Root Mean Square Formula

Root mean square is also known as RMS. We know that the mean is the sum of observations divided by a number of observations. 

After squaring all the observations, we add them, and by dividing all the observations, we get a mean square.

Further, doing the square root of our obtained, i.e., root of the mean of the squares, we get the root mean square.

So, in short, the root mean square formula is the root of the arithmetic mean of the squares.

On this page, we will understand the formula of root mean square along with the calculation of rms.

How to Calculate Root Mean Square?

From the above text, we understand the RMS formula is the root mean square of the set of values or a continuous-time waveform. We also understand the RMS  is the square root of the arithmetic mean of the squares of the values.

Besides this, RMS is the square of the function that specifies the continuous waveform. 

In physics, we use the term RMS current value, where we define it as the "value of the direct current that dissipates the power in a resistor."

Let’s suppose that there are a set of ‘n’ values, say, x1, x2, x3,...., xn. The RMS formula will be:

\[x_{RMS}=\sqrt{\frac{1}{n}(x1^{2}+x2^{2}+.....+xn^{2})}\]

Let’s consider an example for the calculation of RMS:

Calculation of RMS

Assume that there are a set of observations viz: 1, 2, 3, 4, 6, 7, and 9.

Step 1: Naming these values as;

x₁ = 1

x₂ = 2

x₃ = 3

x₄ = 4

x₅ = 6

x₆  =  7

x₇  =  9

Step 2: Now, squaring individual digits:

x₁² =  1²  = 1  

x₂²   =  2² =  4 

x₃²   =3² =  9

x₄² =  4²  = 16

x₅²   = 6² =  36

x₆²   = 7² =  49

 x₇²   =   9² = 81

Step 3: Now, adding these all squares:

x₁² + x₂²  + x₃² +  x₄² + x₅² + x₆² +   x₇²    =   1 + 4 + 9 + 16 + 36 + 49 +81

=>  196

Now, \[\sqrt{x1^{2}+x2^{2}+x3^{2}+x4^{2}+x5^{2}+x6^{2}+x7^{2}}=\sqrt{196}=14\]

Further, applying the root mean square formula:

=>    \[\sqrt{\frac{1}{7}(x1^{2}+x2^{2}+x3^{2}+x4^{2}+x5^{2}+x6^{2}+x7^{2}})=\sqrt{28}\]

So, the square root of \[ \sqrt{28}\] is approximately 5.29.

As per the RMS formula, the RMS value for the above observation is 5.29.

Root Mean Square in Continuous Function

The corresponding formula of root mean square for a continuous function (or waveform) for a defined function f(t) over the interval T1 t T2 is given as;

\[f_{RMS=}\sqrt{\frac{1}{T_{2}-T_{1}}\int_{T1}^{T1}[F(t)]^{2}dt}\]

Now, considering the RMS formula for the overall function, we have:

\[f_{RMS=}\lim_{X\rightarrow \infty }\sqrt{\frac{1}{2T}\int_{-T}^{T1}[F(t)]^{2}dt}\]        

From the above root mean square equation for a continuous function, we understand that The RMS throughout the periodic function is equivalent to the RMS of one period of the function. 

Hence, the RMS value of a continuous function or signal can be estimated by considering the RMS of an example comprising equally spaced observations.

We can also calculate the root mean square of given observations. Now, let’s understand what does RMSE means?

Root Mean Square Error (RMSE)

The Root Mean Square/Error or RMSE is an as often as possible applied proportion of the contrasts between numbers (populace esteems and tests) which is anticipated by an assessor or a mode. 

The RMSE portrays the example standard deviation of the contrasts between the anticipated and noticed qualities. 

Every one of these distinctions is known as residuals when the computations are done over the information test that was utilized to gauge, and known as expectation mistakes when determined out of the test. The RMSE totals the extent of the mistakes in foreseeing various occasions into a solitary proportion of prescient force.

Conclusion

RMS is additionally called a quadratic mean and is an extraordinary instance of the summed up to mean whose type is 2. 

Besides this, the root mean square is characterized as a varying function dependent on a vital of the squares of the qualities which are momentary in a cycle.