Root Mean Square (RMS) Calculator
Calculate the root mean square (RMS) of a set of numbers. Enter your values separated by commas and get the RMS along with intermediate steps. RMS is widely used in statistics, physics, and electrical engineering.
The root mean square (RMS) is a statistical measure that gives the "effective" magnitude of a set of values. It is always greater than or equal to the arithmetic mean for any set containing both positive and negative numbers.
The RMS Formula:
RMS = sqrt((x1² + x2² + ... + xn²) / n)
Three steps: square each value, take the mean of those squares, then take the square root.
RMS vs. Arithmetic Mean:
The arithmetic mean can be zero for values like {-3, 3}, but the RMS captures the actual magnitude: RMS = sqrt((9 + 9)/2) = 3. This is why RMS is preferred when sign does not matter but magnitude does.
Applications:
- Electrical engineering: AC voltage and current are measured as RMS values. A 120V outlet in the US delivers 120V RMS, meaning the effective voltage is 120V even though the peak is about 170V.
- Audio engineering: RMS power gives a more accurate picture of speaker output than peak power.
- Statistics: RMS error (RMSE) measures how far predictions deviate from observed values.
- Physics: RMS speed of gas molecules in the kinetic theory of gases.