About the Root Mean Square Calculator
This root mean square calculator finds the RMS, also called the quadratic mean, of a set of numbers: square every value, average the squares, then take the square root. Paste your data and it returns the RMS along with the mean square, the sum of squares, the ordinary mean, the peak absolute value and the crest factor (peak ÷ RMS). Negative numbers are fine — squaring makes every value count by its size.
RMS is the natural "average size" whenever the sign does not matter or energy and power are involved. Electricians and electronics students use it for AC voltage and current (a 170 V peak sine wave is about 120 V RMS), audio engineers for signal level, physicists for the RMS speed of gas molecules, and data analysts for the typical magnitude of errors or fluctuations around zero. Switch to waveform mode to get the RMS and average rectified value of a sine, square, triangle, sawtooth or rectified wave directly from its peak amplitude.
For sampled data, the RMS of the samples approximates the RMS of the underlying signal only if you sample whole cycles evenly. Waveform mode assumes an ideal, symmetric wave with no DC offset.
With the default inputs, the root mean square (rms) is 3.89444. Change any value above to recalculate instantly.
How to use the root mean square calculator
- 1Choose whether to calculate the RMS of a list of numbers or a waveform.
- 2For data, paste your values separated by commas, spaces or new lines.
- 3For a waveform, pick the wave shape and enter its peak amplitude.
- 4Read the RMS, mean square, peak and crest factor.
Formula and method
Square each value so that negatives and positives both contribute by their magnitude, average the squares to get the mean square, and take the square root to return to the original units. The result is always at least as large as the mean of the absolute values, because squaring gives larger values more weight; the two are equal only when every value has the same magnitude.
For a continuous periodic signal the sum becomes an integral over one period. Integrating sin² over a cycle gives ½, so a sine wave of peak A has RMS A/√2. RMS matters in electricity because a resistor dissipates the same average power from an AC signal as from a DC voltage equal to its RMS value. The useful identity RMS² = mean² + σ² links RMS to the mean and the population standard deviation.
- xᵢ
- Each data value (may be negative)
- n
- Number of values
- A
- Peak amplitude of a waveform
- Crest factor
- Peak absolute value divided by RMS
Worked examples
RMS of six signed values
Squares: 9, 16, 25, 4, 36 and 1 add up to 91. The mean square is 91 ÷ 6 ≈ 15.167 and the RMS is √15.167 ≈ 3.8944 — much larger than the ordinary mean of 1.5, because the negative values no longer cancel the positive ones.
US mains voltage: 170 V peak sine wave
For a sine wave RMS = A/√2 = 170 ÷ 1.4142 ≈ 120.2 V, which is why a 170 V peak supply is called 120 V AC. The crest factor of a pure sine is always √2 ≈ 1.414.
Triangle wave with 5 V peak
A triangle wave has RMS = A/√3 = 5 ÷ 1.7321 ≈ 2.887 V, and its mean square is 25/3 ≈ 8.333. Its crest factor √3 is higher than a sine’s because it spends less time near the peak.
Frequently asked questions
What is the root mean square?+
The root mean square (RMS) is the square root of the average of the squared values. It measures the typical magnitude of a set of numbers regardless of sign, and for electrical signals it gives the equivalent DC value that would deliver the same power.
How do you convert peak voltage to RMS?+
For a pure sine wave, multiply the peak voltage by 1/√2 ≈ 0.7071. A 325 V peak gives about 230 V RMS and a 170 V peak gives about 120 V RMS. Other wave shapes use different factors, such as A/√3 for a triangle wave.
What is the difference between RMS and average?+
The arithmetic average lets positive and negative values cancel, so a symmetric AC signal averages to zero. RMS squares the values first, so it always reflects their size; it is also at least as large as the average of the absolute values.
Is RMS the same as standard deviation?+
Only when the mean is zero. In general RMS² = mean² + population variance, so the RMS measures spread around zero while the standard deviation measures spread around the mean.
What is crest factor?+
Crest factor is the peak absolute value divided by the RMS. It is √2 ≈ 1.414 for a sine wave, 1 for a square wave and √3 ≈ 1.732 for a triangle wave. Audio and power engineers use it to judge how "peaky" a signal is.