Root Mean Square Calculator
Calculate the root mean square of any dataset - or find RMS voltage from peak voltage.
√ What is Root Mean Square (RMS)?
The root mean square (RMS), also called the quadratic mean, is the square root of the mean of the squares of a set of values. The formula is: RMS = √((x12 + x22 + … + xn2) / n). The three-step process - square, mean, root - ensures that negative values contribute positively (since they are squared), making RMS particularly useful for alternating quantities like AC voltage and current.
In electrical engineering, RMS is the fundamental measure for AC circuits. The 230V or 120V rating on your wall outlet is the RMS voltage, not the peak voltage. The peak voltage of a 230V RMS supply is 230 × √2 ≈ 325V. RMS voltage is used because it equals the DC voltage that would dissipate the same power in a resistive load - a 230V RMS AC supply heats a resistor exactly as much as a 230V DC supply. This is known as the equivalent DC voltage interpretation of RMS.
A key mathematical identity connects RMS to more familiar statistics: RMS2 = Mean2 + Variance. This follows from the variance formula: Var = E[x2] − (E[x])2, rearranged as E[x2] = Mean2 + Variance. Since RMS2 = E[x2], the identity follows. Practically, this means a dataset's RMS increases when either its mean or its spread (variance) increases.
Beyond electricity, RMS is ubiquitous in engineering and data science. Root Mean Square Error (RMSE) measures prediction accuracy in machine learning. Acoustic engineers use RMS sound pressure level. Vibration analysts use RMS acceleration to assess machine health. Audio engineers reference RMS signal levels in dBu and dBV. In all these applications, the same principle applies: RMS captures "how large the values are on average" in a way that treats positive and negative deviations symmetrically.
📐 Formula
📖 How to Use This Calculator
Steps to Calculate RMS
💡 Example Calculations
Example 1 — RMS of a Simple Dataset
Find the RMS of 1, 2, 3, 4, 5
Example 2 — AC Mains Voltage
230V RMS mains supply - find the peak voltage
Example 3 — RMS with Negative Values
AC-like signal: −3, −1, 0, 1, 3
Example 4 — RMSE for Predictions
Forecast errors: −4, +2, −1, +3, 0 (predicted − actual)
❓ Frequently Asked Questions
🔗 Related Calculators
What is the formula for root mean square (RMS)?
RMS = √((x₁² + x₂² + ... + xₙ²) / n) - square each value, find their mean, then take the square root. Example: RMS of 3, 4, 5 = √((9+16+25)/3) = √(50/3) = √16.67 ≈ 4.082.
What is RMS voltage and how is it related to peak voltage?
For a sinusoidal AC signal, V_rms = V_peak / √2 ≈ 0.7071 × V_peak. The 230V or 120V printed on your appliances is the RMS voltage. The actual peak voltage is higher: 230V RMS corresponds to a peak of 230×√2 ≈ 325V. RMS is used because it equals the DC voltage that would deliver the same power to a resistive load.
What is the difference between RMS and average?
The arithmetic mean (average) simply sums and divides. RMS squares first, then finds the mean, then takes the square root. RMS ≥ AM for any dataset (by the power mean inequality). For symmetric AC waveforms, the arithmetic mean is zero (positive half cancels negative half), but RMS is non-zero and meaningful - showing why RMS is preferred for AC circuits.
What does RMS² = Mean² + Variance mean?
RMS² = Mean² + Variance is equivalent to: (sum of squares)/n = (mean)² + variance. This follows directly from the variance formula Var = E[x²] - (E[x])². Rearranged: E[x²] = Var + Mean². Since RMS² = E[x²], we get RMS² = Mean² + Variance. This means RMS captures both the mean level and the spread of data.
When is RMS used in engineering?
RMS is the standard measure for: AC voltage and current (V_rms determines power dissipation); audio signal levels (dBu and dBV reference RMS values); vibration analysis (RMS acceleration for machinery health); noise floor in electronics (RMS noise voltage); and error analysis (Root Mean Square Error, RMSE, measures prediction accuracy).
How do I convert peak voltage to RMS?
V_rms = V_peak / √2 for a sinusoidal (pure sine wave) signal. For a square wave: V_rms = V_peak. For a triangle wave: V_rms = V_peak / √3. Example: an audio amplifier with a 20V peak output delivers 20/√2 ≈ 14.14V RMS into a speaker. Power = V²_rms / R = (14.14)² / 8 = 25W.
What is the RMS of a sine wave?
For a sine wave with amplitude A (peak value): RMS = A/√2 ≈ 0.707A. Average (absolute value) = 2A/π ≈ 0.637A. Form factor = RMS/Average = π/(2√2) ≈ 1.1107. These ratios are fundamental constants in AC circuit analysis and appear in every electrical engineering textbook.
What is RMSE (Root Mean Square Error)?
RMSE measures prediction accuracy: RMSE = √(Σ(predicted - actual)² / n). It is the RMS of the error values. RMSE is the most common metric in regression analysis and machine learning because it (1) penalises large errors more than small ones, (2) has the same units as the predicted variable, and (3) is always ≥ MAE (mean absolute error).
How do I calculate RMS in Excel?
Use =SQRT(SUMSQ(values)/COUNT(values)). Or with an array formula: =SQRT(SUMPRODUCT(A1:A10^2)/COUNT(A1:A10)). For AC voltage from peak: =A1/SQRT(2) where A1 contains the peak value. For RMSE: =SQRT(SUMPRODUCT((A1:A10-B1:B10)^2)/COUNT(A1:A10)).
Why is RMS called the quadratic mean?
RMS is a member of the power mean family: quadratic mean (RMS) uses p=2 (squaring), arithmetic mean uses p=1, harmonic mean uses p=−1, and geometric mean is the limit as p→0. The name 'quadratic mean' comes from the squaring step. Power means satisfy: HM ≤ GM ≤ AM ≤ RMS for positive values, so RMS is always the largest of the common means.