I'm an analytical chemist, here's how I would do it. Take a cooking thermometer, calibrate it and check the medical thermometer against it. This is why chemists (or students taking college chemistry) need to get good at algebra. It's not too difficult. But it's non-trivial. I'm going to work in Celsius so set your thermometers to Celsius mode. (It makes the calculations a tiny bit simpler)
Boil some water and check with the cooking thermometer. Record the temp. Now take some ice and stir it with the cooking thermometer until the temp stabilizes. Now take the linear equation:
where A and B are unknown constants your solving for, T is the temperature of the water bath and t is the reading on the thermometer.
You will end up with a system of two equations, in unknowns A and B. One for 100 degrees Celsius, the booking water, and 1 for 0 degrees Celsius, the ice bath. Let's call The the reading in the hot water and Tc the reading in the ice bath.
0A+B=Tc ---> B=Tc We now have pinned down Tc
100A+Tc=Th ----> (Th-Tc)/100= A Now we know A
(Th-Tc)t/100+Tc=T Now we substitute A and B into the original equation, and we have and equation for the measured temp in terms of the actual temp. But we need the actual temp in terms of the measured temp. So we have to rearrange things a bit.
T-Tc=(Th-Tc)t/100 We just move subtract off the Tc moving it to the other side
100(T-Tc)/(Th-Tc)=t Now we have the actual temp in terms of T the measured temp.
So the final equation for calibrating the cooking thermometer is:
T is the temperature you measure, or , read off the cooking thermometer.
Th is the temperature you read off the cooking thermometer in the he boiling water
Tc is the temperature you read of the cooking thermometer when you are stirring the ice bath (make sure you stir it with the thermometer)
t is the actual temperature of whatever the cooking thermometer is measuring, corrected by the calibration you just did.
Now we put it all together.
Take a lot of water and bring it to a good boil.
Measure the temp with the cooking thermometer. This is Th . Write it down, even if it's exactly 100.
Now take a cup, of crushed ice.
Stir the ice and water with the cooking thermometer.
The temp you measure is Tc. Write it down, even if it's exactly 0.
Now take a cup and put some tap water in it. styrofoam cups works best as the temp will stay more stable. Heat the cup in the microwave in 30 second bursts until it's between 37 and 38 degrees C. If you get it to hot you can stir in some cold water until it cools down enough.
Now, measure the temp with the cooking thermometer. This is T, write it down.
Now measure the water with the medical thermometer. Write that down too.
Now, plug those numbers it into the formula we derived,
Compare that to what the medical thermometer says.