91Ó°ÊÓ

Data at this text's website show the number of central public libraries in each of the 50 states and the District of Columbia. A summary of the data is shown in the following table. Should the maximum and minimum values of this data set be considered potential outliers? Why or why not? You can check your answer by using technology to make a boxplot using fences to identify potential outliers. (Source: Institute of Museum and Library Services) $$ \begin{aligned} &\text { Summary statistics }\\\ &\begin{array}{lcccccccc} \text { Column } & \text { n } & \text { Mean } & \text { Std. dev. } & \text { Median } & \text { Min } & \text { Max } & \text { Q1 } & \text { Q3 } \\ \text { Central } & 51 & 175.76471 & 170.37319 & 112 & 1 & 756 & 63 & 237 \\ \text { Public } & & & & & & & \\ \text { Libraries } & & & & & & & & \end{array} \end{aligned} $$

Step by step solution

01

Compute the inter-quartile range (IQR)

The inter-quartile range (IQR) is a measure of statistical dispersion and is calculated as the difference between the upper quartile (Q3) and the lower quartile (Q1). Using the data provided, we find the IQR = Q3 - Q1 = 237 - 63 = 174.

02

Compute the Lower and Upper fences

The lower fence is calculated as Q1 - 1.5 * IQR, and the upper fence is found as Q3 + 1.5 * IQR. This gives Lower fence = 63 - 1.5 * 174 = -198 and Upper fence = 237 + 1.5 * 174 = 498.

03

Determine if the Min and Max are potential outliers

A potential outlier is any data point below the lower fence or above the upper fence. Here Min = 1 and Max = 756. Clearly, Min = 1 above the Lower fence (-198) so Min is not an outlier. However, Max = 756 is above the Upper fence (498), hence Max is a potential outlier.

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Key Concepts

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

Outliers

Outliers are data points that are significantly different from the other values in a data set. They could be extremely high or low compared to the majority of data.
Outliers can greatly affect the results of statistical analysis, making it important to identify and possibly exclude them.
They might indicate a variation in measurement or suggest data entry errors, but can also be a result of naturally occurring variations.

• To identify outliers, we often use visual tools like boxplots and apply calculation methods such as using fences to find outliers.
• An observation is considered an outlier if it lies outside the expected range derived from the data, marked by the lower and upper fences.

In our example, by calculating these fences, we determined that the maximum value of 756 is a potential outlier.

Inter-Quartile Range (IQR)

The inter-quartile range (IQR) is a measure that describes the middle 50% of a dataset, providing insight into the data's spread over the middle half.
It is calculated by subtracting the first quartile (Q1) from the third quartile (Q3).
In our exercise, the IQR is manual computed as: \[ IQR = Q3 - Q1 = 237 - 63 = 174 \]

• The IQR helps statisticians understand the degree of variability in a dataset and identify potential outliers for further analysis.
• This range is less sensitive to extremes or outliers, making it a useful statistic for understanding central data trends.

By using the IQR, we also determine how data are spread around the median, thus providing insights into data consistency.

Boxplot

A boxplot is a graphical representation of a dataset that visually displays the summary information and makes it easy to spot outliers.
It showcases the minimum, first quartile (Q1), median, third quartile (Q3), and maximum in a simple and effective format.
Here are the main components of a boxplot for better understanding:

• The "box" part of the boxplot shows the IQR, which contains the middle 50% of the data.
• The "whiskers" are lines extending from the box to the smallest and largest values that are not considered outliers.
• Any point lying outside these whiskers is flagged as a potential outlier.

In this exercise, creating a boxplot would help to confirm our calculations by showing a data point at 756 well beyond the upper whisker, indicating it as a potential outlier.

Data Dispersion

Data dispersion refers to the spread of data points in a dataset, giving us insight into how much the data values deviate from the mean.
Various statistical measures are used to understand data dispersion, with IQR and standard deviation being the most common.
Here's what you need to know:

• Standard deviation shows how close the data points are to the mean. A high standard deviation indicates more spread out data.
• IQR, on the other hand, focuses on the internal spread, excluding the effects of very high or low values, making it useful for data sets with potential outliers.
• Data dispersion is vital for comparing variability between different data sets or understanding the reliability of data predictions.

In our example, the standard deviation is 170.37319, which is relatively large, indicating notable variability in the data set of libraries in U.S. states.