What Does It "Mean"? A Review of Interpreting and Calculating Different Types of Means and Standard Deviations

Typically, investigations are conducted with the goal of generating inferences about a population (humans or animal). Since it is not feasible to evaluate the entire population, the study is conducted using a randomly selected subset of that population. With the goal of using the results generated from that sample to provide inferences about the true population, it is important to consider the properties of the population distribution and how well they are represented by the sample (the subset of values). Consistent with that study objective, it is necessary to identify and use the most appropriate set of summary statistics to describe the study results. Inherent in that choice is the need to identify the specific question being asked and the assumptions associated with the data analysis. The estimate of a "mean" value is an example of a summary statistic that is sometimes reported without adequate consideration as to its implications or the underlying assumptions associated with the data being evaluated. When ignoring these critical considerations, the method of calculating the variance may be inconsistent with the type of mean being reported. Furthermore, there can be confusion about why a single set of values may be represented by summary statistics that differ across published reports. In an effort to remedy some of this confusion, this manuscript describes the basis for selecting among various ways of representing the mean of a sample, their corresponding methods of calculation, and the appropriate methods for estimating their standard deviations.