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Thermal Physics
Specific Latent Heat of Fusion of Water
 
Assuming that you do the experiment by adding similarly sized pieces of ice at an approximately regular rate then you should have a situation in which the cup of water/ice mixture spends about the same amount of time above room temperature as it spends below room temperature (and with approximately the same average temperature difference between it and the surroundings).
 
This helps to decrease the errors due to energy exchange between the experimental apparatus and the surrounding air: the energy lost during the first half of the experiment is (hopefully) balanced by energy gained from the surroundings in the second half.
 
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