How to estimate displacement error using accelerometer
One way to test it would be to use a potentiometer that has a linear motion of knob. A hack for it could be to use a regular lab rheostat or for better accuracy, you may use a good damper position sensor if you can get one. You need to fix your MPU sensor on the knob (you may need to extend cables here) and also measure the displacement of the knob on potentiometer using ADC. Calibrate the ADC readings in your code using a ruler. Now you need to carefully move the knob, collect data from both ADC and MPU simultaneously and compare.
You need to be very careful about error induced in displacement measurement due to flaws of a setup like free play of knob, motion of sensor wrt knob, ADC calibration and quantization error etc. Try to minimize these. You may use MATLAB and SIMULINK to analyze the experimental data and may be to implement filters for your MPU but I don't think you can estimate the error on these tools without any physical experiment.