Total personal expenditure includes the FCA that is payed to each person in the household.
Therefore to calculate the total tax receipts under the NRST you must add FCA cash flow into the income amount for each quintile. The FCA is received by the citizen before it is expended on consumption and returned to the treasury each month within the total tax receipts.
The net effect is for the NRST to collect an apparent higher monthly tax take accuing to total tax receipts, though the FCA is really only a round robin cash flow.
Thus Total Tax receipts under the NRST are #families * (income + FCA) * 0.23. With each month's FCA returned to the treasury for payout to the following month.
The distribution is calculated from income without the FCA addition (since FCA is a cash payment from government provided to offset quintile tax payments and not counted as income to the family per-se though it is included in expenditure to calculate total tax received).
The net federal tax retained for the federal budget then
Equals (#all_families) * avg_income * 0.23
NOT (#all_families) * avg_income * 0.23 - FCA.
You appear to be calculating the latter as the net total tax and not the former.
Okay, I can see that. Also I used your figure of $5352 for a family of 4. However, with 118.6 families, the avg family size is obviously smaller than 4 and the average FCA must also be smaller. Where can I go to figure out what the FICA is for different family sizes?