by sampling “without replacement”. We draw an element from the population at random, and do not return it to the population (therefore, it can never be chosen again).

Independent and identically distributed samples

In the fertilizer experiment (discussed earlier), the population is arguably infinite, as there are infinitely many possible plots of land that could be included in the study.

Also, based on the day that we start the study, the amount of rainfall in the year that the study is run, and many other factors that we cannot control, the results would presumably vary.

Therefore, it is not easy to imagine how we would obtain a simple random sample in the setting of the fertilizer experiment.

An independent and identically distributed sample, or IID sample is a more abstract type of sample that can be used in cases where it doesn’t make sense to think about an SRS.

Independent and identically distributed samples

In an IID sample x1, …, xn:

The second condition implies that the sample can contain repeated elements.

An IID sample can be generated sequentially by sampling “with replacement”. We draw an element from the population, and after noting what value we obtained, we return it to the population. Therefore, we are just as likely to draw it again when selecting the remainder of the sample.

Example of a non-IID sample

Our goal is to obtain a representative sample of the adult population of Japan. We place an add on Facebook (that runs only in Japan), offering 25 USD to anyone who completes our on-line questionnaire. After someone completes the survey, they are asked to name up to five friends who will then be shown the same ad the next time they log into Facebook.

Why is this not an IID sample (or an SRS), and therefore likely not representative of the population?

Example of a non-IID sample

We are interested in rainfall patterns within Texas during late spring. We selected 50 locations inside Texas, and measured the rainfall at those locations for every day in May, 2019.

We can think of the population as being every possible location on every day of May in every year (say since 1900).

Why is this not an IID sample (or an SRS), and therefore likely not representative of the population?