Why would one use root sum square for a calculation?

The square root of the sum of the squares (RSS) can be used to calculate the aggregate accuracy of a measurement when the accuracies of the all the measuring devices are known. The average accuracy is not merely the arithmetic average of the accuracies (or uncertainties), nor is it the sum of them.

Let's say you are conducting a test to verify the resistance of a coil. The coil is built to have a resistance to within one percent of its nominal value. Further, say you have an ohm meter that is accurate to within 0.5 percent of the measured value, but the test leads introduce an uncertainty of two percent. What is the inherent accuracy of any measurement that you make with that set-up?

Use RSS to figure it out.

RSS = SQRT(0.0052 + 0.022) = 0.0206 = 2.06 percent. Note how the RSS result in this case is greater than the largest of the values under the radical. (BTW, that test rig isn't a very good one for verifying whether the coil is within spec. The rig's uncertainty is more than two times the tolerance of the coil.)