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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

The harmonic mean is often used as a measure of center for data sets consisting of rates of change, such as speeds. It is found by dividing the number of values \(n\) by the sum of the reciprocals of all values, expressed as $$\frac{n}{\sum \frac{1}{x}}$$ (No value can be zero.) The author drove 1163 miles to a conference in Orlando, Florida. For the trip to the conference, the author stopped overnight, and the mean speed from start to finish was \(38 \mathrm{mi} / \mathrm{h}\). For the return trip, the author stopped only for food and fuel, and the mean speed from start to finish was \(56 \mathrm{mi} / \mathrm{h}\). Find the harmonic mean of \(38 \mathrm{mi} / \mathrm{h}\) and \(56 \mathrm{mi} / \mathrm{h}\) to find the true "average" speed for the round trip.

Find the range, variance, and standard deviation for the given sample data. Include appropriate units (such as "minutes") in your results. (The same data were used in Section 3-I, where we found measures of center. Here we find measures of variation.) Then answer the given questions. Listed below are annual U.S. sales of vinyl record albums (millions of units). The numbers of albums sold are listed in chronological order, and the last entry represents the most recent year. Do the measures of variation give us any information about a changing trend over time? $$\begin{array}{r} 0.30 .60 .81 .11 .11 .41 .41 .51 .21 .31 .41 .20 .90 .91 .01 .92 .52 .83 .94 .66 .1 \end{array}$$

In what sense are the mean, median, mode, and midrange measures of "center"?

Use the following cell phone airport data speeds (Mbps) from Sprint. Find the percentile corresponding to the given data speed. $$\begin{array}{cccccccccc} 0.2 & 0.3 & 0.3 & 0.3 & 0.3 & 0.3 & 0.3 & 0.4 & 0.4 & 0.4 \\ 0.5 & 0.5 & 0.5 & 0.5 & 0.5 & 0.6 & 0.6 & 0.7 & 0.8 & 1.0 \\ 1.1 & 1.1 & 1.2 & 1.2 & 1.6 & 1.6 & 2.1 & 2.1 & 2.3 & 2.4 \\ 2.5 & 2.7 & 2.7 & 2.7 & 3.2 & 3.4 & 3.6 & 3.8 & 4.0 & 4.0 \\ 5.0 & 5.6 & 8.2 & 9.6 & 10.6 & 13.0 & 14.1 & 15.1 & 15.2 & 30.4 \end{array}$$ $$13.0 \mathrm{Mbps}$$

Identify the symbols used for each of the following: (a) sample standard deviation; (b) population standard deviation; (c) sample variance; (d) population variance.
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