Technically, the root mean square (RMS) of a variable is the square root of the average of the square of the variable. This kind of average is useful when a simpler type of averaging yields little or no helpful information. The electrical current in an AC circuit, for example, has an average value of zero because it spends as much time going in one direction as the other. By squaring the values that the current takes on over time, averaging these positive values and taking the square root, you can obtain a more meaningful number to describe the current.

- Skill level:
- Moderate

### Other People Are Reading

## Instructions

- 1
Treat a discrete variable by squaring all possible values. Weight each square by multiplying it by the probability of the variable taking that value. Sum the weighted squares and take the square root of the sum. This is the variable's RMS.

Suppose a variable fluctuates as follows: 0, 1, 2, 1, 0, -1, -2, -1, 0, .... The set of possible values is {0, 1, 2, -1, -2}. The variable is 0 a fourth of the time, 1 a fourth of the time, 2 an eighth of the time, and so on. So the squared values are {0, 1, 4, 1, 4}. The corresponding probabilities are {0.25, 0.25, 0.125, 0.25, 0.125}. Weighting the squared values gives {0, 0.25, 0.50, 0.25, 0.50}. Summing these values and taking the square root gives √1.5 = 1.225 after rounding. 1.225 is the RMS. So though the possible variable values are integers, the RMS is non-integer.

- 2
Use calculus to determine the RMS of a continuous variable. The integral to use on a variable x is ∫x^2*f(x)dx, where f(x) is the probability density function (pdf) of x. Here "^2" means that you square x. Take the square root of this integral to solve for the RMS.

For example, if the pdf of x is 5x^4/2 from x=-1 to +1, then RMS is the square root of ∫x^2*f(x)dx = (5/2) ∫x^6 dx = (5/2)(1/7)[1^7 - (-1)^7] = 5/7. The square root is 0.845 after rounding. So the RMS is 0.845.

- 3
Obtain the RMS of a variable that's a sine or cosine function merely by dividing by the square root of 2. This trick applies if the variable varies symmetrically above and below zero.

For example, if the current in a circuit has a maximum value of I and can be described as I*sin ωt, then the RMS of the current is I/√2.

#### Tips and warnings

- To see why the integration in Step 2 works, recall from calculus that the integral of x^n is x^(n+1) / (n+1).
- To see why the trick in Step 3 works, integrate the square of sin θ from θ=0 to 2π. The result is π. Now divide by the length of the interval over which θ varies so the effective weighting is 1. This gives you π/2π = ½. Now take the square root to get the root mean square: 1/√2.