Root Mean Square (RMS) Calculator

Calculate the RMS of a dataset. Compare with arithmetic mean. Used in physics, engineering, and statistics.

Enter Your Numbers

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Result

Root Mean Square (RMS)
--
RMS value
RMS Value--
Arithmetic Mean--
RMS - Mean Difference--
Count (n)--
Sum of Squares--
Mean of Squares--
Standard Deviation--
RMS = sqrt( (1/n) * sum(x_i^2) )

Step-by-Step Solution

What Is Root Mean Square (RMS)?

The root mean square (RMS), also called the quadratic mean, is a statistical measure of the magnitude of a set of numbers. It is calculated by taking the square root of the mean of the squares of the values. The formula is:

RMS = sqrt( (x₁² + x₂² + ... + xₙ²) / n )

RMS is especially useful when values can be positive or negative, because squaring eliminates the sign before averaging.

RMS vs. Arithmetic Mean

The arithmetic mean simply sums values and divides by the count. RMS always produces a result greater than or equal to the arithmetic mean (this follows from the inequality of means). The difference is most significant when values vary widely or include negative numbers.

For a constant set of identical values, RMS equals the arithmetic mean. The more variation in the data, the larger the gap between the two.

Applications of RMS

Electrical Engineering

RMS voltage and current measure the effective (equivalent DC) value of an alternating current signal. A 120V AC outlet has an RMS voltage of 120V.

V_rms = V_peak / sqrt(2)

Audio Engineering

RMS measures the average loudness of an audio signal more accurately than peak values. Loudness meters use RMS calculations.

dB_rms = 20 * log10(V_rms / V_ref)

Physics

RMS speed of gas molecules in kinetic theory. The molecular speed distribution uses RMS to characterize average molecular motion.

v_rms = sqrt(3kT/m)

Statistics

RMS error (RMSE) measures the average magnitude of prediction errors. It penalizes large errors more heavily than mean absolute error.

RMSE = sqrt( sum(e_i^2) / n )

Step-by-Step Example

Find the RMS of the numbers 2, 4, 6:

  1. Square each value: 2² = 4, 4² = 16, 6² = 36
  2. Sum the squares: 4 + 16 + 36 = 56
  3. Divide by the count: 56 / 3 = 18.667
  4. Take the square root: sqrt(18.667) = 4.3205

Compare with the arithmetic mean: (2 + 4 + 6) / 3 = 4. The RMS (4.32) is larger because it gives more weight to larger values.

Why RMS Is Greater Than the Arithmetic Mean

This follows from the Cauchy-Schwarz inequality (or QM-AM inequality). Squaring amplifies larger values proportionally more than smaller ones, so the square root of the mean of squares is always at least as large as the simple mean. Equality holds only when all values are identical.