Root Mean Square Calculator

Root Mean Square, also known as the quadratic mean, is the square root of the arithmetic mean of the squares of a set of values. It is denoted as RMS or x_rms and is especially useful when values can be negative, because squaring removes the sign before averaging. The formula is: RMS = √((x₁² + x₂² + ... + xₙ²) / n). See also our Interquartile Range (IQR).

The arithmetic mean simply sums all values and divides by n, whereas RMS first squares each value, averages those squares, then takes the square root. RMS is always greater than or equal to the arithmetic mean (by the inequality of means), and it gives more weight to larger values. This makes RMS better suited for datasets with large variations or signed quantities.

In electrical engineering, RMS is used to express the effective value of an alternating current (AC) or voltage. An AC signal with an RMS value equal to a DC value delivers the same power to a resistive load. For a pure sine wave, the RMS voltage equals the peak voltage divided by √2 (approximately 0.707 × peak value).

For a pure sine wave, V_rms = V_peak / √2 ≈ 0.707 × V_peak. So a standard 120V AC supply (RMS) has a peak voltage of about 170V, and a 230V RMS supply peaks at around 325V. This relationship only holds for pure sinusoidal waveforms. You might also find our find Absolute Error with Absolute Error Calculator useful.

Yes. Because each value is squared before averaging, negative numbers contribute positively to the RMS. The sign is eliminated by squaring, so RMS is always a non-negative value regardless of whether your dataset contains negative numbers.

RMS is widely used in physics (describing wave amplitudes), electrical engineering (AC voltage and current), signal processing (audio levels and noise measurement), statistics (as the quadratic mean), and finance (volatility calculations). Anywhere you need an effective or representative magnitude of a set of varying quantities, RMS is the appropriate measure.

Follow these three steps: (1) Square each value in your dataset. (2) Calculate the arithmetic mean (average) of those squared values. (3) Take the square root of that mean. For example, for values 4, 2, 6, 8: squares are 16, 4, 36, 64; mean of squares = 120/4 = 30; RMS = √30 ≈ 5.477.

There is a well-known inequality of means: Harmonic Mean ≤ Geometric Mean ≤ Arithmetic Mean ≤ RMS. This means RMS is always the largest of the four classical means for any dataset with positive values (unless all values are equal, in which case all means are the same).