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Chapter 3: Problem 38

Can the standard deviation have a negative value? Explain.

Short Answer

Expert verified
No, the standard deviation cannot be negative. Given the mathematical operations involved in its calculation (squares and square roots), all resulting values must be either zero or positive.

Step by step solution

01

Understand the question

The goal is to determine whether the standard deviation, a statistical measure, can ever be negative. The standard deviation measures the amount of variation or dispersion of a set of values.
02

Recall the formula for standard deviation

The calculation for standard deviation is: \( \sigma = \sqrt{\frac{1}{N} \sum_{i=1}^N (x_i - \mu)^2 }\) where \( \sigma \) is the standard deviation, \( N \) is the size of the sample, \( x_i \) are the individual data points, and \( \mu \) is the mean of the data set.
03

Explain why standard deviation can't be negative

In the formula for standard deviation, \( (x_i - \mu)^2 \) ensures that each term is non-negative because the square of any real number, whether negative or positive, is always non-negative. The sum of non-negative numbers is also non-negative, and the square root of a non-negative number is also non-negative. So, the standard deviation can never be negative since all the operations and terms involved in its calculation can never result in a negative number.

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Most popular questions from this chapter

Nixon Corporation manufactures computer monitors. The following data give the numbers of computer monitors produced at the company for a sample of 30 days. \(\begin{array}{llllllllll}24 & 32 & 27 & 23 & 33 & 33 & 29 & 25 & 23 & 36 \\\ 26 & 26 & 31 & 20 & 27 & 33 & 27 & 23 & 28 & 29 \\ 31 & 35 & 34 & 22 & 37 & 28 & 23 & 35 & 31 & 43\end{array}\) a. Calculate the values of the three quartiles and the interquartile range. Where does the value of 31 lie in relation to these quartiles? b. Find the (approximate) value of the 65 th percentile. Give a brief interpretation of this percentile. c. For what percentage of the days was the number of computer monitors produced 32 or higher? Answer by finding the percentile rank of 32 .

A large population has a bell-shaped distribution with 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 \(\mu \pm 3 \sigma\).

According to the American Time Use Survey conducted by the Bureau of Labor Statistics (www.bls.gov/atus/), Americans spent an average of \(985.50\) hours watching television in \(2010 .\) Suppose that the standard deviation of the distribution of times that Americans spent watching television in 2010 is \(285.20\) hours. a. Using Chebyshev's theorem, find at least what percentage of Americans watched television in 2010 for i. \(272.50\) to \(1698.50\) hours ii. \(129.90\) to \(1841.10\) hours *b. Using Chebyshev's theorem, find the interval that contains the time (in hours) that at least \(75 \%\) of Americans spent watching television in 2010 .

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