Micro Electro Mechanical Systems (MEMS) is one of the fastest growing field in silicon industry. Low cost production is key for any company to improve their market share. MEMS testing is challenging since input to test a MEMS device require physical stimulus like acceleration, pressure etc. Also, MEMS device vary with process and requires calibration to make them reliable. This increases test cost and testing time. This challenge can be overcome by combining electrical stimulus based testing along with statistical analysis on MEMS response for electrical stimulus and also limited physical stimulus response data. This thesis proposes electrical stimulus based built in self test(BIST) which can be used to get MEMS data and later this data can be used for statistical analysis. A capacitive MEMS accelerometer is considered to test this BIST approach. This BIST circuit overhead is less and utilizes most of the standard readout circuit. This thesis discusses accelerometer response for electrical stimulus and BIST architecture. As a part of this BIST circuit, a second order sigma delta modulator has been designed. This modulator has a sampling frequency of 1MHz and bandwidth of 6KHz. SNDR of 60dB is achieved with 1Vpp differential input signal and 3.3V supply