The mean, after all, is the average. The mean is calculated by simply adding all the values that constitute the data set and then dividing the sum by the total number of values. The median, in turn, is simply the middle value: half the values above and half below. Stack the values cardinally low to high and the median is the middle value.
Many who expound in the media for a living fail to realize the distinction. They conflate mean with median. The mean, in their mind, is also the middle value.
The mean is by definition not the middle-value (the median), though it can approximate it. We see that when the variability in the values that compose the data set is slight. If we were to calculate the mean annual income of 100 people and the income range of values reside between $100,000 and $120,000, the odds are high that the mean will approximate the median.
Let's reduce variability further. When the values in the data set are equidistant, the mean and median will be the same number if the data set comprises an odd number of values. We have a data set that runs from 5 to 95 in equal five-number increments (e.i., 5, 10, 15, 20, etc.). That's 19 values total. The mean and median are both 50.
But let's add a value to the top end of our example. The number of values increases to 20 (5 to 100). An even number of values offers no middle value. What's a poor boy to do? Average the two middle values that bifurcate the data equally (50 and 55, in this instance). The median assumes the non-value number of 52.5, which is also the mean.
That the mean and median are the same number is a quirk in my example. Should I then overlook the faux pas when the expounder expounds incorrectly?
If more variability and randomness pervaded the data set, chances are high the mean and the median will diverge. The divergence with expand with variability and randomness. The number of data values in the set also influences proximity. To be sure, the larger the data set, the closer the mean and median will be, but the two can easily diverge to the distance that separates London and Melbourne. Here's what I mean (not what I median).
Ten people occupy a room. You are charged to calculate the mean and median wealth of the room occupants. After a bit of sleuthing and surveying, you find the individual incomes. You find they slot within a range of $500,000 to 750,000. After a quick crunching of the number, you find the the difference between mean and median to be small.
But no sooner than you compete your mean and median calculations, Warren Buffett and Bill Gates stroll in. Watch what happens next.
The average wealth of each individual is measured in billions of dollars, as opposed to hundreds of thousands a minute ago. The mean devolves into a meaningless number to all involved, and more meaningless as it relates to the original 10 occupants. As for the median, it still paints a picture that somewhat resembles reality, just less so as reality pertains to Messrs. Buffett and Gates.