When would we use the 5-number summary to describe the distribution of a data set rather than using the sample mean ##barx## and sample standard deviation ##s##?
The sample mean and SD are fine if you believe your data conform with a parametric model.
Besides that, Standard Deviation is easily interpreted in the case of the normal distribution, otherwise it isn't that easy.Of course, if you are certanly that the data folows one distibution you can just calculate the probability using the density , but it isnt as straightforward as it is with the normal.
On top of that even if you are sure that the data follow a certain distribution, you need enough data to be sure that the sample mean and the sample standard deviation are close enough, in probability, to the real mean and standard deviation.
All those thing combined makes the case to use a five number summary
The five number summary is often used as an exploratory summary of your data, and don't require that you make any assumptions about the way you model the data generating mechanism. take income data as an example – the data are said to be skewed because many people earn small amounts, most earn a modest amount and only one or two earn billions. You get a sense of this shape when you look at the five number summary. However, if you don't acknowledge the skewed / asymmetric distribution of values you could get a very bad impression of the data by looking at just the mean and SD.