Mean & RMS Value Calculator

In many engineering and scientific applications, it's useful to characterize a function over an interval with single representative values. Two such important values are the Mean (or Average) Value and the Root Mean Square (RMS) Value. For a deeper look, see our Mean and RMS Value reference guide.

• The Mean Value gives the average height of the function over the interval. If the function represents a quantity like temperature or velocity, the mean value is its average over that period.
• The RMS Value is particularly important for time-varying signals, especially in electrical engineering (e.g., AC voltage or current). It represents the DC equivalent value that would deliver the same average power to a resistive load.

Both values are determined using integration.

The mean or average value of a function $f(x)$ over an interval $[a, b]$ is defined by the formula:

$$ f_{avg} = \text{Mean Value} = \frac{1}{b-a} \int_a^b f(x) \,dx $$

Geometrically, this means that the area under the curve $y=f(x)$ from $x=a$ to $x=b$ is equal to the area of a rectangle with width $(b-a)$ and height $f_{avg}$.

To find the mean value, we first calculate the definite integral of the function over the interval, and then divide by the length of the interval.

The Root Mean Square (RMS) value of a function $f(x)$ over an interval $[a, b]$ is defined by the formula:

$$ f_{rms} = \text{RMS Value} = \sqrt{\frac{1}{b-a} \int_a^b [f(x)]^2 \,dx} $$

The process to find the RMS value involves three steps:

1. Square: Square the function, $[f(x)]^2$.
2. Mean: Find the mean (average) of this squared function over the interval. This is called the Mean Square value: $$ \text{Mean Square} = \frac{1}{b-a} \int_a^b [f(x)]^2 \,dx $$
3. Root: Take the square root of the Mean Square value.

The RMS value is always non-negative.