Antenna radation pattern measurement requires a test range (best an open area without obstacles), a reference antenna, a transmitter and a calibrated receiver.
Details can be found in antenna textbooks and you can google commercial companies offering such service.
The pattern is measured relative to a maximum directivity in dB, and antenna under test is orientedin small angular steps to obtain a point-to-point pattern. Similarly polarization response can be taken, then the reference antenna must be calibrated for pol/copol response.