Root Mean Square Calculator
Free Root mean square Calculator for arithmetic. Enter values to get step-by-step solutions with formulas and graphs. Free to use with no signup required.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Root Mean Square Calculator
Calculator
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Formula: RMS = √(Σx² / n)
Worked example — RMS ≈ 4.0825 (arithmetic mean = 4.0000)
Formula
RMS = √(Σx² / n)
Square every value in the dataset, average the squares, then take the square root of that average. Because squaring removes the sign, RMS reflects the true magnitude of a dataset even when values are both positive and negative — unlike the arithmetic mean, which can average out to zero.
Worked Examples
Example 1: RMS of a small dataset
Problem:Find the RMS of the values 3, 4, and 5.
Solution:Sum of squares = 3² + 4² + 5² = 9 + 16 + 25 = 50. Mean of squares = 50/3 ≈ 16.667. RMS = √16.667 ≈ 4.0825.
Result:RMS ≈ 4.0825 (arithmetic mean = 4.0000)
Example 2: RMS voltage of a sine wave
Problem:A sinusoidal AC voltage has a peak (amplitude) of 170 volts. Find its RMS voltage.
Solution:For a pure sine wave, RMS = peak / √2 = 170 / 1.4142 ≈ 120.2.
Result:RMS voltage ≈ 120 volts (matches standard US household mains)
Frequently Asked Questions
What is the root mean square (RMS), and how is it different from the arithmetic mean?
The root mean square is found by squaring every value, averaging those squares, and then taking the square root of that average — RMS = √(Σx² / n). Unlike the arithmetic mean, squaring the values before averaging means negative and positive values can never cancel each other out, and larger-magnitude values are weighted more heavily. The RMS of any dataset with more than one distinct value is always greater than or equal to its arithmetic mean.
How is RMS used to measure error or deviation, as in RMSE?
Root Mean Square Error (RMSE) applies the same formula to a set of prediction errors (actual minus predicted values) rather than raw data. Squaring the errors before averaging penalizes large errors disproportionately and prevents positive and negative errors from canceling, making RMSE one of the most widely used accuracy metrics in statistics, machine learning, and weather forecasting.
What is the relationship between RMS and standard deviation?
If a dataset has mean μ and standard deviation σ, its RMS value equals √(μ² + σ²). When the mean is zero (as with a symmetric AC signal or a set of pure deviations from an average), RMS and standard deviation become numerically identical — this is why RMS is often described as 'standard deviation around zero.'
Why does squaring the values before averaging matter mathematically?
Squaring makes every term non-negative, so an equal mix of large positive and negative values (which would average to near zero arithmetically) instead produces a meaningfully large RMS value reflecting their true magnitude. This is precisely why RMS, not the arithmetic mean, is used whenever a dataset's 'sign' fluctuates but its magnitude — like power, energy, or error size — is what actually matters.
Is RMS always larger than the arithmetic mean of the same absolute values?
For any set of values that are not all identical, the quadratic mean (RMS) is always greater than or equal to the arithmetic mean of their absolute values — a consequence of the QM-AM-GM-HM inequality chain in mathematics. The two are exactly equal only when every value in the dataset has the same magnitude.
How do I compute RMS by hand for a small dataset?
Square each value, sum the squares, divide by the count to get the mean of squares, then take the square root. For example, for {3, 4, 5}: squares are 9, 16, 25; sum = 50; mean = 50/3 ≈ 16.67; √16.67 ≈ 4.08 — the RMS, compared to the arithmetic mean of exactly 4.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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