How do I properly install a replacement thermostat in my refrigerator?
This is the simplest thermostat you will get. It has 4 pins on it. 


2 of which will be GND and Vcc
1 will turn on and off (the compressor and other stuff)
1 will be the sensor.




Easiest way to test which is the sensor and which is the switch will be to test continuity on the pins. Set your multimeter to ohm. Place one probe on GND and the other on one of the 2 suspected pins (senors, switch) 


The switch will usually be closed- so no ohm reading (no route)
The sensor will give you some ohm reading (5~50)
GND will give you 0ohm (Short)
Vcc will give you and ohm reading again.(this one can get confused with the sensor.


They should be pretty clearly marked to be honest. Are you sure that the thermostat has not been set to the lowest setting? Maybe the sensor is misinterpreting the thermostat.

The other way to test it is the Live way. This is dangerous so if you do not have gloves and insulated tools, or unsure of the voltage do not attempt this. Have somebody with you to turn off power in case of emergency.


Adjust the thermostat to the lowest level (warmest inside - or like 0/10)
Test voltage on the pins and turn up the thermostat slowly
Keep reading until the voltage goes on or off
Try the next pin the same way.