I offer to you a number of scatter plots that I made, with percent of students passing the various test along the bottom axis, and the percent of economically deprived students along the vertical axis. I think you will find that there is a very strong correlation.
First, for the regular DC public schools, a graph of passing rates in reading against poverty rates:
Then, for the regular DC public schools, a graph of passing rates in math against poverty rates:
and the same thing for the charter schools: