3.7 : Root Mean Square

If in an experiment, data values have a probability of being both positive and negative, neither the arithmetic mean, the geometric mean, nor the harmonic mean can be used to calculate the central tendency of the data set. In particular, if the positive and negative values are equally likely, the arithmetic mean is close to zero.

For example, consider the velocity of gas molecules in a container. The gas molecules are moving in different directions, which might impart positive and negative values to the velocity. Hence, the average speed of all the gas molecules may get close to zero, which is not true.

One alternative, however, is to consider only the absolute values of such a quantity. Another is to calculate its root mean square. Calculating the square of each gas molecule’s speed overcomes the positive or negative signs. The square root of the sum of all the squares divided by the total number of elements is defined as the root mean square.

Calculating the root mean square is often more than just a mathematical exercise. For example, in the case of velocities of gas molecules, it can be shown that the root mean square is directly proportional to the square root of the temperature of gas molecules.