The example you give is true. However, it's a toy example that doesn't usually relate to real world data sets.
In most cases, with the amount of data you have, the rounding washes out any real impact of the effect, and you often have measurement error far larger than it anyway.
But yeah, if you're ranking 9 people on an easily knowable stat, then technical 4/9 are below median, 4/9 above, and 1 is exactly median.
With the IQ example you gave, it's just that people don't speak with that level of precision about most things--also, that level of precision is very rarely useful.
So, you're not wrong, but kind of sideways to the discussion at hand. It's actually a common problem in business, where really detail-oriented analysts miss the forest for the trees over stuff like this.
Regarding median vs mean vs average (as the 1st reply to your comment mentioned), median and mean are generally equal as long the values being measure are symmetrically dispersed around the "average". If you get a fat tail in one direction or the other, the mean will be skewed in the direction of that tail, and you need to be careful.
In most cases, with the amount of data you have, the rounding washes out any real impact of the effect, and you often have measurement error far larger than it anyway.
But yeah, if you're ranking 9 people on an easily knowable stat, then technical 4/9 are below median, 4/9 above, and 1 is exactly median.
With the IQ example you gave, it's just that people don't speak with that level of precision about most things--also, that level of precision is very rarely useful.
So, you're not wrong, but kind of sideways to the discussion at hand. It's actually a common problem in business, where really detail-oriented analysts miss the forest for the trees over stuff like this.
Regarding median vs mean vs average (as the 1st reply to your comment mentioned), median and mean are generally equal as long the values being measure are symmetrically dispersed around the "average". If you get a fat tail in one direction or the other, the mean will be skewed in the direction of that tail, and you need to be careful.