Root Mean Square (RMS) Calculator

For n values \(x_1, x_2, \ldots, x_n\), the root mean square is calculated in three steps:

$$\text{RMS} = \sqrt{\frac{x_1^2 + x_2^2 + \cdots + x_n^2}{n}} = \sqrt{\frac{\sum_{i=1}^{n} x_i^2}{n}}$$

1. Square each value to eliminate negative signs
2. Mean — take the arithmetic average of the squared values
3. Root — take the square root to return to the original scale

The relationship between RMS, mean, and standard deviation is:

$$\text{RMS}^2 = \bar{x}^2 + \sigma^2$$

where \(\bar{x}\) is the arithmetic mean and \(\sigma\) is the population standard deviation. This means RMS captures both the central tendency and the variability of the data in a single number. When the mean is zero (as in a pure AC signal), RMS equals the standard deviation.

An RMS of 4.0 for values {3, -4, 5, -2, 6} indicates that the effective magnitude of these values is 4.0. The arithmetic mean (1.6) is much lower because positive and negative values partially cancel. The RMS preserves the true magnitude by squaring first. In practical terms: if these values represented voltage readings, the RMS (4.0V) is the effective voltage that would deliver the same power as a constant 4.0V DC source.