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Chapter 3: Problem 39

When is the value of the standard deviation for a data set zero? Give one example. Calculate the standard deviation for the example and show that its value is zero.

Short Answer

Expert verified
The standard deviation of a data set is zero when all the numbers in the data set are the same. An example of such a dataset is [5, 5, 5, 5]. Calculating the standard deviation of this data set using the standard deviation formula gives a result of zero, as expected.

Step by step solution

01

Understand the Concept of Standard Deviation

The standard deviation is a measure of how spread out the numbers in a data set are. If the standard deviation is zero, it means all the numbers in the data set are identical because there is no 'deviation' or variation among the numbers.
02

Provide Example

An example of such a data set is [5, 5, 5, 5]. All the numbers in this data set are equal, so the standard deviation should be zero.
03

Calculate Standard Deviation

The formula for the standard deviation \( \sigma \) is: \[ \sigma = \sqrt{\frac{1}{N}\sum_{i=1}^{N}(x_i - \mu)^2} \] where: \(N\) is the number of observations in the dataset; \(x_i\) are the observed values; and \( \mu \) is the mean of the dataset. For the given dataset (5,5,5,5), the mean \( \mu \) is 5. Replace these values into the formula, and you get: \[ \sigma = \sqrt{\frac{1}{4}\sum_{i=1}^{4}(5 - 5)^2} \] which reduces to: \[ \sigma = \sqrt{0} \] So, the standard deviation \( \sigma \) is indeed 0.

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Most popular questions from this chapter

A large population has a mean of 310 and a standard deviation of 37 . Using the empirical rule, find what percentage of the observations fall in the intervals \(\mu \pm 1 \sigma, \mu \pm 2 \sigma\), and \(\mu \pm 3 \sigma\).

The following data represent the numbers of tornadoes that touched down during 1950 to 1994 in the 12 states that had the most tornadoes during this period (Storm Prediction Center, 2009). The data for these states are given in the following order: CO, FL, IA, IL, KS, LA, MO, MS, NE, OK, SD, TX. \(\begin{array}{llllllllllll}1113 & 2009 & 1374 & 1137 & 2110 & 1086 & 1166 & 1039 & 1673 & 2300 & 1139 & 5490\end{array}\) a. Calculate the mean and median for these data. b. Identify the outlier in this data set. Drop the outlier and recalculate the mean and median. Which of these two summary measures changes by a larger amount when you drop the outlier? c. Which is the better summary measure for these data, the mean or the median? Explain.

In some applications, certain values in a data set may be considered more important than others. For example, to determine students' grades in a course, an instructor may assign a weight to the final exam that is twice as much as that to each of the other exams. In such cases, it is more appropriate to use the weighted mean. In general, for a sequence of \(n\) data values \(x_{1}, x_{2}, \ldots, x_{n}\) that are assigned weights \(w_{1}\), \(w_{2}, \ldots, w_{n}\), respectively, the weighted mean is found by the formula $$ \text { Weighted mean }=\frac{\sum x w}{\sum w} $$ where \(\Sigma x w\) is obtained by multiplying each data value by its weight and then adding the products. Suppose an instructor gives two exams and a final, assigning the final exam a weight twice that of each of the other exams. Find the weighted mean for a student who scores 73 and 67 on the first two exams and 85 on the final. (Hint: Here, \(x_{1}=73, x_{2}=67, x_{3}=85, w_{1}=w_{2}=1\), and \(w_{3}=2 .\) )

In a study of distances traveled to a college by commuting students, data from 100 commuters yielded a mean of \(8.73\) miles. After the mean was calculated, data came in late from three students, with respective distances of \(11.5,7.6\), and \(10.0\) miles. Calculate the mean distance for all 103 students.

On a 300 -mile auto trip, Lisa averaged 52 mph for the first 100 miles, 65 mph for the second 100 miles, and 58 mph for the last 100 miles. a. How long did the 300 -mile trip take? b. Could you find Lisa's average speed for the 300 -mile trip by calculating \((52+65+58) / 3 ?\) If not, find the correct average speed for the trip.
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