Suppose a basketball team scored 44,58,50,61,62 in the previous 5 games, and scored 50 in the next game. How to statistically determine if the team is improving or not, apart from comparing the immediately previous games (i.e 62 and 50).
For the 6 games in your sample, you could calculate a Pearson correlation coefficient
between score and number of the game (from game 1 to game 6). This would tell you
whether there was a marked linear trend within your sample of 6. But maybe I did not
understand your question correctely.
@Karabiner is correct. However such an approach does not control for differences between competitor teams, percentage of playing time per player, home/away court advantage, etc. So just keep that in mind. You would be able to note an association, but not ascribe that the approach controlled for endogeneity.