Considerations for Small Samples in Inferential Statistics

Instructor: Maria Airth

Maria has a Doctorate of Education and over 20 years of experience teaching psychology and math related courses at the university level.

Small samples should be avoided in inferential statistics, but there are certain circumstances that require small sample sizes. This lesson reviews those circumstances and procedures to follow when using small samples.

The Reason for Research

What would you say is the reason that researchers conduct studies? Most people would say the reason is to answer questions. If research is conducted to answer questions, it follows that being confident in the resulting answers is important.

Consider how you would feel if you spent months conducting a study on the link between sugar and increased intellectual alertness. At the end of your study, you found that people who consumed moderate amounts of sugar before a learning event were not any better able to retain the information presented than those who did not consume sugar. So you accept the null hypothesis and state that sugar does not have any impact on the retention of new information. Not a very interesting result; but what if you were wrong? What if sugar really does have an impact and for some reason your study just didn't show it? This is called a Type II Error, accepting the null hypothesis when it is false.

Researchers are always trying to reduce Type II errors because they want to be able to confirm that an intervention has an effect. Type II errors cause that effect to be overlooked.

Sample size, the number of participants in a study, is a key factor in Type II errors. Typically, the higher your sample size, the lower your chance of a Type II error. In other words, the more people you have from a population, the more generalizable (able to be applied to the general population) your results are.

So, having large sample sizes is optimal, but it isn't always possible. How can you conduct good research with confident outcomes when using a small sample size? This lesson will discuss special considerations for small samples and will cover how to calculate confidence intervals and degrees of freedom for small sample sizes in inferential statistics, which involves making predictions about a general population based on the outcomes of a sample.

Inferential statistics generalize results from a sample to an entire population.

Why Use Small Samples?

In inferential statistics, samples under the size of 30 are considered small. Why would anyone use small sample sizes if the results are so much less powerful than large sample sizes? Well, when conducting research on serious medical conditions, there may not be enough people with the relevant symptoms to create a large sample. In this case, the knowledge gained from the small sample could be very important even if its overall generalizability is reduced due to the sample size.

When studies are conducted with the target unit being a large group (such as community welfare studies), it is understandable that only small sample sizes can participate in the study. Often, it would not be possible to include large numbers of whole communities in a single study.

Studying animal behavior in endangered species is another scenario that would require a small sample size. It would be hazardously intrusive to attempt to gather a large sample of endangered animals in the wild, thus a small sample size must suffice when studying the creatures.

Small Samples and Statistical Measures

When we do research, we want to have an understanding of how confident we can be that our results, or inference, is correct. Two measures of this are degrees of freedom and confidence intervals.

Degrees of Freedom for Small Samples

Degrees of freedom (the amount of variation allowed in a set of observations) is found by subtracting one from the number of participants (or observations), n - 1. Thus, if you had a sample size of 100, the degrees of freedom would be 100 - 1 = 99. This does not change with small sample sizes.

When conducting specific tests, you will see different results based on small degrees of freedom:

  • 1 sample T-test - the smaller the sample size, the fatter the tails of the test (fewer rejections of the null hypothesis made)
  • Chi-Square test - small sample sizes result in a positively skewed results chart
  • Regression - small sample sizes with large numbers of parameters being tested can result in error reports when the data is analyzed

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