The Box Plot visualization is a convenient graphical tool for displaying descriptive statistical measures such as median, mean, and quartile. We have to look at some statistical theory now because the main reason for using a box plot is to show statistical measures, and it's important that you understand the basic principles. In broad terms, statistical analysis may be divided between parametric and nonparametric approaches.
Parametric statistics are based on an assumption that the data falls into some known probability distribution, and they make inferences about the parameters of that distribution. They can provide good accuracy and precision and, therefore, good statistical power, but they are not robust because they depend on assumptions about distribution. The mean (what people commonly refer to as the average) and standard deviation are parametric statistical measures.