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Interquartile range, commonly abbreviated as IQR, is a vital concept in understanding how to detect the spread within a data set.

The IQR is essentially the range within the middle 50% of a set of numbers, ignoring the lowest 25% and the highest 25%. It is calculated by taking the difference between the third quartile (Q3) and the first quartile (Q1). In simpler terms, the IQR gives us the range between the first group of 25% of the data and the third group of 25% of the data when it’s sorted in ascending order.
To put this into perspective, consider a set of test scores for a class. If the first quartile score is 60 and the third quartile score is 80, then the IQR is 80 - 60 = 20. This tells us that the middle 50% of students scored within a 20-point range. The IQR is particularly useful because it is uninfluenced by extreme values, or outliers, which may skew our understanding of the data range.

When applying the IQR to the exercise provided, we subtract the first quartile value from the third quartile value, resulting in an IQR of 10.8. This measurement is crucial for the next steps, as it helps to determine the boundaries beyond which data points would be considered unusual, or outliers.

A boxplot is a standardized way of displaying the distribution of data based on a five-number summary: minimum, first quartile (Q1), median, third quartile (Q3), and maximum. It can reveal much about a data set with a simple glance.

A boxplot shows the median of the data as a line within a box, which itself represents the interquartile range. Small lines, or 'whiskers,' extend from the box to indicate the variability outside the upper and lower quartiles. Outliers may be plotted as individual points beyond the whiskers.

Creating a boxplot for the gas taxes example highlights the central tendency (median) and the dispersion (IQR) of the data. It also visually indicates if there are any potential outliers when compared to the calculated 'fences'. If we were to construct a boxplot using the provided statistics, it would allow us to quickly assess whether the maximum and minimum values fall within the range typically considered normal for this data set.

Statistical outliers are data points that differ significantly from other observations. They can provide valuable insights or highlight errors in data collection. However, they can also skew the analysis and might need to be handled separately.

Outliers can be identified using the interquartile range by calculating 'fences'. As per statistical convention, lower and upper fences are typically set to 1.5 times the IQR below Q1 and above Q3 respectively.

In our exercise, by extending 1.5 times the IQR from the quartiles, we have set the criteria for outliers. Any value lying beyond these fences can be considered a statistical outlier. Following this, the calculated lower fence is 23.8 and the upper fence is 67.2. Since the maximum gas tax value (68.7) exceeds the upper fence, it is marked as an outlier, providing us with evidence that it is unusually high compared to the rest of the data.