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Chapter 2: Problem 20

To make a boxplot of a distribution, you must know a. all the individual observations. b. the mean and the standard deviation. c. the five-number summary.

Short Answer

Expert verified
To make a boxplot, you need the five-number summary.

Step by step solution

01

Understanding the Question

The question asks what information is needed to create a boxplot, which is a way to graphically represent a data distribution.
02

Identifying Key Features of a Boxplot

A boxplot (or box-and-whisker plot) displays the distribution of data based on a summary of five specific numbers: the minimum, first quartile (Q1), median (Q2), third quartile (Q3), and maximum values. This is known as the five-number summary.
03

Evaluating the Options

You are presented with three options: (a) all individual observations, (b) the mean and standard deviation, and (c) the five-number summary. A true boxplot does not use all individual data points, nor does it require mean and standard deviation, but it does require the five-number summary for its construction.
04

Confirming the Correct Answer

The five-number summary can offer a direct insight into the data's distribution by using key percentiles and extremes. Therefore, the necessary information to draw a boxplot is option (c).

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Key Concepts

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

Five-number summary
The five-number summary is a crucial set of statistics used to summarize a data set. It includes:
  • Minimum - This is the smallest data point in the set.

  • First quartile (Q1) - This is the value below which 25% of the data fall. It marks the boundary of the lowest quarter of the data.

  • Median (Q2) - This middle value divides the data into two equal halves. Fifty percent of the data values are less than or equal to the median.

  • Third quartile (Q3) - This is the value below which 75% of the data fall. It defines the upper boundary of the lower three-quarters of the data.

  • Maximum - The largest data point in the set.

By using the five-number summary, we can get a quick snapshot of the data's range, the spread of the middle half of the data, and the central tendency. This summary is essential for creating a boxplot, as it allows you to visualize these key aspects in one concise graph.
Data distribution
Data distribution describes how data points are spread across a range of values. Understanding data distribution helps to reveal patterns, trends, and potential outliers. There are several shapes your data distribution might take:
  • Symmetrical: The left and right sides of the distribution are approximately mirror images of each other.

  • Skewed: If the data tails off to the right, it's positively skewed, and if it tails off to the left, it's negatively skewed.

  • Uniform: All values have approximately the same frequency of occurrence.

A boxplot is a great way to represent data distribution because it highlights the central 50% of the data (interquartile range), along with potential outliers shown as individual points. This visualization provides a clear overview of how data is distributed around its median, making it easier to spot asymmetry and variability in the dataset.
Quartiles
Quartiles are values that divide a data set into four equal parts. They are essential for interpreting and understanding the distribution within a data set. The three main quartiles used are:
  • First Quartile (Q1): It covers the lower 25% of the data.

  • Second Quartile (Q2): Also known as the median; it divides the data into two 50% parts.

  • Third Quartile (Q3): It comprises the lower 75% of data.

Each quartile marks a percentage boundary within the data set:
  • The interquartile range (IQR) is the range between the first and third quartile. It's calculated as Q3 - Q1, providing a measure of data spread that is less affected by outliers.

Quartiles help to interpret the variability and distribution of the data set, and they are crucial when creating a boxplot, as they define the central box where most of the data points lie. This makes quartiles a potent tool for data analysis.

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Most popular questions from this chapter

Making Resistance Visible. In the Mcar and Med铆an applet, place three observations an the line by clicking below it: two clase together near the center of the line and ane somewhat to the right of these two. 2\. Pull the siagle rightmost observation out to the right. (Place the cursor on the point, hold down a mouse button, and drag the point.) How does the mean behave? How does the median behave? Explain briefly why each measure acts as it does. b. Now drag the single rightmost point to the left as far as you can. What happens to the mean? What happers to the median as yod drag this posint past the other two? (Wat ch carefully)

New House Prices. The mean and median sales prices of new homes sold in the United States in July 2019 were \(\$ 312,800\) and \(\$ 388,000.5\) Which of these numbers is the mean, and which is the median? Explain how you know.

Household Assets. Once every three years, the Board of Governors of the Federal Reserve System collects data on household assets and liabilities through the Survey of Consumer Finances (SCF). 16 Here are some results from the 2016 survey. a. Transaction accounts, which include checking, savings, and money market accounts, are the most commonly held type of financial asset. The mean value of transaction accounts per household for those holding transaction accounts was \(\$ 40,200\), and the median value was \(\$ 4,500\). What explains the differences between the two measures of center? b. The median value of cash value life insurance per household was \(\$ 0\). What does a median of \(\$ 0\) say about the percentage of households with cash value life insurance?

Policy Justification: Pragmatic vs. Moral. How does a leader's just ification of his/her orgamizat ion's policy affect support for the policy? A study compared a moral, pragmatic, and ambiguous justification for three policy proposals: a politician's plan to fund a retirement planning agency, a state governor's plan to repave state highways, and a president's plan to outlaw child labor in a developing country. For example, for the retirement agency proposal, the moral justification was the importance of retirees "to live with dign脿ty and comfort, " the pragmatic was "to not drain public funds," and the ambiguous was "to have sufficient funds." Three hundred seventy-four volunteer subjects were assigned at random to read all three proposals: 122 subjects read the three proposals with a moral justification, 126 subjects read the three proposals with a pragmatic justification, and 126 subjects read the three proposals with an ambiguous justification. Several questions measuring support for each policy proposal were answered by each subject to create a support score for each proposal, and their scores for the three proposals were them averaged to create an index of policy support for each subject, higher values indicating The first individual read the proposals with a pragmatic justification, with a policy support index of 5 , the second with an ambiguous justification and a policy support index of 7 , and so forth. Compare the three distributions. How does the support index vary with the type af justificat ion?

x\( and \)s\( by Hand. Radon is a naturally occurring gas and is the second leading cause of lung cancer in the United States. \)\frac{12}{}\( It comes from the natural breakdown of uranium in the soil and enters buildings through cracks and other holes in foundations. Radon is found throughout the United States, but levels vary considerably from state to state. Several methods can reduce the levels of radon in a home, and the Environmental Protection Agency recommends using one of them if the measured level in a home is above 4 picocuries per liter. Four readings from Franklin County, Ohio, where the county average is \)8.2\( picocuries per liter, were \)3.8,1.9,12.1\(, and \)14.4\(. a. Find the mean step-by-step. That is, find the sum of the four observations and divide by 4 . b. Find the standard deviation step-by-step. That is, find the deviation of each observation from the mean, square the deviations, and obtain the variance and the standard deviation. Example 2.7 (page 57 ) shows the method. c. Now enter the data into your calculator and use the mean and standard deviation buttons to obtain \)x\( and \)s$. Do the results agree with your hand calculations?
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