Question

What is the significance of squaring a calculated average number?

Answer 1

The root-mean-square (RMS) is not a statistic you hear to much about, because it is mostly used as a part of other statistics, such as the standard deviation, which are much more famous. The root mean square is a measure of the magnitude of a set of numbers. It gives a sense for the typical size of the numbers. For example, consider this set of numbers:

-2, 5, -8, 9, -4

We could compute the average, but this doesn't tell us much because the negative values cancel the positive values, leaving an average of zero. What we want is the size of the numbers without regard for positive or negative.

The easiest way to do this is to just erase the signs and compute the average of the new set:

2, 5, 8, 9, 4

Average = 5.6

But ... that's not how statisticians decided to do it. For reasons of their convenience, they chose a different approach. Instead of wiping out the signs, they square every number (which makes them all positive), then take the square root of the average. It's like this:

TO CALCULATE RMS:

SQUARE all the values
Take the average of the squares
Take the square root of the average

For example, the RMS of

-2, 5, -8, 9, -4

is 6.16

The RMS is always the same as or just a little bit larger than the average of the unsigned value