/*! 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 40 Name two measures of the variati... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 40

Name two measures of the variation of a distribution, and state the conditions under which each measure is preferred for measuring the variability of a single data set.

Short Answer

Expert verified
Variance and Standard Deviation are two measures of the variation of a distribution. Variance is preferred when the data set is large and the individual differences aren't necessarily significant while Standard Deviation is preferred when we want a clear and simple representation of the variability in our data set.

Step by step solution

01

Define Variance

Variance is a measure of how spread out the numbers in a data set are. It is calculated as the average of the squared differences from the mean. The formula for variance is \[ \sigma^2 = \frac{\sum (x - \mu)^2}{N} \], where \(x\) is each value in the data set, \(\mu\) is the mean of the data set, and \(N\) is the number of data points. Variance is best used when we want to understand the variability from the average or mean in our data set.
02

Define Standard Deviation

Standard deviation is the measure of the dispersion of a set of data from its mean. It is calculated as the square root of the variance. The formula for standard deviation is \[ \sigma = \sqrt{\frac{\sum (x - \mu)^2}{N}} \]. Standard deviation is preferred when we need to ascertain the variability of the data, but with a unit of measurement similar to the original data units, making it easier to interpret compared to variance.
03

Summarize the application of each measure

Variance and standard deviation are both measures of variability in a data set. Variance is usually used when the data set is large and the individual differences aren't necessarily significant. On the other hand, the standard deviation is more commonly used because it is expressed in the original unit of measure, making it easier to interpret. Therefore, it is best used when we want a clear and simple representation of the variability in our data set.

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!

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

College student Jacqueline Loya asked 50 employed students how many times they went out to eat last week. Half of the students had full-time jobs and half had part-time jobs. Full-time: \(5,3,4,4,4,2,1,5,6,5,6,3,3,2,4,5,2,3,7,5,5,1\), \(4,6,7\) Part-time: \(1,1,5,1,4,2,2,3,3,2,3,2,4,2,1,2,3,2,1,3,3,2\), \(4,2,1\) a. Using the median values, write a sentence comparing the typical numbers of times the two groups ate out. b. Using the interquartile ranges, write a sentence comparing the variability of these two groups.

Babies born after 40 weeks gestation have a mean length of \(52.2\) centimeters (about \(20.6\) inches). Babies born one month early have a mean length of \(47.4 \mathrm{~cm}\). Assume both standard deviations are \(2.5 \mathrm{~cm}\) and the distributions are unimodal and symmetric. (Source: www.babycenter.com) a. Find the standardized score (z-score), relative to all U.S. births, for a baby with a birth length of \(45 \mathrm{~cm}\). b. Find the standardized score of a birth length of \(45 \mathrm{~cm}\) for babies born one month early, using \(47.4\) as the mean. c. For which group is a birth length of \(45 \mathrm{~cm}\) more common? Explain what that means.

In 2011 , the mean rate of violent crime (per 100,000 people) for the 10 northeastern states was 314 , and the standard deviation was 118 . Assume the distribution of violent crime rates is approximately unimodal and symmetric. a. Between what two values would you expect to find about \(95 \%\) of the rates? b. Between what two values would you expect to find about \(68 \%\) of the rates? c. If a northeastern state had a violent crime rate of 896 crimes per 100,000 people, would you consider this unusual? Explain. d. If a northeastern state had a violent crime rate of 403 crimes per 100,000 people, would you consider this unusual? Explain.

In 2010 , the laborers of a major factory in Australia went on strike. At that time, the average salary of a laborer was AU\$ \(21.60\) per hour, and the median salary was AU\$ \(20.00\). If you were representing the owners, which summary would you use to convince the public that a strike was not needed? If you were representing the laborers, which would you use? Why was there a discrepancy between the mean and median hourly rates? Explain. (Source: www.payscale.com)

College students and surfers Rex Robinson and Sandy Hudson collected data on the self-reported numbers of days surfed in a month for 30 longboard surfers and 30 shortboard surfers. Longboard: \(4,9,8,4,8,8,7,9,6,7,10,12,12,10,14,12,15,13\), \(10,11,19,19,14,11,16,19,20,22,20,22\) Shortboard: \(6,4,4,6,8,8,7,9,4,7,8,5,9,8,4,15,12,10,11,12\), \(12,11,14,10,11,13,15,10,20,20\) a. Compare the typical number of days surfing for these two groups by placing the correct numbers in the blanks in the following sentence: The median for the longboards was for the shortboards was \(\quad\) days, showing that those with boards typically surfed more days in this month. b. Compare the interquartile ranges by placing the correct numbers in the blanks in the following sentence: The interquartile range for the longboards was shortboards was \(\quad\) days, showing more variation in the days surfed this month for the boards.
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Pure Maths

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.