CALCULATE WITH CONFIDENCE

Averages & Means

Root Mean Square Calculator

Find the quadratic mean while preserving the exact mean-square value.

Averages & Means

Enter signed values

Exact mean square
Enter one ordinary decimal per line. Commas and scientific notation are not accepted.

Root mean square

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Enter valid values to see the result.

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Quick guide

How to use this calculator

  1. Enter signed values, one per line.
  2. Inspect the exact mean of their squares.
  3. Read an exact rational RMS when possible, otherwise an explicitly approximate square root.

Calculation method

Square, average, then take the non-negative root

Each value is squared exactly, so signs do not cancel. Their exact mean square is formed before its principal square root is evaluated.

Worked example

RMS of 3 and 4

The mean square is (9 + 16) / 2 = 25/2.

RMS = √(25/2) ≈ 3.53553390593274

Supported inputs

Precision and limits

List size

Enter 1–100 values, one per line.

Decimal size

Each value may contain up to 200 digits and 100 decimal places.

Exactness

Exact rational results are never replaced by a rounded decimal; approximations are labelled.

Signed inputs

Positive, negative, and zero values are accepted.

Exact roots

The RMS is labelled exact only when the reduced numerator and denominator are both perfect squares.