Root Mean Square (RMS) Calculator
The root mean square (RMS), also called the quadratic mean, is computed by squaring all values, averaging the squares, and taking the square root of that average. It is larger than or equal to the arithmetic mean and is especially useful for alternating or oscillating quantities because squaring removes the sign, capturing the magnitude regardless of direction. In electrical engineering, RMS gives the effective voltage or current of an AC signal. In statistics, the standard deviation is itself an RMS of deviations from the mean. Enter a comma-separated list of numbers to compute the RMS.
Root mean square formula
RMS = √((x1² + x2² + ... + xn²) / n)
Equivalently: square each value, compute the mean of those squares, then take the square root. For a pure sine wave with peak amplitude A, RMS = A / sqrt(2) = A times 0.7071.
RMS vs arithmetic mean
- The arithmetic mean of 3, 4, 5 is 4.00. The RMS is sqrt((9 + 16 + 25)/3) = sqrt(16.67) = 4.08.
- For negative values: the arithmetic mean of -3 and 3 is 0, but the RMS is sqrt((9 + 9)/2) = 3.00.
- RMS is always greater than or equal to the arithmetic mean (by the QM-AM inequality). They are equal only when all values are identical.
- In AC circuits: mains voltage in the US is specified as 120 V RMS, meaning peak voltage is 120 times sqrt(2), approximately 170 V.
Frequently asked questions
What is the root mean square?
The root mean square (RMS), also called the quadratic mean, is the square root of the arithmetic mean of the squares of a set of values. For a dataset x1, x2, ..., xn, RMS = sqrt((x1 squared plus x2 squared plus ... plus xn squared) divided by n). It is always greater than or equal to the arithmetic mean for non-negative numbers.
When is RMS used?
RMS is widely used in physics and engineering. In AC electricity, the RMS voltage gives the equivalent DC voltage that delivers the same power. In acoustics, RMS sound pressure gives the effective sound level. In statistics, the standard deviation is the RMS of deviations from the mean.
How does RMS differ from the arithmetic mean?
The arithmetic mean adds all values and divides by n. RMS squares all values first, takes the mean, then takes the square root. The RMS gives more weight to larger values (because of squaring), so it is larger than the arithmetic mean unless all values are equal.
What is the RMS voltage for a sinusoidal AC signal?
For a pure sinusoidal signal with peak voltage Vpeak, the RMS voltage = Vpeak divided by the square root of 2, approximately 0.7071 times Vpeak. For US household current with Vpeak approximately 170 V, RMS voltage equals approximately 120 V.
Can RMS be computed for negative values?
Yes. Because values are squared before averaging, negative values contribute positively to the RMS. This makes RMS suitable for alternating signals where values oscillate between positive and negative.
Official sources
- NIST/SEMATECH e-Handbook of Statistical Methods: Measures of Location.
- NIST Digital Library of Mathematical Functions: DLMF Home.
Reviewed by the CalculatorHub team, edited by James Graham, 15 June 2026. See our methodology.