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Chapter 3: Problem 40

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: $$\text { Quadratic mean }=\sqrt{\frac{\Sigma x^{2}}{n}}$$ 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

Expert verified
The R.M.S. of the voltages is approximately 72.34, whereas the arithmetic mean is 0.

Step by step solution

01

- List the values

The voltages measured from household current are 0, 60, 110, -110, -60, and 0.
02

- Calculate the squares of each value

Square each value: 0^2 = 0, 60^2 = 3600, 110^2 = 12100, (-110)^2 = 12100, (-60)^2 = 3600, 0^2 = 0.
03

- Sum the squares

Add the squared values together: 0 + 3600 + 12100 + 12100 + 3600 + 0 = 31400.
04

- Divide the sum by the number of values

There are 6 values. Divide the sum of the squares by 6: \(\frac{31400}{6} = 5233.33\).
05

- Take the square root

Take the square root of 5233.33 to find the R.M.S.: \( \text{R.M.S.} = \sqrt{5233.33} \approx 72.34 \).
06

- Calculate the arithmetic mean for comparison

Sum the original values: 0 + 60 + 110 - 110 - 60 + 0 = 0. Divide by the number of values: \( \frac{0}{6} = 0 \). The arithmetic mean is 0.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

root mean square
The root mean square (RMS) is a very useful statistical measure. It allows us to find an average value of a set, even if some values are negative. For example, in household current voltages, certain values could be negative due to the direction of the current. The RMS method ensures that these negative values do not cancel out positive ones.
To find the RMS, you need to:
  • Square each value in your set. This removes any negative signs.
  • Add up all the squared values to get a sum.
  • Divide this sum by the number of values, giving you the mean of the squares.
  • Take the square root of this mean. This is the root of the mean of the squares鈥攈ence,

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Most popular questions from this chapter

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