/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 38 Can the standard deviation have ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 38

Can the standard deviation have a negative value? Explain.

Short Answer

Expert verified
No, the standard deviation cannot have a negative value. This is due to the fact that it is calculated as the square root of the variance which involve positive elements only.

Step by step solution

01

Understand Standard Deviation

Standard deviation is a measure that is used to quantify the amount of variation or dispersion of a set of values. It's the square root of the Variance. Variance is the average of the squared differences from the Mean.
02

Analyze Variance

Variance involves averaging the squared differences from the Mean. As such, every term in the calculation is positive due to the squaring of the differences. In this sense, variance can never be negative.
03

Understand the Implications for Standard Deviation

Given that the variance can never be negative and the fact that the standard deviation is the square root of the variance, it also cannot be negative. The square root of a positive number is always positive.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Variance: Understanding the Foundation
Variance is a fundamental concept in statistics that measures how spread out a set of numbers is. It is calculated as the average of the squared differences from the mean.
  • The process of squaring differences ensures that negative and positive deviations don't cancel each other out, thereby emphasizing total variation.
  • This calculation results in a value that reflects the overall spread of data points.
By interpreting the variance, you can understand how data points lie in relation to the average, giving insight into the data set鈥檚 diversity.
Mean: The Central Tendency
The mean, often referred to as the average, is a central value that represents the data set. It is found by adding all numbers in the data set and dividing by the count of numbers.
  • The mean provides a straightforward summary of the data.
  • It serves as a reference point from which individual data points are compared when calculating variance.
Understanding the mean is critical, as it lays the groundwork for other statistical measures, including variance and standard deviation.
Dispersion: Measuring the Spread
Dispersion refers to the extent to which a distribution is stretched or squeezed. It indicates how much the values in a data set differ from each other.
  • Standard deviation and variance are key indicators of dispersion. They signify the distribution鈥檚 spread around the mean.
  • A higher measure of dispersion often implies greater unpredictability within the data set.
Analyzing dispersion helps in understanding the relative consistency or volatility of data.
Square Root: Connecting Variance and Standard Deviation
The square root functions as a bridge between variance and standard deviation. The standard deviation is derived by taking the square root of the variance.
  • Squaring deviations during variance calculation ensures non-negative results, aligning with the definition and purpose.
  • Using the square root to determine standard deviation translates this squared measure back into the original unit of the data set.
This process helps in getting a more interpretable measure of spread that reflects the actual data's scale.
Statistical Measures: Informing Data Insights
Statistical measures provide a structured way to interpret and analyze data. They include various calculations like mean, variance, and standard deviation.
  • These measures work together to give a comprehensive understanding of data characteristics.
  • They help in identifying trends, patterns, and potential outliers in data sets.
Employing statistical measures is crucial for drawing accurate conclusions from data, enabling informed decisions based on quantitative analysis.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Nixon Corporation manufactures computer monitors. The following data are the numbers of computer monitors produced at the company for a sample of 10 days. \(\begin{array}{lllllll}24 & 32 & 27 & 23 & 35 & 33 & 29\end{array}\) 40 23 28 Calculate the mean, median, and mode for these data.

Refer to Exercise \(3.115\). Suppose the times taken to learn the basics of this word processor by all students have a bell-shaped distribution with a mean of 200 minutes and a standard deviation of 20 minutes. a. Using the empirical rule, find the percentage of students who will learn the basics of this word processor in i. 180 to 220 minutes ii. 160 to 240 minutes "b. Using the empirical rule, find the interval that contains the time taken by \(99.7 \%\) of all students to learn this word processor.

Suppose the average credit card debt for households currently is \(\$ 9500\) with a standard deviation of \(\$ 2600\). a. Using Chebyshev's theorem, find at least what percentage of current credit card debts for all households are between i. \(\$ 4300\) and \(\$ 14,700\) ii. \(\$ 3000\) and \(\$ 16,000\) :b. Using Chebyshev's theorem, find the interval that contains credit card debts of at least \(89 \%\) of all households.

Refer to the data of Exercise \(3.109\) on the current annual incomes (in thousands of dollars) of the 10 members of the class of 2000 of the Metro Business College who were voted most likely to succeed. \(\begin{array}{llllllllll}59 & 68 & 84 & 78 & 107 & 382 & 56 & 74 & 97 & 60\end{array}\) a. Determine the values of the three quartiles and the interquartile range. Where does the value of 74 fall in relation to these quartiles? b. Calculate the (approximate) value of the 70 th percentile. Give a brief interpretation of this percentile. c. Find the percentile rank of 97 . Give a brief interpretation of this percentile rank.

The following data set belongs to a population: $$ \begin{array}{ccccccc} 5 & -7 & 2 & 0 & -9 & 1 & 61 \end{array} $$ Calculate the range, variance, and standard deviation.
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.