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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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