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Chapter 2: Problem 140

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, using the IQR method. (b) Draw a boxplot. Five number summary: (5,10,12,16,30)\(;\) \(n=40 .\) Tails: \(5,5,6,6,6, \ldots, 22,22,23,28,30 .\)

Short Answer

Expert verified
The outliers identified using the IQR method are the numbers 28 and 30.

Step by step solution

01

Calculating the Interquartile Range (IQR)

The IQR is the 3rd Quartile (Q3) subtract the 1st Quartile (Q1). From the given five-number summary, Q3 = 16 and Q1 = 10. Therefore, \(IQR = Q3 - Q1 = 16 - 10 = 6\).
02

Identifying Outliers

To identify if there are any outliers, calculate the boundaries, which are 1.5 * IQR below Q1 and above Q3. Below Q1 is \(10 - 1.5*6 = -1\) and above Q3 is \(16 + 1.5*6 = 25\). Datas beyond these values are considered to be the outliers. Looking at the numbers in the tails of the distribution, the numbers 28 and 30 are the outliers because they are above 25.
03

Drawing a Boxplot

For drawing the boxplot, mark the minimum, Q1, median, Q3 and the maximum values from the five number summary on the number line. In this case, minimum = 5, Q1 = 10, median = 12, Q3 = 16 and maximum = 30. Next, construct a box from Q1 to Q3 and draw a vertical line at the median. Then, draw lines (whiskers) from the box to the minimum and maximum values not including the outliers. The outliers are represented as individual points beyond the whiskers.

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Key Concepts

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

IQR Method
The IQR Method is a powerful tool for detecting outliers in your data. IQR stands for "Interquartile Range," which is calculated by subtracting the first quartile (\(Q1\)) from the third quartile (\(Q3\)). This method helps you identify the spread of the middle 50% of your data.

Here's how the IQR method works:
  • Calculate the IQR: \(IQR = Q3 - Q1\)
  • Determine the lower boundary for outliers: \(Q1 - 1.5 \times IQR\)
  • Determine the upper boundary for outliers: \(Q3 + 1.5 \times IQR\)
  • Identify any data points falling outside these boundaries as outliers.

In our example, the five-number summary is (5, 10, 12, 16, 30) with \(Q1 = 10\) and \(Q3 = 16\). So, the IQR is \(6\). The lower boundary is \(-1\), and the upper boundary is \(25\). Any data points below \(-1\) or above \(25\) are outliers. Thus, 28 and 30 are identified as outliers.
Boxplot
A boxplot, sometimes called a whisker plot, is a graphical representation of the data distribution based on the five-number summary. It provides a clear visual of the central tendency, spread, and potential outliers.

To draw a boxplot, follow these steps:
  • Mark the minimum, \(Q1\), median, \(Q3\), and maximum values.
  • Draw a box from \(Q1\) to \(Q3\) and a line at the median inside the box.
  • Extend "whiskers" from the box to the smallest and largest data points within the non-outlier range.
  • Plot any outliers as individual points beyond the whiskers.

In this example, the minimum value is 5, and the maximum is 30. The box covers from 10 to 16, with a line at the median, 12. The whiskers stretch to the minimum and maximum but exclude the outliers 28 and 30, which appear as separate points.
Five Number Summary
The five-number summary is a concise way to describe a dataset using five key statistics:
  • Minimum: The smallest value.
  • First Quartile (\(Q1\)): The median of the lower half.
  • Median: The middle value of the dataset.
  • Third Quartile (\(Q3\)): The median of the upper half.
  • Maximum: The largest value.

This summary offers a quick glimpse into the center and spread of the data. In our given data, the summary (5, 10, 12, 16, 30) shows that the data is spread from 5 to 30, with the core 50% between 10 and 16.

By combining this summary with the IQR, you can visually check for symmetry, skewness, and outliers using a boxplot. It's an essential tool in exploratory data analysis.

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Most popular questions from this chapter

Does pre-season success indicate regular season success in the US National Football League? We looked at the number of preseason wins and regular season wins for all 32 NFL teams over a 10 -year span. (a) What would a positive association imply about the relationship between pre-season and regular season success in the NFL? What would a negative association imply? (b) The correlation between these two variables is \(r=0.067\). What does this correlation tell you about the strength of a linear relationship between these two variables?

Laptop Computers and Sperm Count Studies have shown that heating the scrotum by just \(1^{\circ} \mathrm{C}\) can reduce sperm count and sperm quality, so men concerned about fertility are cautioned to avoid too much time in the hot tub or sauna. A new study \(^{44}\) suggests that men also keep their laptop computers off their laps. The study measured scrotal temperature in 29 healthy male volunteers as they sat with legs together and a laptop computer on the lap. Temperature increase in the left scrotum over a 60-minute session is given as \(2.31 \pm 0.96\) and a note tells us that "Temperatures are given as \({ }^{\circ} \mathrm{C}\); values are shown as mean \(\pm \mathrm{SD}\)." The abbreviation SD stands for standard deviation. (Men who sit with their legs together without a laptop computer do not show an increase in temperature.) (a) If we assume that the distribution of the temperature increases for the 29 men is symmetric and bell-shaped, find an interval that we expect to contain about \(95 \%\) of the temperature increases. (b) Find and interpret the \(z\) -score for one of the men, who had a temperature increase of \(4.9^{\circ}\).

Public Expenditure on Education Figure 2.27 shows the public expenditure on education as percentage of Gross Domestic Product (GDP) for all countries. \(^{42}\) The mean expenditure is \(\mu=4.7 \%\) and the standard deviation of the expenditures is \(\sigma=2 \% .\) The data are stored in EducationLiteracy. (a) The United States spends \(5.2 \%\) of it's GDP on education. Without doing any calculations yet, will the \(z\) -score for the US be positive, negative, or zero? Why? (b) Calculate the \(z\) -score for the US. (c) There are two high outliers; Lesotho (a small country completely surrounded by South Africa) spends \(13 \%\) of it's GDP on education and Cuba spends \(12.8 \%\). Equatorial Guinea spends the lowest percentage on education at only \(0.8 \% .\) Calculate the range. (d) The five number summary for this data set is \((0.8,3.2,4.6,5.6,13) .\) Calculate the IQR.

In Example 2.43 on page 127 , we used the approval rating of a president running for re-election to predict the margin of victory or defeat in the election. We saw that the least squares line is \(\widehat{\text { Margin }}=-36.76+0.839\) ( Approval). Interpret the slope and the intercept of the line in context.

Create Your Own: Bubble Plot Using any of the datasets that come with this text that include at least three quantitative variables (or any other dataset that you find interesting and that meets this condition), use statistical software to create a bubble plot of the data. Indicate the dataset, the cases, and the variables that you use. Specify which variable represents the size of the bubble. Comment (in context) about any interesting features revealed in your plot.
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