Root mean square tells you how hard a changing current really works
Household alternating current keeps reversing direction, so its simple average is useless for judging power. Engineers instead square every value, average the squares and take the square root. That root mean square equals the steady direct current that would heat the same resistor just as much.
The recipe works for any list of numbers: square each, find the arithmetic mean of those squares, then take the square root. It is also called the quadratic mean, a special case of the generalized mean. For a continuous signal, the averaging becomes an integral of the squared function over an interval. The value over all time for a periodic wave matches its value over a single cycle, and sampling at evenly spaced moments gives a good approximation. For random processes the expected value replaces the ordinary mean, and estimation theory uses the root-mean-square deviation to measure how far an estimator strays from data.
Wave shape matters. For a pure sine wave centred on zero, the peak-to-peak height is about 2.8 times the RMS value, exactly twice the square root of two. Triangular and sawtooth waves instead run about 3.5 times, twice the square root of three. Arbitrary signals that are neither periodic nor continuous have no fixed relationship at all.
Combining signals follows neat rules. If component waves are orthogonal, meaning the average of any two different ones multiplied together is zero, the total RMS is the square root of the sum of their squared RMS values. Waves perfectly in phase simply add. A signal with both a steady and an alternating part combines the two the same orthogonal way.
In electrical work, average power in a resistor equals the RMS current squared times the resistance, or the RMS voltage squared divided by it, which holds for any periodic waveform. Multiplying RMS voltage by RMS current also yields power. Those shortcuts assume a purely resistive load; loads that store energy as well need the fuller treatment of AC power.
Source: Root mean square
