√ Root Mean Square (RMS) Calculator
By ToolNimba Editorial Team · Updated 2026-06-19
This RMS calculator works out the root mean square (also called the quadratic mean) of any list of numbers. Type your values separated by commas, spaces, or new lines, and you will instantly see the RMS, the ordinary arithmetic mean for comparison, the count, and the sum of squares. The step-by-step working is shown too, so you can follow exactly how each number is squared, averaged, and then square-rooted.
What is the RMS Calculator?
The root mean square (RMS) is a special kind of average that measures the typical magnitude of a set of numbers regardless of their sign. You find it by squaring every value, taking the mean of those squares, and then taking the square root of that mean. In symbols, RMS = √((x₁² + x₂² + ... + xₙ²) / n). The squaring step is what makes RMS different from the ordinary arithmetic mean: it removes negative signs and gives larger values more weight, so big swings count more than small ones.
Because squaring turns every value positive, the RMS is never negative, and it is always greater than or equal to the absolute value of the arithmetic mean. For a list of identical positive numbers the RMS and the mean are equal, but as the spread of the data grows the RMS pulls ahead of the mean. That gap is exactly why RMS is useful: it captures the size of the variation, not just the central tendency. A set like (−5, 5) has an arithmetic mean of 0 but an RMS of 5, which is the more honest description of how far the values sit from zero.
RMS appears constantly in physics and engineering. The RMS of an alternating voltage or current is the value that delivers the same heating power as a steady direct current of that size, which is why mains electricity is quoted as an RMS figure. In statistics, the standard deviation is the RMS of the deviations from the mean, and the root-mean-square error (RMSE) measures how far predictions fall from observed values. In audio, RMS describes the effective loudness of a waveform. In every case the idea is the same: a meaningful average of magnitudes that does not let positives and negatives cancel out.
When to use it
- Finding the effective (RMS) value of an alternating voltage, current, or any oscillating signal.
- Computing the root-mean-square error (RMSE) or standard deviation as the RMS of a set of deviations.
- Summarising the typical magnitude of a dataset that contains both positive and negative values.
- Checking homework or coursework answers in statistics, physics, or signal-processing classes.
How to use the RMS Calculator
- Type or paste your numbers into the box, separated by commas, spaces, or new lines.
- Negative numbers are fine, the squaring step handles signs automatically.
- Read off the RMS (quadratic mean) along with the arithmetic mean, count, and sum of squares.
- Open the step-by-step section to see each value squared, the mean of squares, and the final square root.
Formula & method
Worked examples
Find the RMS of the numbers 1, 2, 3, 4, 5.
- Square each value: 1² = 1, 2² = 4, 3² = 9, 4² = 16, 5² = 25
- Sum of squares = 1 + 4 + 9 + 16 + 25 = 55
- Mean of squares = 55 ÷ 5 = 11
- RMS = √11 = 3.316625
- For comparison the arithmetic mean = (1 + 2 + 3 + 4 + 5) ÷ 5 = 3
Result: RMS ≈ 3.316625, which is larger than the arithmetic mean of 3
Find the RMS of 3, 4, 5 (for example three measured magnitudes).
- Square each value: 3² = 9, 4² = 16, 5² = 25
- Sum of squares = 9 + 16 + 25 = 50
- Mean of squares = 50 ÷ 3 = 16.666667
- RMS = √16.666667 = 4.082483
- Arithmetic mean = (3 + 4 + 5) ÷ 3 = 4
Result: RMS ≈ 4.082483, slightly above the arithmetic mean of 4
Find the RMS of −5 and 5, where the values cancel in an ordinary average.
- Square each value: (−5)² = 25, 5² = 25
- Sum of squares = 25 + 25 = 50
- Mean of squares = 50 ÷ 2 = 25
- RMS = √25 = 5
- Arithmetic mean = (−5 + 5) ÷ 2 = 0
Result: RMS = 5 even though the arithmetic mean is 0, showing how RMS captures magnitude
How RMS compares with the arithmetic mean for sample sets
| Numbers | Arithmetic mean | RMS |
|---|---|---|
| 2, 2, 2, 2 | 2 | 2 |
| 1, 2, 3, 4, 5 | 3 | 3.316625 |
| 3, 4, 5 | 4 | 4.082483 |
| −5, 5 | 0 | 5 |
| 0, 10 | 5 | 7.071068 |
RMS of common waveforms relative to their peak amplitude (A)
| Waveform | RMS value |
|---|---|
| Sine wave | A ÷ √2 ≈ 0.7071 × A |
| Square wave | A |
| Triangle or sawtooth | A ÷ √3 ≈ 0.5774 × A |
| Steady (DC) value | A |
Common mistakes to avoid
- Confusing RMS with the arithmetic mean. The RMS squares the values first, so it is generally larger than the ordinary mean and is never negative. They only match when every value is identical. Use the mean for a simple average and the RMS when you care about magnitude or spread.
- Dividing before squaring, or averaging then squaring. The order matters: square each value, then average, then take the root. Squaring the mean instead of taking the mean of the squares gives a different (wrong) answer for any data that is not all equal.
- Dropping negative numbers. You do not need to remove or flip negative values. Squaring makes them positive automatically, and that is exactly the point of RMS, so leave the signs in.
- Using n − 1 instead of n. Plain RMS divides the sum of squares by n. The n − 1 divisor belongs to the sample standard deviation, a related but different quantity. Use n here unless you are specifically computing a sample standard deviation.
Glossary
- Root mean square (RMS)
- The square root of the mean of the squares of a set of values, a measure of their typical magnitude.
- Quadratic mean
- Another name for the root mean square, contrasting it with the arithmetic and geometric means.
- Arithmetic mean
- The ordinary average: the sum of the values divided by how many there are.
- Sum of squares
- The total you get by squaring every value and adding the results together.
- Standard deviation
- The RMS of the deviations of values from their mean, a measure of spread.
Frequently asked questions
What is the root mean square (RMS)?
The RMS, or quadratic mean, is the square root of the mean of the squares of a set of numbers: RMS = √((x₁² + x₂² + ... + xₙ²) / n). It measures the typical magnitude of the values and is never negative, because every value is squared before averaging.
How do I calculate RMS?
Square each value, add the squares together, divide that sum by the count of values to get the mean of squares, then take the square root. This calculator does all four steps for you and shows the working.
How is RMS different from the average?
The arithmetic average just adds the values and divides by the count. The RMS squares the values first, which removes signs and weights larger numbers more heavily. As a result the RMS is always at least as large as the absolute value of the arithmetic mean, and the two are equal only when every value is identical.
Can RMS be negative?
No. Because every value is squared during the calculation, the sum of squares and its mean are always zero or positive, so the square root, and therefore the RMS, is never negative.
Why is RMS used for AC voltage and current?
The RMS value of an alternating voltage or current is the steady (DC) value that would deliver the same heating power. That makes RMS the most useful single figure for describing AC electricity, which is why mains supply is quoted as an RMS voltage.
Should I divide by n or n − 1?
Plain RMS divides the sum of squares by n, the number of values. The n − 1 divisor is used for the sample standard deviation, a different statistic. This tool computes the RMS, so it divides by n.