How many standard deviations is the average away from the median?
Probably something very closely approximating zero.
That's true of a large amount of data sets, regardless of topic, if the set is large enough. Unless you have a bi-modal distribution, or outliers that significantly skew the mean, or some other very non-normal distribution.
It's also not how mean vs median is usually described.
Mean is what most think of when we say "average". If you have, say, pfe scores of 200, 203, 205, 205, 210, 215, 230, 246, 255, 276, and 285, the mean is the sum of those scores, divided by the number of points. In this case, 230. The median is the middle point of the set. In this example 215. They're far apart because it's a small set of 11 points that I deliberately skewed. With roughly 300 swabs completing the summer training, assuming there aren't a bunch of studs skewing the mean high, or a bunch of pudgy out of shape kids skewing it low, or both, the mean and median scores will be very close.
In statistics, standard deviation describes how far from the mean (not median) a given percentage of data points will be found. In a perfectly normal distribution, in a large data set, +-1 standard deviation will encompass I think 2/3 of the data points. +-3 standard deviations covers something like 95-99%. I'm sure that's a little off, but it is good enough for discussion purposes.
Outliers are usually defined as points beyond 3 standard deviations from the mean.