The poster still thinks she was correct, but she wasn't, and that's not how averages work.
There are three terms to understand here: mean, median and mode.
Mean is the mathematical term for average.
Median is the one right in the middle of any group of numbers, if you lined them up in order.
Mode is the number that occurs most frequently.
Let's suppose we have eleven people and they all have some buttons. The number of buttons they have is:
5,6,8,8,9,9,9,9,10,11,12.
The mean (average) number they have is all those numbers added together (96) divided by the eleven people you have. So 8.72 (for these purposes we can stop at 2 decimal places). Look at the list of numbers above. Four of those people are below the average and seven are above it. The median number is the one in the middle (so 9) and the mode is the one that occurs the most often (also 9).
Now let's say the numbers of buttons are as follows:
2,2,3,3,3,3,8,9,14,18,19.
The mean is 84 divided by 11, so 7.63. Look at the list: 6 people below the mean (average), 5 above. The median is 3 and so is the mode.
One more group of eleven and their buttons:
2,3,4,5,6,8,9,10,11,14,14.
The mean is 86 divided by 11, so 7.81. Five people below that number and six above. The median is 8 and the mode is 14.
It might sometimes be exactly half the people (or numbers, or whatever it is) are above the average and half below. But usually not. Because the people (or numbers, or whatever) don't all deviate from the median by the same amount. That's the key to understanding it. (I think. It can be hard to understand why others don't understand things that seem obvious to oneself. But I think so). And sometimes mean and median can be the same, or median and mode, or mode and mean, or all three, etc. But by no means always.
The median is the one in the middle (i.e. if you could line up everyone in the country by IQ, the one stood at the exact halfway point would have the median IQ). Exactly half of people are above / below the median. Not the mode.