Not OP but I think the 50% of cases above or below only applies if you refer to the median (https://en.wikipedia.org/wiki/Median). If you refer to the average, or mean, the distribution could be different.

But I’ve never mentioned median. I’m stating that by definition, 50% of $ outperforms SP500. Sure that could be skewed, with a median underperforming, where in the extreme 1 participant has $20T outperforming and the other 1,000 collectively have $20T underperforming. But on average your $ has a 50/50 chance of under or over performing total market (and thus basically SP500) by throwing darts at a board.

But you mentioned average (mean). You can have the same average but different median, and only if you would talk about median it would be guaranteed that 50% of the cases are above and 50% under.

In the case you mentioned, with an average, there might be extreme outliers that make the average different from the median, thus rendering the assertion that 50% of the cases are above the average and 50% under false.

Thanks. I couldn't not bring myself to explain what a median is.

Which definition do you mean? SPX is a cap-weighted index, but that's very different from "50% of capital outperforms the index".