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The less wealthy the neighborhood the more likely private schools do better.

That's because private schools act as a filter for the families that can afford them. Kids from poorer families are more likely to have an unstable home life, insufficient access to meals outside of school lunches, and other things which are the biggest factors in educational success. The voucher system takes funding from the public schools used by the poorest kids and gives it to the private schools used by families that could already afford to go elsewhere.

Private schools can also arbitrarily accept/reject students which gives them an advantage in outcome metrics.