The median of a set of 7 distinct observations is \(10.5\) If each of the last 3 observation of the set is increased by 3 then the median of the new set \(=\) (a) in decreased by 2 (b) is two times the original median (c) remain the same as that of the original set (d) is increased by 2

Understanding the concept of data set ordering is central to grasping how medians are determined in statistics. But what exactly does ordering a data set entail? It involves arranging the values from a collection of numbers in ascending or descending order. In statistics, ascending order is typically preferred as it provides a clear view of the distribution from the smallest to the largest value.

When applying data set ordering in calculating medians, it ensures that we can easily spot the central value in an odd-numbered set or find the average of the two central values in an even-numbered set. For example, in the problem provided, the 7 distinct observations would be organized in such a way that each value is greater than its predecessor. Once ordered properly, finding the median—or the middle value—becomes a straightforward task.

Properly ordering a data set is the first step in the process of understanding more complex statistical analyses and is a fundamental skill in the fields of data science and statistics.

The central tendency of a data set is a measure that represents the center point or the typical value of the data. It's a way to summarize a set of data by identifying the single value that best represents the entire set. There are several measures of central tendency, the most common being mean, median, and mode.

The median, in particular, is the value separating the higher half from the lower half of a data set. For an odd number of observations, the median will be the exact middle value, as we saw with the collection of 7 observations in the exercise. If the number of observations is even, the median is found by calculating the average of the two middle values. One of the key properties of the median is that it is not affected by extremely large or small values, making it a robust measure of central tendency for skewed distributions or when outliers are present in the dataset.

The statistical median carries with it some intrinsic properties that make it unique and particularly useful. One of these properties, as demonstrated by our textbook example, is that increasing or decreasing all values above or below the median does not affect the median itself. This holds true as long as there's an odd number of observations, and the value(s) adjusted is not the median.

Also, the median divides the data set into two equal parts but does not depend on the magnitude of the observations, which means it can resist the effects of outliers or skewed data. This makes it a preferred measure in real-world situations where extreme values can throw off the mean. Furthermore, in cases of ordinal data, where we care about the rank order but not the exact values, the median provides a clear picture of the central tendency without necessitating numerical calculations.

Underlying all these properties is the concept that the median, by its very definition, is the 'middle' in a set of ranked observations. This can be used to draw conclusions about the overall distribution and can provide valuable insights, especially in fields like economics, psychology, and epidemiology, where understanding the central trend plays a significant role in decision-making and analysis.