Root Mean Square — Definition, Formula & Examples

Root Mean Square
RMS

A kind of average sometimes used in statistics and engineering, often abbreviated as RMS. To find the root mean square of a set of numbers, square all the numbers in the set and then find the arithmetic mean of the squares. Take the square root of the result. This is the root mean square.

Formula:

Square root of (sum from i=1 to n of a_i squared, divided by n)

Example:

For the numbers 4 and 9,

RMS formula: RMS = sqrt((4² + 9²) / 2) ≈ 6.96, showing example calculation with numbers 4 and 9.

See also

Mean

Key Formula

x_{\text{rms}} = \sqrt{\frac{x_1^2 + x_2^2 + \cdots + x_n^2}{n}}

Where:

$x_1, x_2, \ldots, x_n$ = The individual values in the data set
$n$ = The total number of values in the set
$x_{\text{rms}}$ = The root mean square of the set

Worked Example

Problem: Find the root mean square of the numbers 3, 5, and 7.

Step 1: Square each number in the set.

3^2 = 9, \quad 5^2 = 25, \quad 7^2 = 49

Step 2: Find the arithmetic mean of the squares by adding them and dividing by the count.

\frac{9 + 25 + 49}{3} = \frac{83}{3} \approx 27.667

Step 3: Take the square root of the mean.

\sqrt{27.667} \approx 5.26

Answer: The root mean square of 3, 5, and 7 is approximately 5.26.

Another Example

This example includes negative values, showing that RMS handles them naturally because squaring eliminates the sign. The ordinary arithmetic mean of these four numbers is 0, but the RMS captures the typical magnitude.

Problem: Find the root mean square of the numbers −4, 1, −2, and 5.

Step 1: Square each number. Notice that squaring removes the negative signs.

(-4)^2 = 16, \quad 1^2 = 1, \quad (-2)^2 = 4, \quad 5^2 = 25

Step 2: Find the arithmetic mean of the squares.

\frac{16 + 1 + 4 + 25}{4} = \frac{46}{4} = 11.5

Step 3: Take the square root of the result.

\sqrt{11.5} \approx 3.39

Answer: The root mean square of −4, 1, −2, and 5 is approximately 3.39.

Frequently Asked Questions

What is the difference between root mean square and arithmetic mean?

The arithmetic mean adds the values and divides by the count. The RMS squares the values first, averages those squares, then takes the square root. Because of the squaring step, larger values have a greater influence on the RMS, so the RMS is always greater than or equal to the arithmetic mean for any set that contains at least one nonzero value.

Why is root mean square used instead of a regular average?

RMS is preferred when values can be both positive and negative, because positive and negative values would cancel in a regular average. For example, alternating current (AC) voltage swings positive and negative, so its arithmetic mean is zero — but its RMS gives a meaningful measure of the voltage's effective strength. RMS is also used whenever the quantity of interest relates to the square of the values, such as power or energy.

Is the RMS always larger than the arithmetic mean?

By the quadratic mean-arithmetic mean (QM-AM) inequality, the RMS is always greater than or equal to the arithmetic mean of the same values, with equality only when all values in the set are identical. For example, the RMS of 3, 5, and 7 is approximately 5.26 while their arithmetic mean is 5. When the data set contains negative values the arithmetic mean can be negative while the RMS is always non-negative, so RMS is at least as large as the arithmetic mean in every case.

Root Mean Square (RMS) vs. Arithmetic Mean

	Root Mean Square (RMS)	Arithmetic Mean
Definition	Square the values, average the squares, take the square root	Sum the values and divide by the count
Formula	√((x₁² + x₂² + … + xₙ²) / n)	(x₁ + x₂ + … + xₙ) / n
Handles negatives	Yes — squaring removes signs before averaging	Negatives cancel with positives, which can give a misleading result
Sensitivity to large values	Higher — squaring amplifies large values	Lower — all values contribute equally
Typical use	AC voltage, signal processing, error analysis	General-purpose average for everyday data

Why It Matters

You will encounter RMS in physics when studying AC circuits — the household voltage labeled 120V (in the US) is actually an RMS value, not the peak voltage. In statistics, the standard deviation is closely related to the RMS of deviations from the mean, so understanding RMS helps you grasp how spread is measured. Engineering and data science also use RMS error (RMSE) to evaluate the accuracy of predictions and models.

Common Mistakes

Mistake: Forgetting to take the final square root, which gives the mean of the squares instead of the root mean square.

Correction: Always remember that RMS has three steps in its name: Root → Mean → Square (read in reverse order of operations: square first, then mean, then root). The last step is taking the square root.

Mistake: Taking the square root of each value first, then averaging those roots.

Correction: The correct order is: square all values, compute the mean of those squares, then take one square root at the end. Rooting each value individually is a different calculation entirely.

Related Terms

Arithmetic Mean — The simple average; RMS replaces it when signs matter
Average — General term; RMS is one type of average
Square Root — The final operation in calculating RMS
Mean — Broad concept that includes arithmetic, geometric, and quadratic means
Set — A collection of values from which RMS is calculated
Standard Deviation — Equals the RMS of deviations from the mean
Square (Algebra) — Squaring values is the first step in computing RMS