5.4

Percentile

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Percentiles are a type of quantiles or fractiles that divide the set of observations into a hundred equal parts after arranging the data in ascending order.

Percentiles describe the percentage of data values that fall at or below a particular data point. So, the 50^th percentile has about 50 percent of the data points above and 50 percent below the percentile.

Percentiles are useful for comparing values with larger populations. For example, universities extensively use percentiles to compare students' scores and declare their ranking.

First, arrange all the scores from low to high. Now using the percentile formula, each score can be converted into corresponding percentiles. For instance, if Robert scored 80 marks, what's his percentile score?

First, determine how many students' scores are less than 80. Then divide this number by the total number of students, and multiply this value by 100.

However, a 90^th percentile does not necessarily mean the student has received 90 percent. It means that 90 percent of test scores are lower, and 10 percent are higher than his score.

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A percentile indicates the relative standing of a data value when data are sorted into numerical order from smallest to largest. It represents the percentages of data values that are less than or equal to the p^th percentile. For example, 15% of data values are less than or equal to the 15th percentile.

Low percentiles always correspond to lower data values.
High percentiles always correspond to higher data values.

Percentiles divide ordered data into hundredths. To score in the 90th percentile of an exam does not mean, necessarily, that you received 90% on a test. It means that 90% of test scores are the same or less than your score and 10% of the test scores are the same or greater than your test score.

The following formula is used to find k^th percentile

Mathematical formula calculation: i = (k/100)(n + 1) equation representation.

k = the k^th percentile. It may or may not be part of the data.
i = the index (ranking or position of a data value)
n = the total number of data points or observations

If i is an integer, then the k^th percentile is the data value in the i^th position in the ordered data set. If i is not an integer, then round i up and round i down to the nearest integers. Average the two data values in these two positions in the ordered data set.

A percentile may or may not correspond to a value judgment about whether it is "good" or "bad." The interpretation of whether a certain percentile is "good" or "bad" depends on the context of the situation to which the data applies. In some situations, a low percentile would be considered "good;" in other contexts, a high percentile might be considered "good." In many situations, there is no value judgment that applies.

Understanding how to interpret percentiles properly is important not only when describing data but also when calculating probabilities in later chapters of this text.

This text is adapted from Openstax, Introductory Statistics, Section 2.2 Measures of the Location of the Data

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