Population versus sample

In general, the goal of statistical modeling is to answer a question about a group by making an inference about that group. The group we are making an inference on could be machines in a production factory, people voting in an election, or plants on different plots of land. The entire group, every individual item or entity, is referred to as the population. In most cases, the population of interest is so large that it is not practical or even possible to collect data on every entity in the population. For instance, using the voting example, it would probably not be possible to poll every person that voted in an election. Even if it was possible to reach all the voters for the election of interest, many voters may not consent to polling, which would prevent collection on the entire population. An additional consideration would be the expense of polling such a large group. These factors make it practically impossible to collect population statistics in our example of vote polling. These types of prohibitive factors exist in many cases where we may want to assess a population-level attribute. Fortunately, we do not need to collect data on the entire population of interest. Inferences about a population can be made using a subset of the population. This subset of the population is called a sample. This is the main idea of statistical modeling. A model will be created using a sample and inferences will be made about the population.

In order to make valid inferences about the population of interest using a sample, the sample must be representative of the population of interest, meaning that the sample should contain the variation found in the population. For example, if we were interested in making an inference about plants in a field, it is unlikely that samples from one corner of the field would be sufficient for inferences about the larger population. There would likely be variations in plant characteristics over the entire field. We could think of various reasons why there might be variation. For this example, we will consider some examples from Figure 1.2.

Figure 1.2 – Field of plants

The figure shows that Sample A is near a forest. This sample area may be affected by the presence of the forest; for example, some of the plants in that sample may receive less sunlight than plants in the other sample. Sample B is shown to be in between the main irrigation lines. It’s conceivable that this sample receives more water on average than the other two samples, which may have an effect on the plants in this sample. The final Sample C is near a road. This sample may see other effects that are not seen in Sample A or B.

If samples were only taken from one of those sections, the inferences from those samples would be biased and would not provide valid references about the population. Thus, samples would need to be taken from across the entire field to create a sample that is more likely to be representative of the population of plants. When taking samples from populations, it is critical to ensure the sampling method is robust to possible issues, such as the influence of irrigation and shade in the previous example. Whenever taking a sample from a population, it’s important to identify and mitigate possible influences of bias because biases in data will affect your model and skew your conclusions.

In the next section, various methods for sampling from a dataset will be discussed. An additional consideration is the sample size. The sample size impacts the type of statistical tools we can use, the distributional assumptions that can be made about the sample, and the confidence of inferences and predictions. The impact of sample size will be explored in depth in Chapter 2, Distributions of Data and Chapter 3, Hypothesis Testing.