91Ó°ÊÓ

Annual sales, in millions of dollars, for 21 pharmaceutical companies follow. \\[\begin{array}{rrrrrr} 8408 & 1374 & 1872 & 8879 & 2459 & 11413 \\ 608 & 14138 & 6452 & 1850 & 2818 & 1356 \\ 10498 & 7478 & 4019 & 4341 & 739 & 2127 \\ 3653 & 5794 & 8305 & &\end{array}\\] a. Provide a five-number summary. b. Compute the lower and upper limits. c. Do the data contain any outliers? d. Johnson \& Johnson's sales are the largest on the list at \(\$ 14,138\) million. Suppose a data entry error (a transposition) had been made and the sales had been entered as \(\$ 41,138\) million. Would the method of detecting outliers in part (c) identify this problem and allow for correction of the data entry error? e. Show a box plot.

Step by step solution

01

Arrange the Data

First, list all the annual sales data in increasing order. The data in millions of dollars is: \[608, 739, 1356, 1374, 1850, 1872, 2127, 2459, 2818, 3653, 4019, 4341, 5794, 6452, 7478, 8305, 8408, 8879, 10498, 11413, 14138.\]

02

Calculate the Five-Number Summary

The five-number summary includes the minimum, first quartile (Q1), median, third quartile (Q3), and maximum. - **Minimum**: 608 - **Q1**: The first quartile, which is the median of the first half of the data. For 21 data points, Q1 is the 6th value: 1872. - **Median**: Being the 11th value for 21 data points: 4019. - **Q3**: The third quartile, the median of the second half of the data, is the 16th value: 8305. - **Maximum**: 14138.

03

Compute the Interquartile Range (IQR)

Calculate the IQR, which is Q3 - Q1. \[IQR = 8305 - 1872 = 6433.\]

04

Calculate the Lower and Upper Limits

The lower and upper limits are computed to detect outliers. - **Lower Limit**: Q1 - 1.5 * IQR = 1872 - 1.5 * 6433 = -2638.5. - **Upper Limit**: Q3 + 1.5 * IQR = 8305 + 1.5 * 6433 = 12815.5.

05

Identify Outliers

Outliers are data points outside the lower and upper limits. Since no sales figures are below the lower limit (-2638.5), and only 14138 exceeds the upper limit (12815.5), the latter is an outlier.

06

Consider a Data Entry Error

If Johnson & Johnson's sales were erroneously entered as 41138 instead of 14138, the method in Step 5 would still identify it as an outlier since 41138 exceeds the upper limit significantly.

07

Draw a Box Plot

To create a box plot, draw a number line that accommodates the data range. Mark the five-number summary: minimum (608), Q1 (1872), median (4019), Q3 (8305), and maximum (14138). Outliers, like 14138, should be indicated with an asterisk or different marker.

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Key Concepts

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

Box Plot

A box plot, sometimes called a whisker plot, is an excellent tool for visualizing the distribution of a dataset. It's a simple graphical representation that provides a snapshot of the data's spread and its central ideas. To create a box plot, you need five key pieces of information: the minimum, first quartile (Q1), median, third quartile (Q3), and maximum values of the dataset.
This set of values is known as the five-number summary.

• The box itself shows the interquartile range (IQR), spanning from Q1 to Q3.
• The line inside the box is the median, which divides the data into two equal parts.
• The ends of the "whiskers" extend to the minimum and maximum values, excluding outliers.

Outliers can be displayed as individual points beyond the whiskers, making them easily identifiable.

Interquartile Range

The interquartile range (IQR) is a vital statistic in descriptive statistics, which helps understand data variability. It is calculated as the difference between the third quartile (Q3) and the first quartile (Q1).
The IQR focuses on the central 50% of a dataset, offering insight into its density and spread.

• IQR is a robust measure because it is not affected by extreme values or outliers, making it more reliable than the traditional range.
• In this particular exercise, the IQR is computed as \[IQR = Q3 - Q1 = 8305 - 1872 = 6433\].

Understanding the IQR helps in identifying the range of the "middle" part of the data, ensuring that the analysis isn't skewed by atypical data points.

Outliers

Outliers are data points that differ significantly from the rest of the dataset. They can potentially skew and mislead statistical analysis. The detection and handling of outliers are crucial for obtaining reliable results.
To identify outliers using the IQR, you calculate the lower and upper limits as follows:

• Lower Limit: \( Q1 - 1.5 \times IQR \)
• Upper Limit: \( Q3 + 1.5 \times IQR \)

In this data, any sales figure beyond the calculated upper limit of 12815.5, such as 14138, is deemed an outlier.
Outliers in data can suggest anomalies or errors, and, as such, should be examined closely to determine their validity.

Quartiles

Quartiles are an essential aspect of data analysis, helping to break down a dataset into four equal parts. They offer a clearer picture of the data's spread and center.

• **First Quartile (Q1)**: Represents the 25th percentile, where one-quarter of the data falls below this value.
• **Median**: Also known as the second quartile, it is the midpoint of the dataset.
• **Third Quartile (Q3)**: Marks the 75th percentile, indicating that three-quarters of the data are below this value.

In our context, the key quartile values are 1872 for Q1, 4019 for the median, and 8305 for Q3.
Quartiles are highly informative, especially when crafting a five-number summary and understanding data distribution. They help in identifying the general location of most data points, ensure easy plotting, and quick comparison across datasets.