Question

Quadratic Mean The quadratic mean (or root mean square, or R.M.S.) is used in physical applications, such as power distribution systems. The quadratic mean of a set of values is obtained by squaring each value, adding those squares, dividing the sum by the number of values n, and then taking the square root of that result, as indicated below:

$\mathbf{Quadratic}\mathbf{mean}\mathbf{=}\sqrt{\frac{\sum {\mathbf{x}}^{\mathbf{2}}}{\mathbf{n}}}$

Find the R.M.S. of these voltages measured from household current: 0, 60, 110, -110, -60, 0.

How does the result compare to the mean?

Short answer

The root mean square value is 72.3, which is different from the mean value of 0.

Step-by-step solution

Given information

The voltages measured for five household currents are 0, 60, 110, –110, –60, 0.

Root mean square formula

For n observations, the root mean square is computed in the following steps.

• Obtain the sum of squares observations.
• Find the quotient of the sum over the count of observations.
• Obtain the square root of the quotient.

Mathematically,

$\text{R.M.S}=\sqrt{\frac{\sum x^2}{n}}$ for x observations.

Substitute the values to obtain theroot mean square.

$$\begin{gathered} \text{R.M.S}=\sqrt{\frac{0^2+{60}^2+{110}^2+{\left(-110\right)}^2+{\left(-60\right)}^2+0^2}{6}} \\ =\sqrt{\frac{31400}{6}} \\ =\sqrt{5233.33} \\ =72.3418 \end{gathered}$$

Thus, the root mean square value is 72.3.

Compute the mean value

The formula for mean is stated below.

$\overline{x}=\frac{\sum x}{n}$

Substitute the values.

$$\begin{gathered} \overline{x}=\frac{0+60+110-110-60+0}{6} \\ =0 \end{gathered}$$

Thus, the mean value is 0.0.

Compare the two measures of mean

The value of root mean square is 72.3, which is different from 0. The observations take a negative set of values, which leads to the sum of 0.

On the other hand, the root mean square uses the squared value of observations which nullifies the effect of negative signs.