Root Mean Square Calculator
Calculate RMS (quadratic mean) of any dataset with step-by-step solutions
Enter Your Values
Example Calculation
Example Dataset: [2, 6, 3, -4, 2, 4, -1, 3, 2, -1]
Step 1: Square each value:
2² = 4, 6² = 36, 3² = 9, (-4)² = 16, 2² = 4, 4² = 16, (-1)² = 1, 3² = 9, 2² = 4, (-1)² = 1
Step 2: Sum of squares: 4 + 36 + 9 + 16 + 4 + 16 + 1 + 9 + 4 + 1 = 100
Step 3: Divide by count: 100 ÷ 10 = 10
Step 4: Take square root: √10 ≈ 3.162
Result
RMS = 3.162278
This means the quadratic mean of the dataset is approximately 3.16
RMS Formulas
Standard RMS
Weighted RMS
Where:
x = individual values
w = weights
n = number of values
Σ = sum symbol
RMS Applications
Electrical Engineering
AC voltage and current measurements
Statistics
Standard deviation calculations
Signal Processing
Audio and signal analysis
Physics
Kinetic energy and velocity analysis
Quick Tips
RMS is always ≥ arithmetic mean for any dataset
RMS emphasizes larger values due to squaring
Use weighted RMS when values have different importance
RMS is also known as quadratic mean
Understanding Root Mean Square (RMS)
What is Root Mean Square?
The Root Mean Square (RMS) is a statistical measure that calculates the square root of the arithmetic mean of the squares of a set of values. It's also known as the quadratic mean and is particularly useful for measuring the magnitude of a varying quantity.
Why Use RMS?
- •Provides a meaningful average for quantities that vary in sign
- •Emphasizes larger values in the dataset
- •Commonly used in physics and engineering applications
- •Related to standard deviation calculations
Mathematical Properties
RMS = √[(x₁² + x₂² + ... + xₙ²) / n]
Standard RMS Formula
- Always non-negative: RMS ≥ 0
- RMS ≥ Arithmetic Mean: For any dataset
- Equal when all values are equal: RMS = |x| when all x are the same
- Units: Same units as the original data
Relationship to Standard Deviation:
For a dataset: σ² = RMS² - μ²
where σ is standard deviation and μ is the mean