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When visualizing data, detecting outliers is crucial, as they can significantly influence statistical analyses and visual impressions. Outliers are values in a data set that are significantly higher or lower than the rest of the data.

In the given problem, we calculated outlier boundaries using the 1.5×IQR rule, which is a commonly used method. To detect outliers, subtract 1.5 times the Interquartile Range (IQR) from the first quartile, Q1, for the lower boundary, and add 1.5 times the IQR to the third quartile, Q3, for the upper boundary. Any data points outside these boundaries are considered outliers. In our case, the number 22 was identified as an outlier, being above the upper boundary of 14.5.

Quartiles divide a sorted data set into four equal parts and are essential in describing the spread and center of the data. In the exercise, we calculated three quartiles: Q1, the median (Q2), and Q3.

The median, Q2, divides the data set into two equal halves. For an odd number of data points, it is the middle value; for an even number, the average of the two middle values. In our data, the median was 6.

Q1 is found by taking the median of the first half of data, excluding Q2 if necessary, which was 4.5 for our data. Q3 is similarly the median of the second half; here, it was 8.5. The location of these quartiles provides insight into the distribution of our data and is a key part of constructing a box plot.

The Interquartile Range, or IQR, is the range within which the middle 50% of the data falls. It is calculated by subtracting Q1 from Q3, and it provides a measure of the data's dispersion that is resistant to outliers. The IQR can be particularly useful for comparing differences between the data sets and identifying where the bulk of data points lie.

In the problem, the IQR was 4, derived from Q3 (8.5) minus Q1 (4.5). This measure told us that the central 50% of the data was spread across a range of 4 units. This information, coupled with the outlier detection step, gives a clear picture of the data's variability.

Data visualization is a powerful tool to communicate complex information clearly and effectively. A box plot, or box-and-whisker plot, is a standardized way of displaying the distribution of a data set based on a five-number summary: minimum, first quartile (Q1), median (Q2), third quartile (Q3), and maximum.

In constructing the box plot for this exercise, we used these five key statistics to draw a clear and informative picture of the data distribution. The box plot's edges represent Q1 and Q3, the line inside represents the median, and whiskers extend to the minimum and maximum values that are not outliers. Outliers, such as the value 22 in our problem, are plotted as individual points. This visual representation helps immediately identify key aspects like symmetry, skewness, and where the data is concentrated.