Ch. 3 - Describing, Exploring, and Comparing Data

Triola - Elementary Statistics 14th Edition

Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook

Triola 14th Edition

Ch. 3 - Describing, Exploring, and Comparing Data

Problem 3.1.40

Chapter 3, Problem 3.1.40

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:

Quadratic mean = sqrt(∑x^2/n)

Find the R.M.S. of these voltages measured from household current: 0, 60, 110, 0. How does the result compare to the mean?

Verified step by step guidance

Step 1: Understand the formula for the quadratic mean (R.M.S.), which is given as: Quadratic mean = sqrt(∑x^2 / n). Here, ∑x^2 represents the sum of the squares of the values, and n is the total number of values.

Step 2: Square each of the given voltages: 0, 60, 110, and 0. This means calculating 0^2, 60^2, 110^2, and 0^2.

Step 3: Add the squared values together to compute ∑x^2. This involves summing the results from the previous step.

Step 4: Divide the sum of the squared values (∑x^2) by the total number of values, n. In this case, n = 4 because there are 4 voltage measurements.

Step 5: Take the square root of the result from Step 4 to find the quadratic mean (R.M.S.). Compare this value to the arithmetic mean of the original voltages, which is calculated as the sum of the original values divided by n.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Quadratic Mean (Root Mean Square)

The quadratic mean, also known as the root mean square (R.M.S.), is a statistical measure used to determine the average magnitude of a set of values. It is calculated by squaring each value, averaging these squares, and then taking the square root of that average. This measure is particularly useful in contexts where values can be both positive and negative, as it emphasizes larger values more than the arithmetic mean.

Calculation of R.M.S.

To calculate the R.M.S. of a set of values, you first square each individual value, sum these squared values, and then divide by the total number of values. Finally, you take the square root of this quotient. For example, for the voltages 0, 60, 110, and 0, you would compute the squares (0, 3600, 12100, 0), sum them (15700), divide by 4, and take the square root to find the R.M.S.

Comparison with Arithmetic Mean

The arithmetic mean is calculated by summing all values and dividing by the number of values, providing a simple average. In contrast, the R.M.S. tends to be higher than the arithmetic mean when there are large values in the dataset, as it gives more weight to larger numbers. Understanding the difference between these two means is crucial for interpreting results in contexts like power distribution, where the R.M.S. reflects the effective value of alternating current.

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Calculating the Mean

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Textbook Question

Trimmed Mean Because the mean is very sensitive to extreme values, we say that it is not a resistant measure of center. By deleting some low values and high values, the trimmed mean is more resistant. To find the 10% trimmed mean for a data set, first arrange the data in order, then delete the bottom 10% of the values and delete the top 10% of the values, then calculate the mean of the remaining values. Use the axial loads (pounds) of aluminum cans listed below (from Data Set 41 “Aluminum Cans” in Appendix B) for cans that are 0.0111 in. thick. An axial load is the force at which the top of a can collapses. Identify any outliers, then compare the median, mean, 10% trimmed mean, and 20% trimmed mean.

247 260 268 273 276 279 281 283 284 285 286 288

289 291 293 295 296 299 310 504

Textbook Question

Large Data Sets from Appendix B. In Exercises 25–28, refer to the indicated data set in Appendix B. Use software or a calculator to find the means and medians.

Body Temperatures Refer to Data Set 5 “Body Temperatures” in Appendix B and use the body temperatures for 12:00 AM on day 2. Do the results support or contradict the common belief that the mean body temperature is 98.6oF?

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Textbook Question

In Exercises 5–20, 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 3-1, where we found measures of center. Here we find measures of variation.) Then answer the given questions.

Jaws 3 Listed below are the number of unprovoked shark attacks worldwide for the last several years. What extremely important characteristic of the data is not considered when finding the measures of variation?

70 54 68 82 79 83 76 73 98 81

Textbook Question

In Exercises 21–28, use the same list of cell phone radiation levels given for Exercises 17–20. Find the indicated percentile or quartile.

Textbook Question

Mean Absolute Deviation Use the same population of {9 cigarettes, 10 cigarettes, 20 cigarettes} from Exercise 45. Show that when samples of size 2 are randomly selected with replacement, the samples have mean absolute deviations that do not center about the value of the mean absolute deviation of the population. What does this indicate about a sample mean absolute deviation being used as an estimator of the mean absolute deviation of a population?

Textbook Question

Super Bowl Ages Listed below are the ages of the same 11 players used in the preceding exercise. How are the resulting statistics fundamentally different from those found in the preceding exercise?

41 24 30 31 32 29 25 26 26 25 30