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Some data sets include values so high or so low that they seem to stand apart from the rest of the data. These data are called outliers. Outliers may represent data collection errors, data entry errors, or simply valid but unusual data values. It is important to identify outliers in the data set and examine the outliers carefully to determine if they are in error. One way to detect outliers is to use a box-and-whisker plot. Data values that fall beyond the limits, $$\begin{aligned} &\text { Lower limit: } Q_{1}-1.5 \times(I Q R)\\\ &\text { Upper limit: } Q_{3}+1.5 \times(I Q R) \end{aligned}$$ where \(I Q R\) is the interquartile range, are suspected outliers. In the computer software package Minitab, values beyond these limits are plotted with asterisks (*). Students from a statistics class were asked to record their heights in inches. The heights (as recorded) were $$\begin{array}{cccccccccccc} 65 & 72 & 68 & 64 & 60 & 55 & 73 & 71 & 52 & 63 & 61 & 74 \\ 69 & 67 & 74 & 50 & 4 & 75 & 67 & 62 & 66 & 80 & 64 & 65 \end{array}$$ (a) Make a box-and-whisker plot of the data. (b) Find the value of the interquartile range \((I Q R)\) (c) Multiply the IQR by 1.5 and find the lower and upper limits. (d) Are there any data values below the lower limit? above the upper limit? List any suspected outliers. What might be some explanations for the outliers?

Step by step solution

01

Organize the Data

First, organize the data in ascending order: 4, 50, 52, 55, 60, 61, 62, 63, 64, 64, 65, 65, 66, 67, 67, 68, 69, 71, 72, 73, 74, 74, 75, 80.

02

Identify the Quartiles

Find the first quartile (\(Q_1\)), second quartile (\(Q_2\) or median), and third quartile (\(Q_3\)). There are 24 numbers, so \(Q_2\) is the average of the 12th and 13th numbers (65 and 66): \(Q_2 = 65.5\). \(Q_1\) is the average of the 6th and 7th numbers: \(Q_1 = \frac{61 + 62}{2} = 61.5\). \(Q_3\) is the average of the 18th and 19th numbers: \(Q_3 = \frac{71 + 72}{2} = 71.5\).

03

Calculate the Interquartile Range (IQR)

The interquartile range (IQR) is the difference between \(Q_3\) and \(Q_1\): \(IQR = Q_3 - Q_1 = 71.5 - 61.5 = 10\).

04

Determine the Limits for Outliers

Calculate 1.5 times the IQR: \(1.5 \times 10 = 15\). Then find the lower and upper limits. The lower limit is \(Q_1 - 1.5 \times IQR = 61.5 - 15 = 46.5\). The upper limit is \(Q_3 + 1.5 \times IQR = 71.5 + 15 = 86.5\).

05

Identify Outliers

Examine the data to see if any values are below the lower limit (46.5) or above the upper limit (86.5). The value 4 is below 46.5 and the value 50 is the closest to the lower limit but not an outlier. There are no values above 86.5. Thus, 4 is a suspected outlier, perhaps due to a data entry error.

06

Construct the Box-and-Whisker Plot

Plot a box-and-whisker plot using \(Q_1\) (61.5), median (65.5), and \(Q_3\) (71.5). Mark the calculated lower and upper limits. Plot the value 4 as an asterisk (*) to indicate it is an outlier.

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Key Concepts

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

Box-and-Whisker Plot

A box-and-whisker plot is a graphical tool that provides a visual summary of data. It is very helpful in detecting outliers. This plot displays data using five key numbers: the minimum, first quartile \(Q_1\), median \(Q_2\), third quartile \(Q_3\), and maximum. Here's how it works:

• The box captures the interquartile range (\

Interquartile Range (IQR)

Interquartile Range (IQR) is a crucial statistic employed in data analysis. It helps to determine the spread of the central 50% of the data set. By focusing on the middle part of the dataset, IQR provides a way to understand the variability without being influenced by outliers.
To calculate the IQR, you subtract the first quartile from the third quartile:
\(IQR = Q_3 - Q_1\)
This simple equation highlights data dispersion from the median. The IQR is effective in identifying outliers because it forms the foundation for calculating the lower and upper limits in a box-and-whisker plot, where potential outliers fall beyond these limits.Calculating the IQR is often one of the first steps analysts take to understand data variability.

Data Analysis

Data analysis is the process of examining, cleaning, transforming, and modeling data to discover useful information, draw conclusions, and support decision-making. When dealing with data sets, it's essential to recognize patterns and uncover insights, such as identifying potential outliers.
Using statistical methods like box-and-whisker plots and IQR calculations, you can better understand your data. For instance:

• Sort and plot the data to see its distribution.


• Calculate key statistics (e.g., IQR) to evaluate data spread.


• Identify any outliers, which could indicate potential errors or unique circumstances that require further investigation.

Through these methods, data analysis not only helps in identifying unusual observations but also aids in the accurate interpretation and representation of data, allowing for informed decisions.

Statistical Method

Statistical methods are tools used to collect, review, and interpret data. They allow you to draw conclusions by applying mathematical theories like probability. When analyzing data for outliers, statistical methods such as box-and-whisker plots and the use of IQR are particularly valuable.
Here's why these methods are essential:

• They provide a systematic way to summarize and visualize complex data sets.


• By identifying outliers, they help illuminate anomalies that may warrant further review or correction.


• These methods enable data-driven decision-making by presenting clear and concise visual data representations.

Statistical methods thus serve as a bridge between data and actionable insights, ensuring that decisions are supported by statistical evidence rather than assumptions.