Root Mean Square Calculator

How to Use the Root Mean Square Calculator

This calculator allows you to instantly find the root mean square of any set of data. To use the calculator:

1. Enter the data values into the Data input field. You can manually or copy-paste values from Excel, Google Sheets, or text documents. Make sure to separate values with commas, spaces, tabs, or line breaks.
2. Select the number of Decimal Places you want to display.
3. Click Calculate.

The calculator will instantly return the root mean square (RMS) along with a clear step-by-step solution.

Want a tool that allows you to compute multiple basic statistics with a click of a button? Use the statistics calculator.

What Is Root Mean Square?

The root mean square, commonly abbreviated as RMS, is a statistical measure of the magnitude, which is commonly used to find the average magnitude of a set of values, especially when the values swing between positive and negative numbers.

Because the values are squared first, negative and positive values both contribute positively to the RMS. For example, the values −5 and 5 have an arithmetic mean of 0, but their RMS is 5. This makes RMS useful when the size or magnitude of the values matters more than their direction.

Tip. Root mean square is also known as the quadratic mean.

Root Mean Square Formula

For a set of n values, x₁, x₂, …, x_n, the root mean square formula is:

$RMS=\sqrt{\frac{x_1^2+x_2^2+\cdots+x_n^2}{n}}$

The formula can also be written as:

$RMS=\sqrt{\frac{\sum_{i=1}^{n}x_i^2}{n}}$

Where:

• x_i represents each value in the data set
• Σx² is the sum of the squared values
• n is the number of values
• RMS is the root mean square

How to Find Root Mean Square

To calculate the RMS of a set of numbers manually, follow these steps:

1. Square each data value.
2. Add the squared values to find the sum of squares, Σx².
3. Divide the sum of squares by the number of values, n.
4. Take the square root of the result to get the root mean square

Example

Suppose you have the following values: −4, 2, −6, 8. Find the root mean square

Solution

To find the root mean square of the above data by hand, follow these steps:

Step 1: Square each value.

Squaring each of the numbers gives:

• (−4)² = 16
• 2²= 4
• (−6)² = 36
• 8² = 64

Step 2: Find the sum of squares.

The sum of squares is: Σx² = 16 + 4 + 36 + 64

= 120

You can also verify this result using the sum-of-squares calculator.

Step 3: Find the mean of the squared values.

To find the mean of the squared values, we divide the result in Step 2 by 4 since there are 4 observations.

Thus, Mean of squares = 120 / 4

= 30

Step 4: Take the square root.

Taking the square root of the result from Step 3, we get:

RMS = √30

= 5.4772

Therefore, the root mean square for the data is: 5.4772

Root Mean Square vs Arithmetic Mean

The root mean square and arithmetic mean measure different things. While the arithmetic mean adds the values and divides by the number of observations, the root mean square squares the values before finding their mean, and finally gets the square root.

Because RMS uses squared values, larger values have more influence on the result.

Where Is RMS Used?

Root mean square appears in several areas of mathematics, statistics, science, and engineering.

Common uses include:

• Measuring the effective size of alternating electrical voltage or current
• Comparing the magnitude of values that change between positive and negative
• Measuring signal strength
• Summarizing error values
• Working with physical quantities that vary over time

In each case, squaring prevents positive and negative values from canceling each other out.

Frequently Asked Questions