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ADC161S626: SPI data output unstable even with rock steady analog input voltage

Part Number: ADC161S626
Other Parts Discussed in Thread: LMP91200
Hi,
 
We're using the ADC161S626 16-bit ADC in a recent design and are having issues with the data output being erratic. The data output on the SPI Data line seems to vary even though the input voltage is rock solid. The top few MSBs seem to jump around, causing our readings to vary significantly for no change at the input voltage signal. Over the full 16-bit range we are getting a variance of 4000-5000 in the value which is approx 10% variance, even though the input voltage sitting at a steady voltage. To get a result that even remotely matches the input we have to do approx 4000 value running average, and even then it is not as accurate as we'd like it to be.
 
We using our own custom PCB but it is a copy of the design from the LMP91200 evaluation module, so the schematic in the link below is almost identical to the one on our board (except we use the 16-bit ADC and the evaluation module uses the 14-bit one):
 
 
We have the evaluation module here and it shows similar odd behaviour.  We have tried "bit banging" as suggested in a different post and it doesn't help. The averaging was implemented as the datasheet discusses averaging being necessary if the values are below the noise floor, but we are reading approx values in the range 0.8V-1.7V which should be OK.
 
Any help you could provide would be appreciated.