How this calculator works
The root mean square names its own three steps: square every value, take the mean of those squares, then take the square root. Squaring removes signs, so a negative number and a positive number of the same size contribute equally. Because squaring gives extra weight to large values, the RMS is never smaller than the arithmetic mean when the values are real, and it is equal only when they are all identical in size.
Formula
RMS = √( (x₁² + x₂² + ... + xₙ²) / n ). The arithmetic mean is (x₁ + x₂ + ... + xₙ) / n.
Worked example
For 3, 4 and 5 the squares are 9, 16 and 25, which total 50. The mean square is 50/3 ≈ 16.667, so the RMS is √16.667 ≈ 4.0825. The arithmetic mean of the same numbers is 4.
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Frequently asked questions
When is RMS more useful than the mean?
When the sign of a value should not cancel a value of the opposite sign. Alternating current is the standard example: the average voltage over a full cycle is zero, but the heating effect depends on the RMS value.
Why is RMS at least as large as the arithmetic mean?
This follows from the inequality between the quadratic mean and the arithmetic mean. The gap grows with how spread out the values are, and it is zero only when every value has the same magnitude.
Does the order of the numbers matter?
No. Squaring and addition are commutative, so any order gives the same RMS.
Can I use negative numbers?
Yes. A negative value contributes its square, which is positive. For example -3, 0 and 3 has the same RMS as 3, 0 and 3, namely √6.