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Root Mean Square Calculator

Find the RMS of your values quickly and accurately.

How it is calculated

The root mean square squares every value, averages the squares and takes the square root: RMS = √((x₁² + x₂² + … + xₙ²) ÷ n). You can enter up to five values; the fifth is optional. The tool counts only the boxes you fill in, so n is the number of non-empty values, and it shows that count as "Values used". Boxes left blank are skipped, but a typed 0 is counted and pulls the result down. A single value returns its own absolute size. Because squaring removes the sign, −3 and 3 give the same result. The result is rounded to two decimals and has the same unit as your values. At least one value is needed.

Worked example

Take the values 2, 4, 6 and 8, with the fifth box empty so n = 4. Squares: 4, 16, 36 and 64. Their sum is 120, and the mean square is 120 ÷ 4 = 30. Square root: √30 = 5.48. The tool reports 4 values used. For comparison the ordinary average of the same numbers is 5, so the RMS is higher because the larger values count for more after squaring.

Try this example →

Common mistakes

  • Taking the square root of each value or dividing by the sum instead of squaring first. The order is square, average, then root.
  • Typing 0 into unused boxes. A 0 is counted as a real value, so with 2, 4, 6 and 0 you get n = 4 rather than 3 and a smaller result. Leave unused boxes empty.
  • Entering only the peak of a sine wave and expecting its RMS. A single value just returns itself; the RMS of a pure sine wave is its peak divided by √2, which is not what five readings taken at random points will give.
  • Confusing RMS with the plain average. They are equal only when all values have the same size; otherwise RMS is larger, and a mix of positive and negative numbers cancels in an average but not in RMS.
  • Entering more than five numbers. The tool has five boxes, so for larger data sets, compute the sum of squares yourself first.

About this calculator

The root mean square (RMS), also called the quadratic mean, is calculated by squaring every value, averaging those squares, and taking the square root of the result: RMS = √((x₁² + x₂² + ... + xₙ²) / n). Because squaring removes the sign of negative values before averaging, RMS gives a magnitude-based average that's always at least as large as the simple arithmetic mean — it's especially sensitive to values with large magnitude, which is exactly the property that makes it useful.

RMS is the standard way electrical engineers describe alternating current and voltage: household "120V AC" is actually an RMS value — the voltage that would deliver the same power to a resistive load as a constant 120V DC source, even though the instantaneous voltage swings both above and below that number in a sine wave. Beyond electronics, RMS is used in audio engineering to measure signal loudness and speaker power ratings, in statistics and signal processing to quantify error magnitude (root mean square error, RMSE, is a standard model-accuracy metric), and in mechanical engineering to characterize vibration and surface roughness.

Up to five values can be entered, and their root mean square is computed directly, giving you the magnitude-based average used across these engineering and statistical applications.

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