The Root mean square (RMS) average [1] of a set of numbers is the square root of the arithmetic mean of the squares of the values.

**RMS Value Formula**

The formula for this is

**x _{RMS} = √(1/n)(x_{1}^{2} + x_{2}^{2} + … + x_{n}^{2})**

where **x _{RMS}** is the RMS value of the sequence of numbers x

_{1}, x

_{2}… x

_{n}.

**RMS Value Calculator**

Use the calculator below to calculate the RMS value.

Enter the numbers **x _{1}, x_{2}, x_{3},** … each separated by a comma.

**Example Calculation**

The RMS average of two numbers **1** and **4** is **2.915**

**Background**

Root-mean-square can be applied to any sequence of numbers.

In electronics, the RMS current value is defined as the “*value of direct current that dissipates the same amount of power in a resistor*.” [1]

For a DC current **I**, through a resistor **R**, the power can be calculated as

**P = I ^{2}R**

However, if the current **I** varies with time it’s referred to as an alternating current. This is represented as **I(t)**. The average power can be written as

**P _{avg} = (I(t)^{2}R)_{avg}**

Since the resistance value R does not change with time, this can simply be written as

**P**_{avg}** = (I(t)**^{2}**)**_{avg}***R**

I(t) can be represented as a sequence of numbers that represent current measurements in time I_{1}, I_{2}, … I_{n}. The average of the sum of squares is

**(I(t)**^{2}**)**_{avg} = **√(1/n)(I _{1}^{2} +I_{2}^{2} + … + I_{n}^{2})**

which is the RMS value of current or **I _{RMS}**

**I**_{RMS} = **√(1/n)(I _{1}^{2} +I_{2}^{2} + … + I_{n}^{2})**

The RMS value of the corresponding alternating voltage that causes this time varying current is

**V _{RMS} = I_{RMS} * R**

It is similarly defined as

**V**_{RMS} = **√(1/n)(V _{1}^{2} +V_{2}^{2} + … + V_{n}^{2})**

where **V _{1}, V_{2}, … V_{n}** are the corresponding values of voltage.

**Where is RMS used?**

**Electrical Outlets**

Electrical outlets in the USA provide 120 Volt AC while those in Europe provide 220 Volt or 230 Volt. A complete list of mains voltage for countries around the world.

While it’s not stated explicitly, these are RMS values for the voltage

The peak to peak value is much higher and calculated to be 340 V in the USA and 650 Volt in Europe.

**Power Detectors**

RF power detectors are differentiated by their ability to detect RMS. A detector is able to produce a DC level that is proportional to the RMS value of the input voltage.

**Common Waveforms**

This post provides the RMS values for different waveforms such as triangle, sine, square, etc. In each case uses the peak, peak-to-peak or average values of the waveform to compute the RMS equivalent.

One of the most common waveforms is a sine wave. The mains voltage in our homes is a sine wave shown below.

Amplitude levels are labelled as follows:

- Peak Voltage (Vpeak)
- Peak to Peak Voltage (in this case, twice the value of the peak voltage)
- RMS Voltage – for a sine wave, it is 0.707*Vpeak
- Period

**Frequently Asked Questions**

**Is the RMS value same as Average value?**

No they are different. In the examples provided above, for a sine wave, the average value is 0. The RMS value is 0.707*Vpeak.

Mathematically

**x _{RMS} = √(1/n)(x_{1}^{2} + x_{2}^{2} + … + x_{n}^{2})**

**x _{AVG} = (1/n)(x_{1} + x_{2} + … + x_{n})**

and **x _{RMS}** is not the same as

**x**in general. There are however some exceptions. In the case of a square wave, the RMS and Average values are the same.

_{AVG}**How to measure RMS Voltage using Oscilloscope?**

Modern oscilloscopes like this one from Siglent have the ability to compute the RMS values using the sampled data values over either the first cycle or all values.

With older analog scopes the RMS value can be calculated from the peak to peak value taking the waveform type into consideration. For instance, VRMS for a triangle wave can be calculated from the peak-to-peak voltage – measured using horizontal bars on the screen.

**What is the RMS Value of 120 VAC?**

120 VAC represents an Alternating Current Voltage of 120 V. This is in fact the RMS value. Within the context of voltage mains, 120 VAC is a sinusoidal waveform with peak values of +170 Volt and -170 Volt or a peak-to-peak value of 340 Volt. The DC value is zero.

**Related Calculators**

**References**

[1] RMS on Wikipedia