But where do the white students who do worse, live?
When you look at white students alone, you'll see that class/socioeconomic background is a major indicator of educational attainment. In areas that are predominantly white, like the North generally, the postcode criteria may cover them.
Secondly, you'll see white immigrants students do better than their native peers.
The other problem with the statistics that you cite is that they didn't breakdown the category of "Black". You'll often see notable differences between Black African and Black Caribbean people.
The emergence of a Black British 3rd and 4th generation demographic of mixed Black heritage will change statistics in the future. We will likely be comparing the outcomes of Black immigrants to Black British people and seeing if and why there is a difference in outcomes.
If things stay as they are, what we will see is that our Black British students have worse outcomes than their Black immigrant and 2nd generation counterparts. Pretty much like white immigrants do better than their native counterparts.
That is a failing of our society.