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**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.

## Root Mean Square Formula

The root mean square formula gives the square root of the total sum of squares of each data in an observation. Root mean square, abbreviated as RMS is the square root of the arithmetic mean of the squares of a group of values. It is also called quadratic mean. Root mean square value for a function can also be defined for a continuously varying function in terms of an integral of the squares of the instantaneous values during a cycle. Here, the root mean square formula calculates the square root of the arithmetic mean of the square of the function that defines the continuous waveform.

The **formula for Root Mean Square** is given below to get the RMS value of a set of data values.

For a group of n values involving {x_{1}, x_{2}, x_{3},…. X_{n}}, the RMS is given by:

The formula for a continuous function f(t), defined for the interval **T _{1} ≤ t ≤ T_{2} **is given by:

The RMS of a periodic function is always equivalent to the RMS of a function’s single period. The continuous function’s RMS value can be considered approximately by taking the RMS of a sequence of evenly spaced entities. Also, the RMS value of different waveforms can also be calculated without calculus.

## What is Root Mean Square Formula?

**Formula 1**

For a group of “n” values involving x1,x2,x3,….xn, the root mean square formula is given as,

**where Xrms = Root mean square value of given “n” observations.**

**Formula 2**

The root mean square formula for a continuous function f(t), defined for the interval T_{1} ≤ t ≤ T_{2}is given as,

## Solved Examples Using Root Mean Square Formula

**Example 1: ** Calculate the root mean square of the following observations: 6, 5, 4, 2, 7? **Solution:** To find: Root mean square of the given observations. Using the root mean square formula,

**Answer:** The root mean square of given observations = 5.196.

**Example 2: **

Find the root mean square value of f(t) = t over the interval 2 ≤ t ≤ 5.

**Solution:**

To find: The root mean square value of f(t) = t over the interval 2 ≤ t ≤ 5.

Using the root mean square value formula for th given function f(t),

**Answer: **Root mean square value of the given function, f(t) = 13.

### How to Calculate the Root Mean Square

Steps to Find the Root mean square for a given set of values are given below:

**Step 1: **Get the squares of all the values

**Step 2:** Calculate the average of the obtained squares

**Step 3:** Finally, take the square root of the average

### Solved Example

**Question: **

Calculate the root mean square (RMS) of the data set: 1, 3, 5, 7, 9

**Solution:**

Given set of data values:

1, 3, 5, 7, 9

Step 1: Squares of these values

1^{2}, 3^{2}, 5^{2}, 7^{2}, 9^{2}

Or

1, 9, 25, 49, 81

Step 2: Average of the squares

(1 + 9 + 25 + 49 + 81)/5

= 165/5

= 33

Step 3: Take the square root of the average.

RMS = √33 = 5.745 (approx)

### Root Mean Square Error (RMSE)

The Root Mean Square Error or RMSE is a frequently applied measure of the differences between numbers (population values and samples) which is predicted by an estimator or a mode. The RMSE describes the sample standard deviation of the differences between the predicted and observed values. Each of these differences is known as residuals when the calculations are done over the data sample that was used to estimate, and known as prediction errors when calculated out of sample. The RMSE aggregates the magnitudes of the errors in predicting different times into a single measure of predictive power.

### Root Mean Square Error Formula

The RMSE of a predicted model with respect to the estimated variable x_{model} is defined as the square root of the mean squared error.

Where, x_{obs} is observed values, x_{model} is modelled values at time i.

## Frequently Asked Questions – FAQs

### What is the root mean square?

The root mean square (RMS or rms) is defined as the square root of the mean square, i.e. the arithmetic mean of the squares of a given set of numbers.

### How is RMS calculated?

RMS or Root Mean Square value can be calculated by taking the square root of arithmetic mean of squared observations.

### What is the RMSE value?

The root-mean-square deviation (RMSD) or root-mean-square error (RMSE) is a frequently used measure of the differences between values (sample or population values) predicted by a model or an estimator and the values observed.

### Is a higher or lower RMSE better?

Lower values of RMSE indicate better fit.

### What is the value of RMS for the data set 2, 3, 5, 7, 11?

Average of squares of given data values = (4 + 9 + 25 + 49 + 121)/5 = 208/5 = 41.6

RMS = square root(41.6) or √41.6 = 6.45

### Applications

Root mean square is frequently used in various mathematical and scientific applications. One common application is in calculating the mean square value of a waveform. This is important because it tells us how much energy is contained in it as a waveform. Its one of the most popular application is in the field of electrical engineering.

For example, it is often used to calculate the magnitude of an alternating current or voltage. It can also be used to determine the power dissipation of a resistor. Another application is in calculating the power required to drive a certain load. It can also be used to measure the variability of a data set. Finally, in physics, it is sometimes used to calculate the average kinetic energy of particles.

The root mean square error is used to measure how the magnitude of dispersion of residuals or prediction errors in a calculation. It denotes the difference between the predicted and observed results.

### Calculation Example

Now let us see how root mean square is calculated with the help of an example.

Let the values be 2, 4, and 6,8,10.

**Step1:**

The square of these values is taken.

22=4

42=8

62=36

82=64,

102=100

The new values that arrived are 4, 8, 36, and 64,100.

**Step 2:**

The averages of the squares shall be taken

Average=(4+8+36+64, 100)/5= 212/5= 42.4

**Step 3:**

The final step shall be to take the square root of the average. Therefore the root mean square value will be = 6.5115.

### Frequently Asked Questions (FAQs)

**How to calculate the root mean square?**

True root mean square of a given set of observations can be calculated in three easy steps. First, squares for each value given shall be calculated. Then, the average of the acquired squares shall be determined. Finally, in the third step, we can compute the square root of the averages.

**Why use root mean square instead of average?**

RMS is utilized when the variables provided in the data set are both positive and negative. At the same time, an average as a function is used to determine the central tendency of a specific data set.

**Can root mean square to be negative?**

The values given are squared in order to remove any negative numbers. The root mean square will always be positive since the squared sum is positive.

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

The arithmetic mean of the squares of a group of numbers or a random variable is the mean square. The true RMS is the square root of a mean square and can be used to calculate its deviation.

### Sample Problems

**Problem 1. Calculate the root mean square of the data set: 2, 7, 3, 5, 1.**

**Solution:**

Using the formula we get,

= √(88/5)

= √(17.6)

= 4.2

**Problem 2. Calculate the root mean square of the data set: 10, 12, 9, 3, 6.**

**Solution:**

Using the formula we get,

= √(370/5)

= √(74)

= 8.6

**Problem 3. Calculate the root mean square of the data set: 5, 7, 2, 4, 3, 9.**

**Solution:**

Using the formula we get,

= √(184/6)

= √(30.66)

= 5.53

**Problem 4. Calculate the root mean square of the data set: 3, 6, 9, 12, 15, 18, 20.**

**Solution:**

Using the formula we get,

= √(1219/7)

= √(174.14)

= 13.196

**Problem 5. Calculate the root mean square of the data set if sum of squares of data set observations is 216 and number of observations is 6.**

**Solution:**

We have,

S = 216

n = 6

Using the formula we get,

R = √(S/n)

= √(216/6)

= √(36)

= 6

**Problem 6. Calculate the root mean square of the data set if sum of squares of data set observations is 5832 and number of observations is 18.**

**Solution:**

We have,

S = 5832

n = 18

Using the formula we get,

R = √(S/n)

= √(5832/18)

= √(324)

= 18

**Problem 7. Calculate the root mean square value of the continuous function f(t) = t over the interval [4, 7].**

**Solution:**

We have,

f(t) = t and 4 ≤ t ≤ 7.

Using the formula we get,

= √(343/9 – 64/9)

= √(279/9)

= √(31)

= 5.56

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