[Jack of all Trades]

Wrong. Exactly 50% of the population is above the median IQ. It's entirely possible for more than 50% of a sample to be either above or below the average value of any metric you choose to apply. For example, the average of 3,3,3,1 is 2.5. 75% of the set is above average.

[Sherman Logan]

Sorry. You’re entirely correct for a small group, or when speaking of statistics in general. In this particular case, the IQ scale was specifically designed so that 50% of the population was above the mean as well as the median.

Where the mean and median fall within subgroups of the population may of course vary significantly.

[Sherman Logan]

BTW, the term “average” may refer to the median, the mode, or most commonly, the mean, which is apparently what you assume is the only meaning of the term.

You are, of course, quite correct that for many data sets these can be very different values. That which use of the term is meant is not commonly stated is one of the most significant failure of popular writing about statistics.