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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
You need the five-number summary.

Step by step solution

01

Understand the Elements of a Boxplot

A boxplot is a graphical representation that summarizes data using five significant values: minimum, lower quartile (Q1), median (Q2), upper quartile (Q3), and maximum. These are collectively known as the "five-number summary."
02

Analyze Each Option

Review each option given in the exercise: Option (a) asks for all individual observations, which are not necessary for a boxplot because it only requires the five-number summary. Option (b) asks for the mean and standard deviation, which are not used in constructing a boxplot as it focuses on medians and quartiles. Option (c) asks for the five-number summary, which is exactly what is needed to create a boxplot.
03

Identify the Correct Requirement

Based on the analysis in Step 2, option (c) is the correct requirement for constructing a boxplot, as it provides the necessary information to draw the data鈥檚 spread and central tendency without calculating mean or standard deviation.

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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 simple yet essential tool in summarizing a large set of numerical data. It consists of five key data points:
  • Minimum: This is the smallest value in the data set.
  • First Quartile (Q1): This is the median of the lower half of the data set. It represents the 25th percentile.
  • Median (Q2): This is the middle value when the data set is arranged in ascending order. It represents the 50th percentile of the data.
  • Third Quartile (Q3): This is the median of the upper half of the data set, representing the 75th percentile.
  • Maximum: This is the largest value in the data set.
Using these five values, you get a concise summary that helps in understanding the distribution of the data without having to evaluate every single data point.
This is particularly useful in statistics when you need to quickly identify the central tendency, spread, and variability of your data.
Quartiles
Quartiles are valuable statistical tools that divide a data set into four equal parts. Each quartile represents a specific percentile and helps to identify the distribution of data. Understanding quartiles is crucial for analyzing data:
  • First Quartile (Q1): Marks the 25th percentile. This means 25% of the data lies below this point.
  • Second Quartile (Q2), also known as the median: Represents the halfway point or 50th percentile of the data.
  • Third Quartile (Q3): Marks the 75th percentile, indicating that 75% of the data is below this point.
These quartiles help to measure the spread of a data set and find outliers.
They are instrumental in identifying skewness and the potential equality or inequality within the data set.
Graphical representation
Graphical representation of data refers to presenting data in visual formats like graphs or charts. A boxplot is one of these graphical tools. A boxplot, also known as a whisker plot, offers a quick visual summary of a data set鈥檚 central tendency and variability using the five-number summary. Here's how it works:
  • The box part of the plot shows the interquartile range (IQR), which is the distance between the first (Q1) and third quartiles (Q3). This range represents the middle 50% of the data.
  • The line inside the box indicates the median value, providing an insight into the data鈥檚 center.
  • The "whiskers" of the plot extend to the minimum and maximum values, showing the full range of the data.
Boxplots are particularly effective for comparing distributions between different data sets, indicating outliers, and revealing any potential skewness in the data.

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Most popular questions from this chapter

Which of the following is most affected if an extreme high outlier is added to your data? (a) The median (b) The mean (c) The first quartile

Fuel Economy for Midsize Cars. The Department of Energy provides fuel economy ratings for all cars and light trucks sold in the United States. Here are the estimated miles per gallon for city driving for the 186 cars classified as midsize in 2016, arranged in increasing order: 9 \(\begin{array}{llllllllllllllllll}11 & 11 & 11 & 12 & 13 & 13 & 13 & 14 & 14 & 14 & 14 & 14 & 15 & 15 & 15 & 15 & 15 & 15 \\ 16 & 16 & 16 & 16 & 16 & 16 & 16 & 16 & 16 & 16 & 16 & 16 & 16 & 17 & 17 & 17 & 17 & 17 \\ 17 & 18 & 18 & 18 & 18 & 18 & 18 & 18 & 18 & 18 & 19 & 19 & 19 & 19 & 19 & 19 & 19 & 19 \\\ 20 & 20 & 20 & 20 & 20 & 20 & 20 & 20 & 20 & 20 & 20 & 20 & 20 & 20 & 20 & 20 & 20 & 20 \\ 21 & 21 & 21 & 21 & 21 & 21 & 21 & 21 & 21 & 22 & 22 & 22 & 22 & 22 & 22 & 22 & 22 & 22 \\ 22 & 22 & 22 & 22 & 22 & 22 & 22 & 23 & 23 & 23 & 23 & 23 & 23 & 23 & 24 & 24 & 24 & 24 \\ 24 & 24 & 24 & 24 & 24 & 25 & 25 & 25 & 25 & 25 & 25 & 25 & 25 & 25 & 25 & 25 & 25 & 25 \\ 25 & 25 & 26 & 26 & 26 & 26 & 26 & 26 & 26 & 26 & 26 & 26 & 26 & 27 & 27 & 27 & 27 & 27 \\ 27 & 27 & 27 & 27 & 27 & 27 & 27 & 27 & 28 & 28 & 28 & 28 & 28 & 28 & 28 & 28 & 28 & 28 \\ 29 & 29 & 29 & 29 & 29 & 29 & 30 & 30 & 30 & 30 & 31 & 31 & 35 & 36 & 39 & 40 & 40 & 40 \\ 40 & 41 & 43 & 44 & 54 & 58 & & & & & & & & & & & & \end{array}\) (a) Give the five-number summary of this distribution. (b) Draw a boxplot of these data. What is the shape of the distribution shown by the boxplot? Which features of the boxplot led you to this conclusion? Are any observations unusually small or large?

Returns on stocks. How well have stocks done over the past generation? The Wilshire 5000 index describes the average performance of all U.S. stocks. The average is weighted by the total market value of each company's stock, so think of the index as measuring the performance of the average investor. Here are the percent returns on the Wilshire 5000 index for the years from 19712015: 22 ? WILSHIRE $$ \begin{array}{lc|cc|cc} \hline \text { Year } & \text { Return } & \text { Year } & \text { Return } & \text { Year } & \text { Return } \\ \hline 1971 & 17.68 & 1986 & 16.09 & 2001 & -10.97 \\ \hline 1972 & 17.98 & 1987 & 2.27 & 2002 & -20.86 \\ \hline 1973 & -18.52 & 1988 & 17.94 & 2003 & 31.64 \\ \hline 1974 & -28.39 & 1989 & 29.17 & 2004 & 12.62 \\ \hline 1975 & 38.47 & 1990 & -6.18 & 2005 & 6.32 \\ \hline 1976 & 26.59 & 1991 & 34.20 & 2006 & 15.88 \\ \hline 1977 & -2.64 & 1992 & 8.97 & 2007 & 5.73 \\ \hline 1978 & 9.27 & 1993 & 11.28 & 2008 & -37.34 \\ \hline 1979 & 25.56 & 1994 & -0.06 & 2009 & 29.42 \\ \hline 1980 & 33.67 & 1995 & 36.45 & 2010 & 17.87 \\ \hline 1981 & -3.75 & 1996 & 21.21 & 2011 & 0.59 \\ \hline 1982 & 18.71 & 1997 & 31.29 & 2012 & 16.12 \\ \hline & & & & & \end{array} $$ $$ \begin{array}{lc|cc|ll} 1983 & 23.47 & 1998 & 23.43 & 2013 & 34.02 \\ \hline 1984 & 3.05 & 1999 & 23.56 & 2014 & 12.07 \\ \hline 1985 & 32.56 & 2000 & -10.89 & 2015 & -0.24 \\ \hline \end{array} $$ What can you say about the distribution of yearly returns on stocks?

Guinea pig survival times. Here are the survival times in days of 72 guinea pigs after they were injected with infectious bacteria in a medical experiment. \({ }^{16}\) Survival times, whether of machines under stress or cancer patients after treatment, usually have distributions that are skewed to the right. \(\begin{array}{cccccccccccc}43 & 45 & 53 & 56 & 56 & 57 & 58 & 66 & 67 & 73 & 74 & 79 \\ 80 & 80 & 81 & 81 & 81 & 82 & 83 & 83 & 84 & 88 & 89 & 91 \\ 91 & 92 & 92 & 97 & 99 & 99 & 100 & 100 & 101 & 102 & 102 & 102 \\ 103 & 104 & 107 & 108 & 109 & 113 & 114 & 118 & 121 & 123 & 126 & 128 \\ 137 & 138 & 139 & 144 & 145 & 147 & 156 & 162 & 174 & 178 & 179 & 184 \\ 191 & 198 & 211 & 214 & 243 & 249 & 329 & 380 & 403 & 511 & 522 & 598\end{array}\) (a) Graph the distribution and describe its main features. Does it show the expected right-skew? (b) Which numerical summary would you choose for these data? Calculate your chosen summary. How does it reflect the skewness of the distribution?

What percent of the observations in a distribution are greater than the first quartile? (a) \(25 \%\) (b) \(50 \%\) (c) \(75 \%\)
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