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Central tendency is a statistical measure that identifies the point or value that best represents a set of data. It’s a way to summarize a dataset with a single value that reflects the center of its distribution. The most common measures of central tendency are the mean, median, and mode.

In the context of boxplots, the median is of particular interest as it splits the dataset in half, with 50% of the data falling below and above this value. In our example exercise, the medians are 25 and 75 for the two different datasets. The position of the median within the boxplot is crucial as it gives a visual representation of where the 'middle' of the data lies. It’s especially useful in highlighting the distribution's skewness. If the median is closer to the lower quartile, the dataset is positively skewed; if it’s closer to the upper quartile, it’s negatively skewed. The difference in medians between datasets can also provide insights into their relative positions on the scale of measurement, indicating shifts in central tendency.

The interquartile range (IQR) is a measure of statistical dispersion and represents the range within which the middle 50% of data values lie. Specifically, it's the difference between the third quartile (Q3), which marks the top of the box in a boxplot, and the first quartile (Q1), which marks the bottom of the box.

In the step-by-step solution provided, the IQR is calculated for both datasets as 20. This means that for both sets of data, half of all the observations fall within a 20-unit range. The IQR is a robust measure of spread because it’s not affected by outliers or extreme values. In our exercise, the IQR informs us that despite the different medians, the middle 50% of the values in the datasets are spread across an equal range. Graphically, on the boxplot, a larger IQR would make for a taller box, and conversely, a smaller IQR results in a shorter box, giving a direct visual cue to the dataset's compactness or sparsity.

Statistical variability, or spread, is a critical concept in statistics as it measures how much the data values diverge from each other and from the measures of central tendency. There are several ways to assess variability, including the range, variance, and standard deviation. In the boxplot exercise, we're provided with the range of each dataset, which is the simplest measure of variability—it’s the difference between the highest and lowest values.

The range for both datasets is 30 units, indicating that from the smallest to the largest number, data points are spread over a 30-unit interval. This, however, does not tell us everything about the distribution of individual values within that interval. Hence, alongside the range, the IQR gives a clearer picture of variability, specifically within the middle portion of the data. Boxplots excel at offering a visual summary of variability. A wider box and longer whiskers in a boxplot suggest more variability, while a narrower box and shorter whiskers show less variability. When constructing or interpreting boxplots, observing both the span of the whiskers and the size of the box is critical for understanding the full story of the data's dispersion.