The pullup resistor is needed because the data line to the sensor is bidirectional.
The sensor only sends data after its been told to by the Micro .
However, depending on the code that actually reads the Sensor, some Micros have weak pullups in them
that means that the resistor may not be needed.
You would have to study the code thats actually doing the sensor reading to see if this is the case.
The resistor value isnt critical, 10K is a good value.
Also, some suppliers of the sensor send it on a small breakout board, which already has the resistor installed.
If the Sensor is jumping back and forth between degree readings 3 D apart, then there is something wrong with either the code thats reading it
or the wiring , or the sensor itself.
The Sensors are pretty rugged so its not likley to be the Sensor.
They either work properly or not at all.