Root Mean Square Calculator
Find the quadratic mean (RMS) of a data set, with every step shown.
Updated · How we check our maths
- Root mean square
- 4.242641
- Count n
- 5
- Sum of squares Σx²
- 90
- Mean of squares
- 18
- Arithmetic mean
- 1.6
- Mean of absolute values
- 4
Quick answer
RMS = √[(x₁² + x₂² + … + xₙ²) / n]. For 3, −4, 5, 6, −2: squares sum to 90, the mean of squares is 18 and the RMS is √18 ≈ 4.243.
What the Root Mean Square Calculator does
The root mean square — also called the quadratic mean — squares each value, averages the squares and takes the square root. Because squaring removes signs, RMS measures the typical size of values that swing positive and negative, which is why it is used for AC voltage and current, signal noise and prediction errors (RMSE).
RMS is always at least as large as the mean of the absolute values.
The formula
x_rms = √( (1/n) Σ xᵢ² ). For a sine wave with peak A, RMS = A/√2 ≈ 0.707A.
Worked example
A 170 V peak sine wave has RMS ≈ 170 × 0.707 = 120 V — the figure quoted for US mains.
How to use the Root Mean Square Calculator
- Paste or type your numbers.
- Read the RMS and the intermediate sums.
Frequently asked questions
What is the difference between RMS and average?
The average can be zero for values that cancel out; RMS measures their magnitude and is never negative.
What is RMSE?
Root mean square error: the RMS of the differences between predicted and actual values.
Why is RMS used for AC power?
Power depends on voltage squared, so the RMS voltage gives the same heating effect as an equal DC voltage.
Results are estimates for planning and learning. Check important figures with a qualified professional. How we build and check our calculators.