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Statistics is a field of study that involves collecting, analyzing, interpreting, presenting, and organizing data. It is a science that provides us ways to understand and utilize data to draw meaningful conclusions about a variety of real-world phenomena. This field is especially valuable because it enables us to manage and analyze large sets of data
and to extract insights that can help with decision-making or predictions.

• Descriptive statistics summarize data points from a sample or entire population.
• Inferential statistics take data from a sample and make inferences about the larger population.

In the context of percentiles, we use statistics to identify relative standings in a data set, helping us understand how a particular observation compares to others.

A data set is a collection of data, which can be numbers, characters, or other forms of observations. In our example, each data set consists of a sequence of numbers, listed in ascending order. Understanding the structure and characteristics of data sets is crucial, as it forms the basis for statistical analysis.
In the given exercise, the data sets are:

• Data set (a): Numbers from 1 to 100.
• Data set (b): Numbers from 0 to 100.
• Data set (c): Numbers from 1 to 99.
• Data set (d): Numbers from 1 to 98.

The size of each data set affects the calculation of statistics such as percentiles, and knowing the exact elements in a data set determines what statistical approaches we apply.

Percentile calculation is a statistical measure that indicates the value below which a given percentage of observations fall in a data set. To find a percentile, we use a specific formula:\[ I = \left(\frac{n}{100}\right) \times (N+1) \]where:

• \(I\) is the position of the percentile in the ordered data set.
• \(N\) is the total number of observations.
• \(n\) is the desired percentile.

Typically, for values of \(I\) that are not integers, we round to the nearest whole number
or interpolate between the two closest observations, depending on the precision required.
For instance, to find the 75th percentile in a data set with 100 numbers, we calculate \(I\) using \(n=75\) and \(N=100\), giving us the 76th data point, corresponding to 76.
The process involves precisely identifying positions within a data set
and, crucially, applying consistent rounding or interpolative methods to ensure accuracy.