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Chapter 2: Problem 146

Examine issues of location and spread for boxplots. In each case, draw sideby- side boxplots of the datasets on the same scale. There are many possible answers. One dataset has median 50, interquartile range \(20,\) and range \(40 .\) A second dataset has median 50 , interquartile range 50 , and range 100 . A third dataset has median 50 , interquartile range 50 , and range 60 .

Short Answer

Expert verified
Boxplot 1 has a smaller interquartile range and range compared to the other boxplots, while boxplots 2 and 3 have larger, same interquartile ranges but different ranges, implying different total spreads of their respective datasets.

Step by step solution

01

Plot the first box

We will start by plotting the boxplot for the first dataset. We consider the median to be 50, so the line in the middle of the box will be at 50. Given that the interquartile range is 20, we divide it by 2 to get the lengths of the upper and lower boxes at 60 and 40 respectively. The range is 40, thus the whiskers will end at 70 and 30.
02

Plot the second box

Proceeding to draw the boxplot for the second dataset. The median is the same as the first box, 50. With the interquartile range 50, the upper and lower boxes will lie at 75 and 25 respectively. Considering the range 100, whiskers will end at 100 and 0.
03

Plot the third box

Finally, we draw the boxplot for the third dataset. The median is again 50, with interquartile range 50, therefore the boxes will be at 75 and 25 similar to the second boxplot. The range, however, is only at 60, placing the whiskers at 80 and 20.

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Key Concepts

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

Interquartile Range
The interquartile range (IQR) is fundamental in statistical data visualization, particularly in boxplots. It offers insight into the middle spread of a dataset by measuring the distance between the first quartile (Q1) and the third quartile (Q3). Essentially, it captures the range of the middle 50% of the data. The IQR is resistant to outliers, making it a robust statistic for describing variability.

For instance, a dataset with an IQR of 20 means that the distance between Q1 and Q3 is 20 units. This is relatively narrow and suggests that the middle half of the data points are closely packed around the median. Conversely, an IQR of 50 signifies a wider spread, indicating greater variability in the middle half of the dataset. Understanding the IQR is crucial as it influences how boxplots are drawn, with the 'box' stretching from Q1 to Q3.
Median
Median is the central value of a dataset when it is ordered from the smallest to the largest value. It divides the dataset into two equal halves. In the context of boxplots, the median is represented by the line inside the box. This line is crucial as it provides a quick visualization of the middle of the distribution of the data.

When multiple datasets have the same median, as in our exercise where each dataset has a median of 50, their boxplots will have the line in the middle of the box at the same point on the scale, making it straightforward to compare their spreads and ranges around a common center.
Range
The range of a dataset is the difference between the highest and the lowest values. It's a measure of how spread apart the data is and gives a quick sense of the overall variability. In a boxplot, this is shown by the whiskers that extend from the lower to the upper extremes.

However, the range is sensitive to outliers, meaning a single extreme value can skew the range significantly. In our exercise examples, ranges vary from 40 to 100, indicating very different levels of overall spread across different datasets. It is essential to note that the range provides a broader picture of dispersion compared to the more outlier-resistant IQR.
Statistical Data Visualization
Statistical data visualization involves graphically representing data to uncover underlying patterns, trends, and correlations. Boxplots are a classic example of this, offering a five-number summary (minimum, Q1, median, Q3, and maximum) of a dataset.

Boxplots are especially effective for comparing distributions across different datasets or conditions. They allow viewers to rapidly assess central tendency, variability, and potential outliers. A well-designed visualization communicates complex data insights in a clear and concise manner, enabling students and researchers alike to make informed decisions based on the data.

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Most popular questions from this chapter

Exercises 2.126 to 2.129 each describe a sample. The information given includes the five number summary, the sample size, and the largest and smallest data values in the tails of the distribution. In each case: (a) Clearly identify any outliers. (b) Draw a boxplot. Five number summary: \((210,260,270,300,\) 320)\(; n=500\) Tails: \(210,215,217,221,225, \ldots, 318,319,319,319,\) 320,320

The survey students consisted of 169 females and 193 males. Find \(\hat{p},\) the proportion who are female.

According to the \(95 \%\) rule, the largest value in a sample from a distribution which is approximately symmetric and bell-shaped should be between 2 and 3 standard deviations above the mean, while the smallest value should be between 2 and 3 standard deviations below the mean. Thus the range should be roughly 4 to 6 times the standard deviation. As a rough rule of thumb, we can get a quick estimate of the standard deviation for a bell-shaped distribution by dividing the range by \(5 .\) Check how well this quick estimate works in the following situations. (a) Pulse rates from the StudentSurvey dataset discussed in Example 2.17 on page \(77 .\) The five number summary of pulse rates is \((35,62,70,\) 78,130) and the standard deviation is \(s=12.2\) bpm. Find the rough estimate using all the data, and then excluding the two outliers at 120 and \(130,\) which leaves the maximum at 96 . (b) Number of hours a week spent exercising from the StudentSurvey dataset discussed in Example 2.21 on page 81 . The five number summary of this dataset is (0,5,8,12,40) and the standard deviation is \(s=5.741\) hours. (c) Longevity of mammals from the MammalLongevity dataset discussed in Example 2.22 on page 82 . The five number summary of the longevity values is (1,8,12,16,40) and the standard deviation is \(s=7.24\) years.

In Exercises 2.11 and 2.12, cases are classified according to one variable, with categories \(\mathrm{A}\) and \(\mathrm{B},\) and also classified according to a second variable with categories \(1,2,\) and 3 . The cases are shown, with the first digit indicating the value of the first variable and the second digit indicating the value of the second variable. (So "A1" represents a case in category \(\mathrm{A}\) for the first variable and category 1 for the second variable.) Construct a two-way table of the data. Twenty cases: \(\begin{array}{llllllllll}\mathrm{A} 1 & \mathrm{~A} 1 & \mathrm{~A} 1 & \mathrm{~A} 2 & \mathrm{~A} 3 & \mathrm{~A} 3 & \mathrm{~A} 3 & \mathrm{~A} 3 & \mathrm{~A} 3 & \mathrm{~A} 3 \\ \mathrm{~A} 3 & \mathrm{~A} 3 & \mathrm{~B} 1 & \mathrm{~B} 1 & \mathrm{~B} 1 & \mathrm{~B} 1 & \mathrm{~B} 2 & \mathrm{~B} 2 & \mathrm{~B} 2 & \mathrm{~B} 3\end{array}\)

If we have learned to solve problems by one method, we often have difficulty bringing new insight to similar problems. However, electrical stimulation of the brain appears to help subjects come up with fresh insight. In a recent experiment \(^{16}\) conducted at the University of Sydney in Australia, 40 participants were trained to solve problems in a certain way and then asked to solve an unfamiliar problem that required fresh insight. Half of the participants were randomly assigned to receive non-invasive electrical stimulation of the brain while the other half (control group) received sham stimulation as a placebo. The participants did not know which group they were in. In the control group, \(20 \%\) of the participants successfully solved the problem while \(60 \%\) of the participants who received brain stimulation solved the problem. (a) Is this an experiment or an observational study? Explain. (b) From the description, does it appear that the study is double-blind, single-blind, or not blind? (c) What are the variables? Indicate whether each is categorical or quantitative. (d) Make a two-way table of the data. (e) What percent of the people who correctly solved the problem had the electrical stimulation? (f) Give values for \(\hat{p}_{E},\) the proportion of people in the electrical stimulation group to solve the problem, and \(\hat{p}_{s},\) the proportion of people in the sham stimulation group to solve the problem. What is the difference in proportions \(\hat{p}_{E}-\hat{p} s\) ? (g) Does electrical stimulation of the brain appear to help insight?
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